Additivity model, Free-Wilson

Data preparation and quality control is a key step in applying Free-Wilson methodology to model biological data. Care must be taken to make sure the underlying data complies with F-W additive assumption. 

When structural modifications are made at various positions in a set of complex drug molecules, this approach combined with the Free-Wilson mathematical model would be a suitable tool (68). If the mathematical contributions of substituents and parts of molecules are additive for a certain biological effect of a series of drugs, the contribution values of substituents at each position can be analyzed further with our approach. Examples are discussed by Paul Craig in Chapter 8. 

Tihe two methods of structure-activity correlation which have received the most application in the past decade are the Hansch multiple parameter method, or the so-called extrathermodynamic approach, and the Free-Wilson, or additive model. The basic differences and similarities of these methods are discussed in this presentation. 

Detailed Discussion of the Free-Wilson Additivity Model... 

The epoch of QSAR (Quantitative Structure-Activity Relationships) studies began in 1963-1964 with two seminal approaches the a-p-7i analysis of Hansch and Fujita " and the Free-Wilson method. The former approach involves three types of descriptors related to electronic, steric and hydrophobic characteristics of substituents, whereas the latter considers the substituents themselves as descriptors. Both approaches are confined to strictly congeneric series of compounds. The Free Wilson method additionally requires all types of substituents to be suflficiently present in the training set. A combination of these two approaches has led to QSAR models involving indicator variables, which indicate the presence of some structural fragments in molecules. 

The Free-Wilson model (also called additivity model) is defined as ... 

The Free-Wilson method (198) and its modifications (212, 213) are all based on the linear additivity assumption. This was criticized by Bocek et al. (215, 216), and the possibility of interactions between substituents was introduced. For two substituents the Bocek-Kopecky interaction model can be expressed by equation 102... 

Statistical methods. Certainly one of the most important considerations in QSAR is the statistical analysis of the correlation of the observed biological activity with structural parameters - either the extrathermodynamic (Hansch) or the indicator variables (Free-Wilson). The coefficients of the structural parameters that establish the correlation with the biological activity can be obtained by a regression analysis. Since the models are constructed in terms of multiple additive contributions the method of solution is also called multiple linear regression analysis. This method is based on three requirements (223) i) the independent variables (structural parameters) are fixed variates and the dependent variable (biological activity) is randomly produced, ii) the dependent variable is normally and independently distributed for any set of independent variables, and iii) the variance of the dependent variable must be the same for any set of independent variables. 

Free-Wilson model The Free-Wilson model is a mathematical approach for QSAR and is based on the hypothesis that the biological activity within a series of molecules arises from the constant and additive contributions of the various substituents, without determining their physicochemical basis. 

In Equations 8-11 a, b, c, and d represent the substituent contribution to the total activity of the compounds while the subscripts x and y denote the substituent positions on the parent molecule. Neither the additive model (noticeably similar to the Free-Wilson model) (Equation 8), the multiplicative model (Equation 9), nor the combined model (Equation 10) described the biological activity significantly. The combined expression (Equation 11), however, gave a statistically significant correlation between substituent activity and biological response for both the m- (35) and p- (34) disubstituted series. 

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. 

By its definition the Free Wilson analysis is limited to linear additive structure-activity relationships (its application to nonlinear relationships and the combination with Hansch analysis to a mixed approach are described in chapter 4.3). A detailed discussion of the scope and limitations of the Free Wilson model is given in refs. [390, 391] some applications are discussed in chapter 8. 

A comparison of the results from Hansch and Free Wilson analyses offers some information, whether a certain Hansch model can be considered to be acceptable or not. In most cases the Free Wilson analysis of a data set shows whether a linear additive model is suited for the analysis only in certain cases is a good fit obtained for nonlinear relationships, especially if there are only few degrees of freedom [22, 390, 391]. 

It should be mentioned that Bocek and Kopecky, independently and at the same time as Free and Wilson, proposed an additive model with additional interaction terms (eq. 80, reformulated exCy = interaction term) [429, 430]. 

The width of the confidence interval for the activity increments and the calculated activity values allow the identification of conspicuous substructures, which deviate with respect to the compounds overall activity. Also it reveals in which position the activity contributions obey the underlying additivity assumption (independent contributions) or not. This obvious advantage of the Free-Wilson method is also its major weakness its predictive use is strictly limited to substances with substructures in the data set used to derive the model. The occurrence of other structural moieties may affect the activity strongly. The neglect of this basic principle is likely to result in erroneous estimates. 

Independently of Hansch and Fujita, Free and Wilson formulated an additive model (2 equation 4) C = molar dose, Oi = group contribution of substituent X, p = biological activity of the parent structure). ... 

The above techniques have been used in numerous calculations of solute free energy profiles. Wilson and Pohorille [52] and Benjamin[53] have determined the free energy profiles for small ions at the water liquid/vapor interface and compared the results to predictions of continuum electrostatic models. The transfer of small ions to the interface involves a monotonic increase in the free energy which is in qualitative agreement with the continuum model. This behavior is consistent with the increase in the surface tension of water with the increase in the concentration of a very dilute salt solution, and it represents the fact that small ions are repelled from the liquid/vapor interface. On the other hand, calculations of the free energy profile at the water liquid/vapor interface of hydrophobic molecules, such as phenol[54] and pentyl phenol[57] and even molecules such as ethanol [58], show that these molecules are attracted to the surface region and lower the surface tension of water. In addition, the adsorption free energy of solutes at liquid/liquid interfaces[59,60] and at water/metal interfaces[61-64] have been reported. 

Higher Order Approximations - In the early attempts to correlate structure with activity mathematically, simple linear combinations of physicochemical parameters were usually considered. It has become evident that the addition of interaction terms to such equations can in some Instances yield sharper correlations.Singer and Purcellhave discussed this problem and compared the models of Free and Wilson, Kopecky and Bocek and Hansch and his colleagues. They point out that in view of the many Instances where BR is not linearly > related to log P and where BR is also not linearly related to electronic effects 30 that the model of Free and Wilson will not hold, but that the Kopecky-Bocek model should apply. 

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