Linear Hansch equations

Px and Ph represent the partition coefficients of a derivative and the parent molecule, respectively. Fujita and Hansch then combined these hydrophobic constants with Hammett s electronic constants to yield the linear Hansch equation and its many extended forms (19). 

Hansch analysis and the Free Wilson method differ in their application, but they are nevertheless closely related [390, 391, 394]. From the general formulation of a linear Hansch equation (eq. 71 is any physicochemical property) group contributions a can be derived for each substituent under consideration (eq. 72 4>ij is the physicochemical property j of the substituent Xj). 

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. 

At this point, a considerable amount of theory on Hansch analysis has been presented with almost no examples of practice. The next three Case Studies will hopefully solidify ideas on Hansch analysis that have already been discussed. Each Case Study introduces a different idea. The first is an example of a very simple Hansch equation with a small data set. The second demonstrates the use of squared parameters in Hansch equations. The third and final Case Study shows how indicator variables are used in QSAR studies. If you are unfamiliar with performing linear regressions, be sure to read Appendix B on performing a regression analysis with the LINEST function in almost any common spreadsheet software. A section in the appendix describes in great detail how to derive Equations 12.20 through 12.22 in the first Case Study. 

Hansch analysis Hansch analysis is a common quantitative structure-activity relationship approach in which a Hansch equation predicting biological activity is constructed. The equation arises from a multiple linear regression analysis of both observed biological activities and various molecular property parameters (Hammett, Hansch, and Taft parameters). 

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 Hansch linear model is related to the - Fujita-Ban model when, in both models, the hydrogen substituted compound is taken as the reference compound each Fujita-Ban regression coefficient b s corresponds to the Hansch equation for a single substituent ... 

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. 

Yes, and the predictions are quite good. Two different methods emerge from a host of others and are most commonly used to predict octanol-water partition coefficients for the many organic compounds that exist. One approach is to calculate from a knowledge of structural constants, whereas the second approach requires that a chemical s partition coefficient be measured between a solvent other than octanol and water, Kf, . Kf, can then be calculated from linear regression equations that relate log (for a particular solvent) and log Kf, . Two forms of the structural constant approach are most popular (1) the Hansch n hydrophobic character of... 

Quantitative structure-activity relationships (QSAR), a concept introduced by Hansch and Fujita (1964) is a kind of formal system based on a kinetic model, which in turn is expressed in term of a first-order linear differential equation. Solution of the differential equation leads to a linear equation ( Hansch-Fujita equation ), the coefficients of which are determined by regression analysis resulting in a QSAR equation of a particular compound series. For a prediction, the dependent variable of the QSAR equation is calculated by algebraic operations. 

The fundamental assumption of SAR and QSAR (Structure-Activity Relationships and Quantitative Structure-Activity Relationships) is that the activity of a compound is related to its structural and/or physicochemical properties. In a classic article Corwin Hansch formulated Eq. (15) as a linear frcc-cncrgy related model for the biological activity (e.g.. toxicity) of a group of congeneric chemicals [37, in which the inverse of C, the concentration effect of the toxicant, is related to a hy-drophobidty term, FI, an electronic term, a (the Hammett substituent constant). Stcric terms can be added to this equation (typically Taft s steric parameter, E,). 

Further studies of the 2-substituted 5-nitroanilines, conducted by Kier and coworkers, searched for a linear combination of structural variables that describes a line, plane, or surface that separates the molecule classes in the optimum manner. They found that sweetness correlated very well with the substituent polarizability-constants for the 2-substituent, implicating the involvement of the 2-substituents in dispersive-binding interactions at the receptor. This is in agreement with the results of Hansch " and McFarland. The correlation equation was not, however, reported. 

Quantitative structure-activity relationship (QSAR) (Hansch and Klein, 1986 Hansch and Leo, 1995) represents an attempt to correlate structural descriptors of compounds with activities. The physicochemical descriptors include numerical parameters to account for electronic properties, steric effect, topology, and hydrophobicity of analogous compounds. In its simplest form, the biochemical activities are correlated to the numerical substituent descriptors of analogous compounds tested by a linear equation such as... 

Hansch et al. (9, 10) have developed linear free energy relations which can correlate biological activity and chemical structure. Equations 1 and 2, together with simplified versions, have been proposed to relate the molar concentration, Cx, of a substituted compound of a series, which all cause an equivalent biological response, to the hydrophobic bonding or partition constant, r, and the Hammett constant, [Pg.137]

As the equations show, linear correlations with the variables tr and a gave satisfactory results. This is certainly a simplification resulting from limited variance in the substituents. One would assume that square terms of the hydrophobic parameter are necessary in every correlation with biological activity not only to account for the random walk penetration process as in the original derivation of his equation by Hansch, but also, or even predominantly, as a description of the fact that numerous indifferent hydrophobic sites within the biological system compete with the site of action for the active molecule. In a first attempt we calculated regression equations for our hydrazones with the molecular parameter... 

LogP has been introduced as an additional descriptor in the new release of VolSurf A training set of 7871 diverse chemical structures was used to build a linear equation to calculate the logP values by fitting the structures with the other VolSurf descriptors. Using a five-component PLS regression, statistics give an r = 0.82, a = 0.82, and a SDEC [13] value = 0.74. The structures and data stem from Hansch et al. [18]. 

Hundreds of equations later, the failure of linear equations in cases with extended hydrophobicity ranges led to the development of the Hansch parabolic equation (20) ... 

The correlation of biological activity with physicochemical properties is often termed an extrathermodynamic relationship. Because it follows in the line of Hammett and Taft equations that correlate thermodynamic and related parameters, it is appropriately labeled. The Hammett equation represents relationships between the logarithms of rate or equilibrium constants and substituent constants. The linearity of many of these relationships led to their designation as linear free energy relationships. The Hansch approach represents an extension of the Hammett equation from physical organic systems to a biological milieu. It should be noted that the simplicity... 

---