drug molecule’s solubility. It reflects the ability of the drug to partition itself into the
lipid surroundings of the receptor microenvironment.
Introduced by Corwin Hansch in the early 1960s, Hansch analysis considers both the
physicochemical aspects of drug distribution from the point of application to the point
of effect and the drug–receptor interaction. In a given group of drugs that have analogous
structures and act by the same mechanism, three parameters seem to play a major role:
- The substituent hydrophobicity constant,based on partition coefficients analogs to
Hammet constants:
wherePXis the partition coefficient of the molecule carrying substituent X, and PH
is the partition coefficient of the unsubstituted molecule (i.e., substituted by hydro-
genonly). More positive πvalues indicate higher lipophilicity of the substituent.
Since these values are additive,Pvalues measured on standard molecules permit
prediction of hydrophobicity of novel molecules.- The Hammet substituent constant σ
- Steric effects,described by the Taft ESvalues
Theσandπconstants of substituents are often useful when correlated to biological
activity in the statistical procedure known as multivariate regression analysis. As is well
known from pharmacological testing of various drug series, such correlations can be
either linear or parabolic. The linear relationship is described by the equation
whereCis the drug concentration for a chosen standard biological effect, and a,b,c,
anddare regression coefficients to be determined by iterative curve fitting. The para-
bolic relationship fits the equation
The coefficients a,b,c,d, and eare fitted to the curve by the least-squares procedure,
using regression methods for which computer programs are readily available. The
extent of the fit is judged by the correlation coefficient ror the multiple regression coef-
ficient r^2 , which is proportional to the variance. A perfect fit gives r^2 =1.00. Once the
best fit has been achieved and rorr^2 has been maximized by using a reasonable number
of known compounds (15−20 is an advisable number, depending on the number of vari-
ables tested, with even more compounds being even better), the curve can be used to
predict the biological activity of compounds that have not been tested or, indeed, have
not even been synthesized. This requires only the substitution of the optimized regres-
sion coefficient constants into the equation, and the use of π,σ, and ESvalues, which
are usually available for just about any substituent. Naturally, independent variables
other than πorσ—including ionization constants, activity coefficients, molar volumes,
or molecular orbital parameters—can also be used.
To achieve these various “best fits,” statistical methods are employed. A regression
analysis of the effects of various substituents on a molecule using the Hansch approach
DESIGNING DRUG MOLECULES TO FIT RECEPTORS 141πX=logPX−logPH (3.1)log 1/C=aπ+bES+cσ+d (3.2)log 1/C=−aπ^2 +bπ+cES+dσ+e (3.3)