Quantitative Structure–Activity Relationship (QSAR) Studies The relationship between chemical structure and biological activity has always been at

the center of drug research. In the past, drug structures were modified intuitively and empirically, depending on the imagination and experience of the synthesizing chemist, and were based on analogies. Surprisingly, the results were often gratifying, even if obtained only serendipitously or on the basis of the wrong hypothesis. However, this hit-or-miss approach, practiced even now, is enormously wasteful. Considering that only one of several thousand synthesized compounds will reach the pharmacy shelves, and that the development of a single drug can cost millions of dollars, it is imperative that rational short-cuts to drug design be found. Quantitative structure–activity rela-tionship (QSAR) studies represent this important rational short-cut. QSAR endeavors to elevate drug design from an art to a science.

QSAR methods are in part retrospective as well as predictive, since a “training set”

of compounds of known pharmacological activity must first be established. The pur-pose of such methods is to increase the probability of finding active compounds among those eventually synthesized, thus keeping synthetic and screening efforts within rea-sonable limits in relation to the success rate. There are three main classifications of QSAR methods:

1. 1D-QSAR (e.g., Hansch analysis)

2. 2D-QSAR (e.g., pattern recognition analysis)

3. 3D-QSAR (e.g., comparative molecular field analysis) Each method has its own strengths and weaknesses.

3.4.2.1 1D-QSAR — Hansch Analysis

Historically, this is the most popular mathematical approach to QSAR. The major con-tribution of Hansch analysis is in recognizing the importance of logP, where P is the octanol–water partition coefficient. LogP is perhaps the most important measure of a

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:

1. The substituent hydrophobicity constant, based on partition coefficients analogs to Hammet constants:

where P_Xis the partition coefficient of the molecule carrying substituent X, and P_H is the partition coefficient of the unsubstituted molecule (i.e., substituted by hydro-gen only). More positive π values indicate higher lipophilicity of the substituent.

Since these values are additive, P values measured on standard molecules permit prediction of hydrophobicity of novel molecules.

2. The Hammet substituent constant σ

3. Steric effects, described by the Taft E_Svalues

Theσ ^andπ constants 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

where C is the drug concentration for a chosen standard biological effect, and a, b, c, and d are regression coefficients to be determined by iterative curve fitting. The para-bolic relationship fits the equation

The coefficients a, b, c, d, and e are 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 r or the multiple regression coef-ficient r², which is proportional to the variance. A perfect fit gives r²= 1.00. Once the best fit has been achieved and r or r²has 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 E}Svalues, which are usually available for just about any substituent. Naturally, independent variables other than π^orσ—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

πX= log P_X− log P_H (3.1)

log 1/C = aπ + bES+ cσ + d (3.2)

log 1/C = −aπ²+ bπ + cE_S+ dσ + e (3.3)

is very useful, saving much time and effort in the synthesis and testing of new drugs.

Hundreds of examples of such analyses are available in the literature; many show pos-itive predictive values for drug activity, whereas some other drug series cannot be inter-preted by this method.

Regression analysis is currently the most widely used correlative method in drug design. This is because it simplifies problems within a set of compounds by using a lim-ited number of descriptors, notably the Hansch hydrophobic constant π, Hammet con-stants, or other electronic characteristics of substituents, and the Taft steric constant E_S. Nevertheless, there are several difficulties and pitfalls in using the Hansch method.

First, the inherent disadvantage of regression analysis is that one can obtain good fits (r²> 0.9) simply by manipulating the constants. Therefore, curve fitting must be done for a relatively large number of compounds to ensure that all predictors are considered.

Second, the mode of action may change for drugs within a seemingly continuous series, invalidating the comparison of some compounds in the series with the predictor com-pounds. The Hansch method cannot anticipate such a change.

Other problems with the Hansch method are that biological systems are often too crude as models for its application, or the electronic effects operative in a drug mole-cule are not sufficiently understood or precise. Finally, the method requires consider-able time and expense, even in the hands of an expert. Difficulties notwithstanding, the Hansch approach took both chemists and pharmacologists out of the dark age of pure empiricism and allowed them to consider simultaneously the effects of a large number of variables of drug activity—a feat unattainable with classical methods.

Nevertheless, Hansch analysis revolutionized drug molecule optimization and directly led to two other strategies for molecule optimization: the Free–Wilson method and the Topliss decision tree.

The Free–Wilson Method. This method also assumes that biological activity can be described by the additive properties of the substituents on a basic molecular structure.

In the Fujita–Ban modification of this method

where C is the drug concentration for a standardized effect, a_iis the group contribution of the ith substituent to the pharmacological activity of the substituted molecule, X is unity if substituent i is present and zero otherwise, and µ0 = 1/C for the parent com-pound. Regression analysis is used to determine a_iand µ. In the Fujita–Ban modifica-tion of the Free–Wilson method, no assumpmodifica-tions are made about the relevance of the model parameters to the biological activity of the molecule. The effect of each sub-stituent is considered to be independent of any other, and each makes a constant con-tribution to the overall activity of the molecule. Therefore the method is applicable to compounds with more than one variable group. The result is a data matrix that shows the contribution of each substituent in each position to the overall biological effect of the molecule. The Free–Wilson equation bears close similarities to the linear Hansch equa-tion, and the results of the two can be comparable. The Free–Wilson method, however, cannot predict the activities of compounds that have substituents not included in the matrix. Consequently, this method has found only limited application in drug series where many close analogs are already available but physicochemical data are lacking.

log 1/C =

a_iX_i+ µ0 (3.4)

Topliss Decision Tree Method. This method is quicker and easier to use than the Hansch method. The Topliss scheme is an empirical method in which each compound is tested before an analog is planned, and is compared in terms of its physical proper-ties with analogs already planned. Like the Free–Wilson method, the Topliss decision tree is no longer extensively used. The 2D- and 3D-QSAR methods are gradually sup-planting the 1D methods.

3.4.2.2 2D-QSAR — Pattern Recognition Analysis

2D-QSAR is a somewhat more advanced method for correlating activity and struc-ture. The first step in performing a 2D-QSAR is to select the training set. This is a subset of molecules that are diverse in terms of both structure and bioactivity. Ideally, the compounds that are available cover the full spectrum of bioactivity, ranging from active (fully and partially, covering a 10³-fold range in receptor binding affinities) to inactive. It is difficult to determine what makes a molecule bioactive (or conversely what makes a molecule bioinactive) if all of the compounds tested have similar bioac-tivities. The more molecules the better, but a reasonable start can be made with as few as ten compounds. It is important not to use all available molecules, since another subset is held back and retained as a test set. This test set will ultimately be used to validate any prediction algorithm that is developed through the study of the training set.

Next, every molecule in the training set, regardless of its pharmacological activity, is characterized by a series of descriptors:

1. Geometric descriptors

Energy of the highest occupied molecular orbital Energy of the lowest unoccupied molecular orbital Molecular dipole

Number of rings in the molecule

Number of aromatic rings in the molecule 4. Physicochemical descriptors

Octanol–water partition coefficients LogP

(LogP)²

Hydrogen bonding number

Number of hydrogen bonding donor sites Number of hydrogen bonding acceptor sites

These various descriptors may be calculated using various molecular mechanics and quantum mechanics approaches, as discussed in chapter 1.

The geometric descriptors reflect molecular geometry and are conceptually straight-forward. Electronic descriptors reflect properties arising from variations in electron dis-tribution throughout the drug molecule framework. Topological descriptors endeavor to describe molecular branching and complexity through the notion of molecular connec-tivity. The concept of molecular connectivity, introduced by Kier and Hall in 1976, describes compounds in topological terms. Branching, unsaturation, and molecular shape are all represented in the purely empirical connectivity index ¹χ, which correlates surprisingly well with a number of physicochemical properties including the partition coefficients, molar refractivity, or boiling point. These graph theory indices are useful to differentiate between an n-butyl substituent and a tert-butyl substituent. The physico-chemical indices reflect the ability of the drug to partition itself into the lipid surroundings of the receptor microenvironment.

All of these descriptors are calculated for every compound within the training set.

Next, a 2D data array is constructed. Along the vertical axis, all of the training set com-pounds are listed in descending order of bioactivity. Along the horizontal axis, all of the descriptors are arranged for every training set compound. This data array is then probed with statistical calculations to ascertain the minimum number of descriptors that differ-entiate active compounds from inactive compounds. In order to probe the data array, several methods are available. Pattern recognition and cluster analysis, two recent quantitative methods, make use of sophisticated statistics and computer software.

Pattern recognition can be used to deal with a large number of compounds, each char-acterized by many parameters. First, however, these raw data must be processed by scaling and normalization—the conversion of diverse units and orders of magnitude from many sources — so that the chosen parameters become comparable. Feature selection methods exist for weeding out irrelevant “descriptors” and obtaining those that are potentially most useful. By using “eigenvector” or “principal component”

analysis algorithms, these multidimensional data are then projected two-dimensionally onto a plot whose axes are the two principal components or two (transformed and normalized) parameters that account for most of the variance; these are the two eigen-vectors with the highest values. Previously unrecognized relational patterns between large numbers of compounds characterized by multidimensional descriptors will thus emerge in a new, comprehensible, two-dimensional plot. The projection of unknowns onto this eigenvector plot will determine their relationship to active and inactive compounds.

Cluster analysis is similar in concept to pattern recognition. It can define the simi-larity or dissimisimi-larity of observations or can reveal the number of groups formed by a collection of data. The distance between clusters of data points is defined either by the distance between the two closest members of two different clusters or by the distances between the centers of clusters.

Once the data array has been probed and the minimum number of descriptors that dif-ferentiate activity from inactivity has been ascertained, a prediction algorithm is deduced. This algorithm attempts to quantify the bioactivity in terms of the relevant descriptors. The predictive usefulness of this algorithm is then validated by being applied to the test set compounds. If the prediction algorithm is sufficiently robust, it can be used to direct the syntheses of optimized compounds.

3.4.2.3 3D-QSAR — Comparative Molecular Field Analysis

Like other forms of QSAR, 3D-QSAR starts with a series of compounds with known structures and known biological activities. The first step is to align the molecular struc-tures. This is done with alignment algorithms that rotate and translate the molecule in Cartesian coordinate space so that it aligns with another molecule. The work starts with the most rigid analogs and then progresses to conformationally flexible molecules that are aligned with the more rigid ones. The end result is that all the molecules are even-tually aligned, each on top of another.

Once the molecules of the training set have been aligned, a molecular field is com-puted around each molecule, based upon a grid of points in space. Various molecular fields are composed of field descriptors that reflect properties such as steric factors or electrostatic potential. The field points are then fitted to predict the bioactivity. A par-tial least-squares algorithm (PLS) is used for this form of fitting. Based upon this PLS calculation, two pieces of information are deduced for every region of space within the molecular field about the molecule: the first piece of information states whether that region of space correlates with biological activity; the second piece of information determines whether the functional group on the molecule within that region of space should be bulky, aromatic, electron donating, electron withdrawing, and so on. The pre-dictions from these molecular field calculations are then validated by being applied to a test set of compounds.

3.4.2.4 Pharmacophore Identification — A Corollary of QSAR

All drugs have pharmacological activity as a result of stereoelectronic interaction with a receptor. The receptor macromolecule “recognizes” the arrangement of certain func-tional groups in three-dimensional space and their electron density. It is the recognition of these groups rather than the structure of the entire drug molecule that results in an interaction, normally consisting of noncovalent binding. The collection of relevant groups responsible for the effect is the pharmacophore, and their geometric arrange-ment is called the pharmacophoric pattern, whereas the position of their complemen-tary structures on the receptor is the receptor map. Over the years, many attempts have been made to define the pharmacophores and their pattern on many drugs. The first attempts were rather naive and simplistic, but the recent use of QSAR has contributed greatly to the evolution of sophisticated methods of practical significance.

The identification of the pharmacophore is a logical corollary of a QSAR calculation.

If the minimum number of descriptors that differentiate activity from inactivity is known, it is possible to deduce the bioactive face of the molecule — that part of the molecule around which all of the relevant descriptors are focused. This bioactive face logically defines the pharmacophoric pattern of the bioactive molecules.

When using QSAR calculations to optimize a drug for the pharmacodynamic phase, it is important to use relevant biological activities. If in vivo activities are used, the bioactivities will be influenced by pharmacokinetic and pharmaceutical factors. In order for QSAR calculations to reflect the pharmacodynamic phase, the bioactivities should be based on in vitro data — optimally, receptor binding studies.

QSAR studies are not restricted to the optimization of biological activity at the phar-macodynamic phase. Since toxicity also arises from drug–receptor interactions, the QSAR

method can be used to identify the biotoxic face of the molecule (i.e., the toxicophore), which could then be engineered out of the molecular structure.

Once QSAR calculations have been used to optimize the pharmacodynamic interac-tions of the drug molecule, the next step is to optimize the pharmacokinetic and phar-maceutical phases of drug action.

3.5 OPTIMIZING THE LEAD COMPOUND: PHARMACOKINETIC