An Introduction to QSAR Methodology
Allen B. Richon
Network Science Corporation
4411 Connecticut Avenue NW, STE 514
Washington, DC 20008-8677
Stanley S. Young
Glaxo Wellcome Research
Five Moore Drive
Research Triangle Park, NC 27709
Drug design is an iterative process which begins with a compound that displays an interesting biological profile and ends with optimizing both the activity profile for the molecule and its chemical synthesis. The process is initiated when the chemist conceives a hypothesis which relates the chemical features of the molecule (or series of molecules) to the biological activity. Without a detailed understanding of the biochemical process(es) responsible for activity, the hypothesis generally is refined by examining structural similarities and differences for active and inactive molecules. Compounds are selected for synthesis which maximize the presence of functional groups or features believed to be responsible for activity.
The combinatorial possibilities of this strategy for even simple systems can be explosive. As an example, the number of compounds required for synthesis in order to place 10 substituents on the four open positions of an asymmetrically disubstituted benzene ring system is approximately 10,000. The alternative to this labor intensive approach to compound optimization is to develop a theory that quantitatively relates variations in biological activity to changes in molecular descriptors which can easily be obtained for each compound. A Quantitative Structure Activity Relationship (QSAR) can then be utilized to help guide chemical synthesis. This chapter develops the concepts used to derive a QSAR and reviews the application of these techniques to medicinal research.
Computational chemistry represents molecular structures as a numerical models and simulates their behavior with the equations of quantum and classical physics. Available programs enable scientists to easily generate and present molecular data including geometries, energies and associated properties (electronic, spectroscopic and bulk). The usual paradigm for displaying and manipulating these data is a table in which compounds are defined by individual rows and molecular properties (or descriptors) are defined by the associated columns. A QSAR attempts to find consistent relationships between the variations in the values of molecular properties and the biological activity for a series of compounds so that these "rules" can be used to evaluate new chemical entities.
A QSAR generally takes the form of a linear equation
Biological Activity = Const + (C1 P1) + (C2 P2) + (C3 P3) + ...
where the parameters P1 through Pn are computed for each molecule in the series and the coefficients C1 through Cn are calculated by fitting variations in the parameters and the biological activity. Since these relationships are generally discovered through the application of statistical techniques, a brief introduction to the principles behind the derivation of a QSAR follows.
The work reported from The Sandoz Institute for Medical Research on the development of novel analgesic agents1 can be used as an example of a simple QSAR. In this study, vanillylamides and vanillylthioureas related to capsaicin were prepared and their activity was tested in an in vitro assay which measured 45Ca2+ influx into dorsal root ganglia neurons. The data, which was reported as the EC50 (µM), is shown in Table 1 (note that compound 6f is the most active of the series).
Capsaicin Analogs Activity Data
|1||6a||H||11.80 ± 1.90|
|2||6b||Cl||1.24 ± 0.11|
|3||6d||NO2||4.58 ± 0.29|
|4||6e||CN||26.50 ± 5.87|
|5||6f||C6H5||0.24 ± 0.30|
|6||6g||N(CH3)2||4.39 ± 0.67|
|7||6h||I||0.35 ± 0.05|
In the absence of additional information, the only way to derive a best "guess" for the activity of 6i is to calculate the average of the values for the current compounds in the series. The average, 7.24, provides a guess for the value of compound 8 but, how good is this guess? The graphical presentation of the data points is shown in Graph 1.
Capsaicin Analogs Activity Data
The standard deviation of the data, s, shows how far the activity values are spread about their average. This value provides an indication of the quality of the guess by showing the amount of variability inherent in the data. The standard deviation is calculated as shown below.
Rather than relying on this limited analysis, one would like to develop an understanding of the factors that influence activity within this series and use this understanding to predict activity for new compounds. In order to accomplish this objective, one needs:
- binding data measured with sufficient precision to distinguish
- a set of parameters which can be easily obtained and which are
likely to be related to receptor affinity;
- a method for detecting a relationship between the parameters
and binding data (the QSAR) and
- a method for validating the QSAR.
The QSAR equation is a linear model which relates variations in biological activity to variations in the values of computed (or measured) properties for a series of molecules. For the method to work efficiently, the compounds selected to describe the "chemical space" of the experiments (the training set) should be diverse. In many synthesis campaigns, compounds are prepared which are structurally similar to the lead structure. Not surprisingly, the activity values for this series of compounds will frequently span a limited range as well. In these cases, additional compounds must be made and tested to fill out the training set.
The quality of any QSAR will only be as good as the quality of the data which is used to derive the model. Dose-response curves need to be smooth, contain enough points to assure accuracy and should span two or more orders of magnitude. Multiple readings for a given observation should be reproducible and have relatively smaller errors. The issue being addressed is the signal-to-noise ratio. The variation of the readings obtained by repeatedly testing the same compound should be much smaller than the variation over the series. In cases where the data collected from biological experiments do not follow these guidelines, other methods of data analysis should be utilized since the QSAR models derived from the data will be questionable.
Once biological data has been collected, it is often found that the data is expressed in terms which cannot be used in a QSAR analysis. Since QSAR is based on the relationship of free energy to equilibrium constants, the data for a QSAR study must be expressed in terms of the free energy changes that occur during the biological response. When examining the potency of a drug (the dosage required to produce a biological effect), the change in free energy can be calculated to be proportional to the inverse logarithm of the concentration of the compound.
G0 = - 2.3RTlogK = log 1/[S]
Further, since biological data are generally found to be skewed, the log transformation moves the data to a nearly normal distribution. Thus, when measuring responses under equilibrium conditions, the most frequent transformation used is to express concentration values (such as IC50, EC50, etc.) as log[C] or log 1/[C]. The transformed data for the capsaicin agonists are shown in Table 2.
|X||EC50||Log EC50||Log 1/EC50|
|1||6a||H||11.80 ± 1.90||1.07||-1.07|
|2||6b||Cl||1.24 ± 0.11||0.09||-0.09|
|3||6d||NO2||4.58 ± 0.29||0.66||-0.66|
|4||6e||CN||26.50 ± 5.87||1.42||-1.42|
|5||6f||C6H5||0.24 ± 0.30||- 0.62||0.62|
|6||6g||N(CH3)2||4.39 ± 0.67||0.64||-0.64|
|7||6h||I||0.35 ± 0.05||- 0.46||0.46|
|8||6i||NHCHO||?? ± ??||??||??|
The effect of this transformation on the spread of the data relative to the average is shown in Graph 2. Note that the data points, projected onto the Y-axis, have become more uniformly distributed.
Given the transformed data, our best guess for the activity of 6i is still the average of the data set (or 0.40). As before, the error associated with this guess is calculated as the square root of the average of the squares of the deviations from the average.
This is an example data set intended to show the general approach; real data sets would have many more compounds and descriptors. Since the purpose of a QSAR is to highlight relationships between activity and structural features, we would like to find one or more structural features which relate these molecules and their associated activity. Additionally, we would like to find a parameter that works consistently for all of the molecules in the series.
There are several potential classes of parameters used in QSAR studies. Substituent constants and other physico-chemical parameters (such as Hammett sigma constants) measure the electronic effects of a group on the molecule. Fragment counts are used to enumerate the presence of specific substructures. Other parameters can include topological descriptors and values derived from quantum chemical calculations.
The selection of parameters is an important first step in any QSAR study. If the association between the parameter(s) selected and activity is strong, then activity predictions will be possible. If there is only weak association, knowing the value of the parameter(s) will not help in predicting activity. Thus, for a given study, parameters should be selected which are relevant to the activity for the series of molecules under investigation and these parameters should have values which are obtained in a consistent manner.
The Sandoz group divided their analysis of capsaicin analogs into three regions: the A-region which was occupied by an aromatic ring; the B-region which was defined by an amide bond; and the C-region which was occupied by a hydrophobic side-chain (See figure in Table 1). The hypothesis for the C-region assumed that a small, hydrophobic substituent would increase activity. Given this assumption, the parameters selected to best define this characteristic were molar refractivity (size) and , the hydrophobic substituent constant. These values are given in Table 3.
The data above can be analyzed for relationships by two means: graphically and statistically. The most visual approach to a problem with a limited number of variables is graphical. In this case, a plot of activity versus either molar refractivity or hydrophobicity gives some insight into the relationship between the parameters and activity. The plots derived by the Sandoz group are reproduced in Graph 3.
Does the graph provide insight into the the activity for compound 6i? Does knowing the value for either the hydrophobicity or molar refractivity parameters for this compound provide a good estimate for activity?
Since this is a simple example where only two values are examined, the answers to these questions are a qualified yes. In more complex situations however, where multiple parameters are correlated to activity, statistics is used to derive an equation which relates activity to the parameter set. The linear equation which defines the best model for this set of data is
Log EC50 = 0.764 - (0.817)
How much confidence should we place in this model? The first step to answering this question is to determine how well the equation predicts activities for known compounds in the series. The equation above estimates the average value for the EC50 based on the value for ; because assays vary, it is not surprising that individual values will differ from the regression estimate. The difference between the calculated values and the actual (or measured) values for each compound is termed the residual from the model. The calculated values for activity and their residuals (or the errors of the estimate for individual values) are shown in Table 4.
|3||6d||NO2||0.66||- 0.28||1.02||- 0.36|
|5||6f||C6H5||- 0.62||1.96||- 0.81||0.19|
|7||6h||I||- 0.46||1.12||- 0.12||- 0.34|
The residuals are one way to quantify the error in the estimate for individual values calculated by the regression equation for this data set. The standard error for the residuals is calculated by taking the root-mean-square of the residuals (in this calculation, the denominator shown as decremented by two to reflect the estimation of two parameters).
In order to be an improved model, the standard deviation of the residuals calculated from the model should be smaller than the standard deviation of the original data. The standard error about the mean was previously calculated to be 0.76 whereas the standard error from the QSAR model is 0.28. Clearly, the the use of linear regression has improved the accuracy of our analysis. The plot of measured values versus calculated is shown in Graph 4 with a 45° line.
Predicted Versus Actual EC50 Values
There are several assumptions inherent in deriving a QSAR model for a series of compounds. First, it is assumed that parameters can be calculated (or measured in some cases) more accurately and cheaply than activity can be measured. Second, it is assumed that deviations from the best fit line follow a normal (Gaussian) distribution. Finally, it is assumed that any variation in the line described by the QSAR equation is independent of the magnitude of both the activity and the parameters. Given these assumptions, the quality of the model can be gauged using a variety of techniques.
Variation in the data is quantified by the correlation coefficient, r, which measures how closely the observed data tracks the fitted regression line. Errors in either the model or in the data will lead to a bad fit. This indicator of fit to the regression line is calculated as:
where the Regression Variance is defined as the Original Variance minus the Variance around the regression line. The Original Variance is the sum-of-the-squares distances of the original data from the mean. This can be viewed graphically as shown in Graph 5.
The calculation is carried out as follows:
- Original Variance = (1.07 - 0.40)2 + (0.09 - 0.40)2 + ...
- Original Variance = 3.49
- Variance around the line = (0.28)2 + (-
0.12)2 + (- 0.36)2 + ...
- Variance around the line = 0.40
- Regression Variance = Original Variance - Variance around the
- Regression Variance = 3.49 - 0.40 = 3.09
- r2 = Regression Variance/Original Variance
- r2 = 3.09/3.49
- r2 = 0.89
Possible values reported for r2 fall between 0 and 1. An r2 of 0 means that there is no relationship between activity and the parameter(s) selected for the study. An r2 of 1 means there is perfect correlation. The interpretation of the r2 value for the capsaicin analogs is that 89% of the variation in the value of the Log EC50 is explained by variation in the value of , the hydrophobicity parameter.
Derivation of r2 values
While the fit of the data to the regression line is excellent, how can one decide if this correlation is based purely on chance? The higher the value for r2 the less likely that the relationship is due to chance. If many explanatory variables are used in a regression equation, it is possible to get a good fit to the data due to the flexibility of the fitting process; a line will fit two points perfectly, a quadratic curve will fit three, multiple linear regression will fit the observed data if there are enough explanatory variables2. Given the assumption that the data has a Gaussian distribution, the F statistic below assesses the statistical significance of the regression equation.
The F statistic is calculated from r2 and the number of data points (or degrees of freedom) in the data set. The F ratio for the capsaicin analogs is calculated as:
This value often appears as standard output from statistical programs or it can be checked in statistical tables to determine the significance of the regression equation. In this case, the probability that there is no relationship between activity and the value is less than 1% (p=0.01).
We have found that hydrophobicity values correlate well with biological activity. Does the addition of a size parameter (MR) improve our model? In order to analyze a relationship which is possibly influenced by several variables (or properties), it is useful to assess the contribution of each variable. and MR appear to be somewhat correlated in this data set so the order of fitting can influence how much the second variable helps the first. Multiple linear regression is used to determine the relative importance of multiple variables to the overall fit of the data.
Multiple linear regression attempts to maximize the fit of the data to a regression equation (minimize the squared deviations from the regression equation) for the biological activity (maximize the r2 value) by adjusting each of the available parameters up or down. Regression programs often approach this task in a stepwise fashion. That is, successive regression equations will be derived in which parameters will be either added or removed until the r2 and s values are optimized. The magnitude of the coefficients derived in this manner indicate the relative contribution of the associated parameter to biological activity.
There are two important caveats in applying multiple regression analysis. The first is based on the fact that, given enough parameters any data set can be fitted to a regression line. The consequence of this is that regression analysis generally requires significantly more compounds than parameters; a useful rule of thumb is three to six times the number of parameters under consideration. The difficulty is that regression analysis is most effective for interpolation and it is extrapolation that is most useful in a synthesis campaign (i.e., the region of experimental space described by the regression analysis has been explained, but projecting to a new, unanalyzed region can be problematic).
Using multiple regression for the capsaicin analogs, one can derive the following equation which relates hydrophobicity and molar refractivity to biological activity.
Log EC50 = 0.762 - (0.819) +
s = 0.313, r2 = 0.888
To judge the importance of a regression term, three items need to be considered.
- Statistical significance of the regression coefficient.
- The magnitude of the typical effect bixi
(in this case, 0.011 25.36).
- Any cross-correlation with other terms.
As more terms are added to multiple linear regression, r2 always gets larger. We recompute the previous calculations (r2 = 0.89) carrying three significant figures so that rounding does not lead to confusion.
These results of this analysis indicate that, within this series, steric bulk is not an important factor in activity. The influence of the hydrophobicity constant confirms the presence of a hydrophobic binding site. Given the limited number of substituents in this analysis, it is unlikely that more can be learned from further analysis.
This section has developed the fundamental mathematics of QSAR studies. Several authors have published reviews of QSAR and have discussed various aspects of the methods3-8. Each of the examples to follow uses these techniques to derive information about the chemical factors which are important for activity.
Approaches to Developing a QSAR
Drugs exert their biological effects by participating in a series of events which include transport, binding with the receptor and metabolism to an inactive species. Since the interaction mechanisms between the molecule and the putative receptor are unknown in most cases (i.e., no bound crystal structures), one is reduced to making inferences from properties which can easily be obtained (molecular properties and descriptors) to explain these interactions for known molecules. Once the relationship is defined, it can be used to aid in the prediction of new or unknown molecules.
The first approach to developing quantitative relationships which described activity as a function of chemical structure relied on the principles of thermodynamics. The free-energy terms E, H and S were represented by a series of parameters which could be derived for a given molecule.
Electronic effects such as electron donating and withdrawing tendencies, partial atomic charges and electrostatic field densities were defined by Hammett sigma () values, resonance parameters (R values), inductive parameters (F values) and Taft substituent values (*, *, Es). Steric effects such as molecular volume and surface area were represented by values calculated for Molar Refractivity and the Taft steric parameter. Enthalpic effects were calculated using partition coefficients (LogP) or the hydrophobic parameter, , which was derived from the partition coefficient. In addition, an assortment of structural indices were used to describe the presence of specific functional groups at positions within the molecule. The linear equation which described the relationship between activity and this parameter set was the Hansch equation
log 1/[C] = A(logP) - B(logP)2 + C(Es) + D() + E + ...
Multiple linear regression analysis was used to derive the values of the coefficients. In general, Hansch type studies were performed on compounds which contained a common template (usually a rigid one such as an aromatic ring) with structural variation limited to functional group changes at specific sites.
Hansch utilized this approach in his analysis of 256 4,6-diamino-1,2-dihydro-2,3-dimethyl-1-(X-phenyl)-s-triazines which were active against tumor dihydrofolate reductase9. It was demonstrated that for 244 of the compounds, activity could be correlated to the presence of hydrophobic groups at the three and four positions of the N-phenyl ring. The parameters used to derive this correlation were the hydrophobic constant () and molar refractivity constant (MR) for meta and para substituents on the N-phenyl ring and six indicator variables I1-I6 which were used to indicate the presence (a value of 1) or absence (a value of 0) of specific structural features. The equation which was formulated from these data using the method of least squares is shown below.
Analysis of the Baker Triazines
log 1/[C] = 0.680(3) - 0.118(3)2 + 0.230(MR4) - 0.024(MR4)2 + 0.238(I1) - 2.530(I2) - 1.991(I3) + 0.877(I4) + 0.686(I5) + 0.704(I6) + 6.489
n = 244, r = 0.923, s = 0.377
The optimal values for MR4 (4.7) and 3 (2.9) were obtained from the partial derivatives of the equation. Note that the number of compounds in the data set was reduced to 244. Hansch and Silipo reported improvements in the value for r and s by removing 12 compounds which were incorrectly predicted by a factor of 10 or more.
While there are limits to the Hansch approach, it permitted complex biological systems to be modeled successfully using simple parameters. The approach has been used successfully to predict substituent effects in a wide number of biological assays. The main problem with the approach was the large number of compounds which were required to adequately explore all structural combinations. Further, the analysis methods did not lend themselves to the consideration of conformational effects. Several authors have published articles which provide additional background on the Hansch approach10-11.
Alternative approaches to compound design have been suggested which avoid the combinatorial problem found in Hansch type analyses. Free and Wilson used a series of substituent constants which related biological activity to the presence of a specific functional group at a specific location on the parent molecule12. The relationship between biological activity and the presence or absence of a substituent was then expressed by the following equation:
Activity = A + ijGijXij
where A was defined as the average biological activity for the series, Gij the contribution to activity of a functional group i in the jth position and Xij the presence (1.0) or absence (0.0) of the functional group i in the jth position.
The procedure used the equation above to build a matrix for the series and represented this matrix as a series of equations. Substituent constants then were derived for every functional group at every position. Statistical tests were used to test the importance of the constants. If the models were shown to be valid, the model was used to predict activity values for compounds which had not been prepared. In general, while a large number of compounds are required to explore the effects of multiple substitution patterns, the Free-Wilson approach substantially reduces the number of analogs required. However, the method demands that the effects of substituents are additive.
In 1972, John Topliss published a paper which detailed methodology to automate the Hansch approach2. The method assumed that the lead compound of interest contained at least one phenyl ring which could serve as the template for functional group modifications. The first modification to the template was preparation of the para-chloro derivative to examine lipophilicity. Additional substitution patterns were then made sequentially in an attempt to explore and optimize the relationship between activity and the hydrophobic and electronic character of the molecule. While the Topliss approach is easy to follow, it has several drawbacks. The primary problems are that the procedure is not applicable to all types of studies and that there is a high degree of risk associated with its use (it essentially ignores the possibility of interactions between substituents as it changes one substituent at a time).
The use of classical QSAR was expanded during the 1960's as a means of correlating observed activity to chemical properties. However, there are many areas where these techniques could not be used or where they failed to provide useful correlations. These included situations in which activity was found to be determined by 3-dimensional geometry, where poor training sets of compounds were used or the set of compounds were too small or insufficiently diverse and cases where biological activity could not be well quantified. Many of these problems were addressed by extensions to the Hansch method and the development of alternative approaches to QSAR.
There are cases where biological activity values cannot be determined accurately for a variety of reasons, e.g. lack of sensitivity of a particular test system. Alternative statistical techniques can be used in these cases; the problem is simplified to a classification scheme in which compounds are labeled as active, partially active, inactive, etc. The resulting data set is then searched for patterns which predict these categories. The methods which have been used for this type of analysis include SIMCA (Soft, Independent Modeling of Class Analogy)13, ADAPT (Automated Data Analysis by Pattern recognition Techniques)14, CASE (Computer Automated Structure Evaluation)15 and CSA (Cluster Significance Analysis)16.
Pattern recognition methods17 attempt to define the set of parameter values which will result in clustering compounds of similar activity into regions of n-dimensional space. The methods used to accomplish this goal can be parametric or nonparametric. Parametric methods search the n-dimensional space for clusters of compounds based on their calculated properties. These methods do not use derived values (e.g., mean vectors and covariance matrices), but instead use the original data to find clustering definitions and apply iterative procedures to find the linear set of parameters which best define the classification scheme.
Where the methods described above develop discriminant functions, SIMCA methods use Principal Component Analysis (PCA) to describe the data set. The objective of PCA is to create a reduced number variables which describe biological activity or chemical properties into a relatively few independent ones. This is accomplished through an analysis of the correlation matrix of biological or chemical properties.
Principle component analysis can be used to create derived variables for each class (e.g., active and inactive) separately by decomposing the correlation matrix; this method is useful to point out redundancies or interrelationships among the variables. PCA seeks to find simplified relationships in data by transforming the original parameters into a new set of uncorrelated variables which are termed principal components. The symmetric correlation matrix is decomposed by an eigenvalue decomposition. The largest eigenvalue and its eigenvector are used to form a linear combination of the original variables with maximum variance. Successively smaller eigenvalues and vectors produce linear combinations of the original variables with diminishing variance. Successive eigenvectors are independent of one another. The simplification is derived by disregarding eigenvectors associated with small eigenvalues. In summary, the procedure finds the set of orthogonal axes for the data which decompose variance in the data.
Another approach to examining the effects of chemical structure on activity was developed by the Jurs' group. Rather than rely on multivariant statistics to highlight these relationships, Jurs used the combination of cluster analysis and pattern recognition techniques as a tool to develop these correlations. The ADAPT program generated a data set of molecular descriptors (topological, geometrical and physicochemical) derived from three dimensional model building, projected these data points onto an n-dimensional surface and analyzed them using pattern recognition methods. The goal of this analysis was to discriminate between active and inactive compounds in a series.
Jurs has reported several applications of the methodology contained in ADAPT. In one study of chemical carcinogens18, a linear discriminant function was derived from a set of 28 calculated structure features including fragment descriptors, substructure descriptors, environment descriptors, molecular connectivity descriptors and geometric descriptors. Two hundred and nine compounds from twelve structural classes (130 carcinogens, 79 noncarcinogens) were selected for this study. The program was used to identify a training set of 192 compounds which was used to find the best set of descriptors and analyze the entire data set. A predictive success of 90% for carcinogenic compounds and 78% for noncarcinogenic compounds was obtained in randomized testing.
The CASE program extended the techniques in ADAPT by using topological methods to define substructural fragments which were essential for activity. CASE was able to differentiate between positional isomers. Both CASE and ADAPT are limited to analyzing structurally similar data sets.
The analysis methods described to this point have not explicitly incorporated the contribution of three dimensional shape in the analysis of the activity of a molecule. While the use of chemical graph indexes18, intermolecular binding distances19, molecular surface areas20 and electrostatic potentials21 contain some information about the 3-D shape of molecules, the Hopfinger22 and Marshall23 groups were the first to exhaustively analyze these effects.
In 1979, Marshall extended the 2-D approach to QSAR by explicitly considering the conformational flexibility of a series as reflected by their 3-D shape23. The first step of the Active Analog Approach was to exhaustively search the conformations of a compound which was highly active in a particular biological assay. The result of the search was a map of interatomic distances which was used to filter the conformational searches of subsequent molecules in the series. The implicit assumption of the method was that all compounds which display similar activity profiles were able to adopt similar conformations. Once the "active conformation" was determined, molecular volumes for each molecule were calculated and superimposed. Regression analysis of the volumes was used to establish a relationship to biological activity. Marshall and co-workers commercialized the Active Analog Approach and a suite of other drug design techniques in the SYBYL molecular modeling program.
Hopfinger and co-workers also used 3-D shape in QSAR. In molecular shape analysis24 of the Baker Triazines, the common space shared by all molecules of a series and the differences in their potential energy fields were computed. When these calculations were combined with a set of rules for overlapping the series, comparative indicies of the shape of different molecules were obtained. Inclusion of these shape descriptors in standard Hansch analysis schemes lead to improved descriptions relating computed parameters to biological activity such that no compounds in the original data set had to be eliminated from the calculations. The techniques developed by Hopfinger and co-workers were made available in the CAMSEQ, CAMSEQ-II, CHEMLAB and CAMSEQ-M computer programs.
In 1988, Richard Cramer proposed that biological activity could be analyzed by relating the shape-dependent steric and electrostatic fields for molecules to their biological activity25. Additionally, rather than limiting the analysis to fitting data to a regression line, CoMFA (Comparative Molecular Field Analysis) utilized new methods of data analysis, PLS (Partial Least Squares) and cross-validation, to develop models for activity predictions.
The approach used in the CoMFA procedure requires that the scientist define alignment rules for the series which overlap the putative pharmacophore for each molecule; the active conformation and alignment rule must be specified. Once aligned, each molecule is fixed into a three-dimensional grid by the program and the electrostatic and steric components of the molecular mechanics force field, arising from interaction with a probe atom (e.g., an SP3 C atom), are calculated at intersecting lattice points within the 3-D grid. The equations which result from this exercise have the form
Act1 = Const1 +
b1(stericxyz) + ... +
b'1(estaticxyz) + ...
Act2 = Const2 + a2(stericxyz) + b2(stericxyz) + ... + a'2(estaticxyz) + b'2(estaticxyz) + ...
Actn = Constn + an(stericxyz) + bn(stericxyz) + ... + a'n(estaticxyz) + b'n(estaticxyz) + ...
Traditional regression methods require that the number of parameters must be considerably smaller than the number of compounds in the data set (or the number of degrees of freedom in the data). The data tables which result from CoMFA analysis have far more parameters than compounds. PLS, which removes this limitation, is used to derive the coefficients for all of the steric and electrostatic terms. PLS essentially relies upon the fact that the correlations among near parts of a molecule are similar so that the real dimensionality is smaller that the number of grid points. Since these coefficients are position dependant, substituent patterns for the series are elucidated which define regions of steric bulk and electrostatic charge associated with increased or decreased activity. The size of the model (the number of components27 needed for the best model) and the validity of the model as a predictive tool are assessed using cross-validation.
As opposed to traditional regression methods, cross-validation evaluates the validity of a model by how well it predicts data rather than how well it fits data. The analysis uses a "leave-one-out" scheme; a model is built with N-1 compounds and the Nth compound is predicted. Each compound is left out of the model derivation and predicted in turn. An indication of the performance of the model is obtained from the cross-validated (or predictive) r2 which is defined as
r2 (cross-validated) = (SD - Press)/SD
SD is the Sum-of-Squares deviation for each activity from the mean. Press (or Predictive Sum of Squares) is the sum of the squared differences between the actual and that predicted when the compound is omitted from the fitting process.
As we have discussed, values for conventional r2 range from 0 to 1. Values for the cross-validated r2 are reported by the method to range from -1 to 1. Negative values indicate that biological activity values are estimated by the mean of the activity values better than they are by the model (i.e, the predictions derived from the model are worse than no model). Once a model is developed which has the highest cross-validated r2, this model is used to derive the conventional QSAR equation and conventional r2 and s values. The results of the final model are then visualized as contour maps of the coefficients.
The first CoMFA study reported analyzed the binding affinities of 21 steroid structures to human corticosteroid-binding globulins and testosterone-binding globulins. This class of compounds is rigid and was selected to eliminate conformationally dependant effects from the study. The models for each steroid were built from coordinates from the Cambridge Crystallographic Database which were minimized using the Tripos force field. Side chain positioning was accomplished using systematic conformational searching. The Field Fit algorithm was used to align each structure within the fixed lattice (the 3-D grid used to calculate the CoMFA field effects). The fit of the regression line for the predicted versus actual binding values for the corticosteroids showed a cross-validated r2 of 0.65 (conventional r2 = 0.897, s = 0.397). For the testosterone-binding steroids, the cross-validated r2 was 0.555 (conventional r2 = 0.873, s = 0.453).
As noted, CoMFA starts with defined pharmacophore and overlap rules and derives a 3-D model which can be used to predict activity for new chemical entities. The Apex and Catalyst (Accelrys, Formerly Molecular Simulations Incorporated) programs are used to identify pharmacophores from databases of chemical structures and biological activity. These models are then used to predict activities for novel compounds.
Apex-3D is an automated pharmacophore identification system which can identify possible pharmacophores from a set of biologically active molecules using statistical techniques and 3-D pattern matching algorithms. The program classifies molecular structures using three methods: the agreement inductive method identifies common structural patterns in compounds having similar activity; the difference inductive method identifies structural patterns which differentiate active and inactive compounds and the concomitant variations inductive method highlights variations in structural features that explain changes in biological activity for sets of compounds.
The methods defined above follow logic similar to that used by a practicing medicinal chemist: What pharmacophoric patterns are present in the active molecule which are not present in the inactive ones? Pharmacophores are defined by different chemical centers (atom centered functional groups) and the distances between these centers. These descriptor centers can include such things as aromatic ring centers, electron donor ability, hydrogen bonding sites, lipophilic regions, and partial atomic charge. The information for each molecule is stored in a knowledge base in the form of rules which can be used to predict the activity of novel structures.
Apex-3D contains an expert system which automatically selects the best conformation and alignment for structures based on identified pharmacophores. When quantitative biological data is available, a 3D-QSAR model can be developed for any of the possible identified pharmacophores. Depending on the type of biological activity available, it is possible to identify pharmacophores for different binding orientations, receptor subtypes, or agonist versus antagonist activity.
To use this approach, the scientist is required to assign the training set of compounds to one or more activity classes Specific descriptor centers also can be defined if desired. The automated pharmacophore identification portion of the program then builds the knowledge base using the following steps:
- identify all possible binding interaction centers for each
compound in the data set;
- generate topological (2D) or topographical (3D) distance
matrices based on the set of descriptors;
- identify possible pharmacophores from all pairs of molecules
using clique selection algorithms;
- classify these pharmacophores based upon their occurance in
compounds in each activity class using Bayesian statistics and
their nonchance occurance;
- set thresholds for probability and reliability statistics
associated with a pharmacophore so that all training set molecules
are properly classified by the pharmacophore rules;
- align compounds containing high probability pharmacophores on
Once the knowledge base has been constructed, the scientist can use it to predict biological activity for compounds not included in the training set.
The pharmacophores defined above can be used to build 3D QSAR models by correlating indexes calculated for biophore sites, secondary sites or whole molecule properties. 3-D QSAR models in Apex are generated and screened using a modified scheme of multiple linear regression analysis with variable selection. Special randomization checks are made to estimate the chance of fortuitous correlation. The steps involved are:
- Interactively select the pharmacophore(s) to use in the
- Interactively choose parameters to include in the pool of
possible parameters to be selected by the program and
- Calculate the best 3D QSAR models for each pharmacophore using
stepwise multiple regression and analysis of statistics to assess
the validity and predictive power of the model.
At this point the knowledge base with 3D QSAR models can be used to calculate activities of novel compounds. Given a set of conformations for a novel structure, an activity range will be calculated based on each conformation which contains one of the pharmacophores and fits the 3D QSAR model.
Few applications of Apex have appeared to date in the literature. One example applied to nonpeptide Angiotensin II antagonists will be discussed for its heuristic value. Several structurally diverse compounds in this activity class have been reported in the literature31-44. Pharmacophore models have been postulated with some disagreement about whether all of the highly active molecules are binding at the same site30. Automated pharmacophore identification can be used to analyze these compounds and to assess the probability that they could be acting at the same site. A set of 55 compounds with specific binding activity (IC50) values ranging over 6 orders of magnitude was used. Multiple conformations were included using 3D structures whose geometries were optimized by AMPAC. Four activity classes were defined; the most active class (< 100 nM) included 27 compounds. Apex-3D was able to generate rules which properly classified all compounds in the most active class without false negatives or positives. The fact that several biophores were required is consistent with the exsistence of multiple binding sites.
Quantitative 3D QSAR models also were developed for some of these pharmacophores. One model which contained 48 compounds had the following statistical parameters: predicted R2 = 0.83, predicted RMSE = 0.86. Predicted activities for compounds excluded from the training set were within statistical boundaries.
Like Apex, Catalyst generates structure-activity hypotheses from a set of molecules of various activities. Once molecular connectivity and activity values are specified for all molecules, Catalyst derives a hypotheses which consist of a set of generalized chemical functions (regions of hydrophobic surface, hydrogen bond vectors, charge centers, or other user-defined features) at specified relative positions. Up to ten hypotheses are produced and ranked by estimated statistical significance. The hypotheses can be examined graphically, fit to new molecules, or fed directly to flexible 3D database search.
In the first step of the process, a set of representative conformers is found that covers the low-energy conformational space of each molecule. Representative structures are chosen to maximize the sampling of conformational space. The second step locates a list of candidate hypotheses that are common among active and rare among inactive compounds (the presence of all identified features is not required for inclusion in the active class since it is unlikely that all active members of a training set possess all the important binding features). The cost of a hypothesis is defined as the number of bits needed to describe the hypothesis as well as the errors in activities as estimated by the hypothesis. The theory of minimum complexity estimation indicates that a predictive hypothesis will minimize this cost. Minimization is carried out over the space of hypotheses covered by the candidates identified above. Statistical significance of the results (low probability of having found a chance correlation) on a variety of medicinal training sets is verified by a non-parametric randomization test.
The Genetic Function Approximation (GFA) algorithm is a novel technique for constructing QSAR models45. It was specifically developed for use with data sets containing many more variables than samples, or data sets which contain nonlinear relationships between the variables and the activity.
GFA begins with a population of randomly-constructed QSAR models; these models are rated using an error measure which estimates each model's relative predictiveness. The population is evolved by repeatedly selecting two better-rated models to serve as parents, then creating a new child model by using terms from each of the parent models. The worst-rated model in the population is replaced by this new model. As evolution proceeds, the population becomes enriched with higher and higher quality models.
The different models are, in effect, multiple fits to the data. Scientists can then use their scientific knowledge and intuition to select among the final models. By studying the similarities and differences among the models, the scientist may be provoked to consider alternative mechanisms which explain the data or may plan new experiments to decide between different possible models. Since the models usually use only a subset of the total number of variables in the data set, GFA behaves as a member of the class of variable reduction algorithms and is most useful if the critical information in the data set is concentrated in a few variables.
Experimental results against published data sets demonstrate that GFA discovers models which are comparable to, and in some cases superior to, models discovered using standard techniques such as stepwise regression, linear regression, or partial least-squares regression. As the number of computational and instrumentational sources of experimental data increase, the ability of GFA to perform variable reduction, to discover nonlinear relationships, and to present the user with multiple models representing multiple interpretations of the data set, may become increasingly useful in data analysis.
There are many additional properties beyond binding which contribute to the biological activity of a molecule including transportation, distribution, metabolism and elimination. Since the techniques described above address binding and typically do not account for any of these other processes, they often fail. However, one should note that their utility is one of efficiency and probability. They are generally more successful and efficient than are the ad hoc/intuitive methods. Because no single method works for all cases, many groups are examining alternative approaches to developing SAR equations including electronic based descriptors and topological indicies.
Peter Goodford has reported an energy based grid approach to compound design. The GRID program46 is a computational procedure for detecting energetically favorable binding sites on molecules of known structure. It has been used to study arrays of molecules in membranes and crystals or proteins. The energies are calculated as the electrostatic, hydrogen-bond and Lennard-Jones interactions of a specific probe group with the target structure. GRID has also been used to distinguish between selective binding sites for different probes.
In general, topological approaches start with a very different graph theoretic representation of molecular structure in which atoms are represented by vertexes and bonds by edges. Numerical indexes for the structure are then defined which abstract information including atom descriptors (atom types, atomic weight, atomic number, ratio of valance electrons to core electrons, etc.) and sub-group descriptors. In addition, a series of indexes (the indexes) are developed which describe the molecule as a set of fragments of varying size and complexity. Regression analysis of equations which have chi indexes as parameters has been used to correlate chemical structure to physicochemical behavior in applications such as chromatographic retention times, molar refractivity, ionization potential and heats of atomization. While the application of topological indicies has been widely reported in QSAR47-49, the utility of these methods generally has been limited to predicting structure-property relationships for polymers and hydrocarbons.
Developing a quantitative structure activity relationship is difficult. Molecules are typically flexible and it is possible to compute many possibly useful properties that might relate to activity. Early in a research program there are typically few compounds to model. Thus we have a few compounds in a very high dimensional descriptor space. Which are the important variables and how do we optimize them? It is clear that many training compounds need to span through the space and model fitting techniques need to address not only deriving a fit, but the predictive quality of the fit. While these methods have not discovered a new compound, they have aided scientists in examining the volumes of data generated in a research program. As the methods evolve, they will find broader application in areas such as combinatorial chemistry.
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