The Maximum Likelihood Method for Pharmacophore Mapping

The maximum likelihood method Catalyst/HipHop [Barnum et al. 1996; Catalyst] also uses a pre-calculated set of low-energy conformations. Typically, these are obtained using poling, a conformational search method designed to generate a relatively small set of conformations that "covers" pharmacophore space [Smellie et al. 1995a, b]. The poling method adds an additional penalty term to the energy function during the minimisation part of the conformational analysis. This penalty term has the effect of "pushing" the conformation away from those found previously.

Having generated a set of conformations for each molecule, the first step in the maximum likelihood method is to identify all configurations of pharmacophoric groups that are present in the molecules. Each molecule is taken in turn as a reference structure and its conformations examined. All possible combinations of pharmacophoric groups contained within it are generated exhaustively (e.g. donor-acceptor-aromatic ring centroid; acceptor-acceptor-hydrophobe-donor). Each of these configurations of pharmacophoric groups in 3D space is then compared to the other molecules in the set in order to determine whether they possess a conformation that can be successfully superimposed on the configuration. In this step it is not required that all molecules match all of the features (i.e. a molecule can be active despite lacking a feature that is present in the binding motif of other active molecules).

This first step can generate a large number of possible configurations which are then scored and ranked according to how well the molecules map onto them and also according to their "rarity". The general strategy is to score more highly those configurations (referred to as hypotheses) that are well matched by the active molecules in the set but which are less likely to be matched by a large set of arbitrary molecules. The scoring function used is:

where M is the number of active molecules in the set and K is the number of pharmacophoric groups in the hypothesis. The summation in Equation 2.2 takes account not only of the hypothesis itself but also other configurations containing K—1 features (there are K such configurations). An active molecule is assigned to the class x = K + 1 if it matches all the features, to one of the classes x = 1 through x = K if it matches one of the configurations with one feature removed, or to the class x = 0 if it matches neither the full hypothesis nor one of the subconfigurations. q (x) is the fraction of active molecules that matches each of the classes x. p(x) corresponds to the fraction of a large set of arbitrary molecules that would match the class x configuration (i.e. the "rarity" value mentioned above). Pharmacophores that contain relatively uncommon features such as positively ionisable groups are less likely to be matched by chance than pharmacophores that contain more common features such as hydrophobic features and so have a smaller value of p(x). In addition, the greater the geometrical distribution of features (dependent on the sum of the squares of the distances between the features and the common centroid) the higher the rarity value. A higher score for an hypothesis is thus associated with a higher value of q (x) (i.e. more of the active molecules match) and lower values of p(x) (i.e. it is less likely that an arbitrary, non-active molecule would match). The values of p(x) are obtained using a mathematical model previously derived from an analysis of a set of biologically active molecules.

Another feature of the maximum likelihood method that contrasts with the constrained systematic search and the clique detection methods is its use of location constraints to define the pharmacophore rather than distance ranges between the features [Greene et al. 1994]. These location constraints usually correspond to a spherical region in 3D space, centred on a point, within which the relevant pharmacophoric feature should be positioned. The radius of the spherical region may vary; the different values reflect differences in the dependence of interaction energy on distance for different types of interaction. For example, an ionic interaction is typically more sensitive to distance changes than a hydrogen bond and so the tolerance on a charged feature would be correspondingly smaller. Another requirement is that the pharmacophoric features must be "accessible", rather than being buried inside the molecule.

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