An inequality for 3D database searching and its use in evaluating the treatment of conformational flexibility

A mathematical formula is introduced for predicting the number of hits that should be observed in a flexible 3D database search, based on the results of a set of related queries. The projected number of hits is always greater than or equal to the actual number of hits, the discrepancy being due to imperfect treatment of conformational flexibility of the molecules. Hence, the difference between the projected and actual number of hits, delta, serves to measure how well conformational flexibility is being treated, in a manner that is objective, easy for a user to quickly verify, and independent of the particular algorithm for flexible 3D database search. It is shown that delta is a function both of how well conformational flexibility is treated and of the precision of the query. When the distance constraint is defined only to a precision of +/-2.0 Angstrom, in a single-conformer database of drug-like molecules delta values of only 0.03 are found, while in a single-conformer database of di- and tripeptides, delta is 0.15. At increased precision, a flexible 3D database search becomes critical. For a single-conformer database, using a query of precision +/- 0.2 Angstrom, applied to a database of drug-like molecules, delta is 0.97; applied to a database of di- and tripeptides, delta is 2.21. By contrast, treating conformational flexibility by storing up to 100 conformers per molecule, at this precision, applied to a database of drug-like molecules, delta is 0.002; applied to a database of di- and tripeptides, delta is 0.07. This inequality, and hence delta, is defined only for database queries containing a single distance constraint; how the inequality may generalize to higher-dimensional queries is still unclear.