A New Class of Molecular Shape Descriptors. 1. Theory and Properties

The integrals V (n₁, n₂, n₃) = ∫ dr x^n₁ y^n₂ z^n₃, where ∫ dr represents integration over the volume of a body, such as a molecule, where x, y, and z are Cartesian coordinates of a point in the interior of the body relative to an arbitrary reference frame, and where n₁, n₂, and n₃ are integers greater than or equal to zero, constitute moments of the volume distribution of the body. Considering all such quantities for which 0 ≤ n₁ + n₂ + n₃ ≤ 6 gives a set of 84 independent numbers which characterize the shape of the body and constitute a very useful set of shape descriptors. They also carry information about the absolute orientation and position of the body, and because their behavior under rotations and translations can be calculated quickly, they provide a fast, robust algorithm for the alignment of two similar molecules as well as a qualitative measure of their similarity. This paper reports the performance of the alignment algorithm on a learning set of about 80 different shapes. The algorithm is further tested against a set of small drug-like compounds that have been screened as anticancer agents. In both cases, excellent alignments of “shape-similar” molecules are obtained. Discussions are provided on many basic properties of these moments, e.g., their behavior under translations and rotations of the reference frame and their symmetry properties.