Higher-Order and Mixed Discrete Derivatives like a Novel Graph-Theoretical Invariant for Generating New Molecular Descriptors.

Abstract

Here, an extension of the previously defined Graph Derivative Indices (GDIs) is presented, and for this purpose, the concept of Higher Order Derivatives and Mixed Derivatives are introduced. These novel approaches to obtaining Molecular Descriptors (MDs) based on the concepts of discrete derivatives (finite difference) of the molecular graphs use the elements of the hypermatrices conceived from 12 different ways (12 events) fragmenting the molecular structures. The result of applying the higher order and mixed GDIs over any molecular structure allow finding Local Vertex Invariants (LOVIs) for atom-pairs, for atoms-pairs-pairs and so on. All new families of GDIs are implemented in a computational software denominated DIVATI (acronym for Discrete DeriVAtive Type Indices), a module of TOMOCOMD-CARDD program (KeysFinder Framework). QSAR modeling of the biological activity (Log 1/K) of 31 steroids reveals that the GDIs obtained using the higher order and mixed GDIs approaches yield slightly higher performance compare to previous reported approaches based on the duplex, triplex and quadruplex matrix. In fact, the statistical parameters for models obtained with the higher order and mixed GDI method are superior to those reported in the literature by using other 0-3D QSAR methods. Therefore, it can be suggested that the higher order and mixed GDIs method, appear as a promissory tool in QSAR/QSPRs, similarity/dissimilarity analysis and virtual screening studies.