Background:
Recently, some authors have defined new molecular descriptors (MDs) based on the
use of the Graph Discrete Derivative, known as Graph Derivative Indices (GDI). This new approach about discrete
derivatives over various elements from a graph takes as outset the formation of subgraphs. Previously,
these definitions were extended into the chemical context (N-tuples) and interpreted in structural/physicalchemical
terms as well as applied into the description of several endpoints, with good results.
Objective:
A generalization of GDIs using the definitions of Higher Order and Mixed Derivative for molecular
graphs is proposed as a generalization of the previous works, allowing the generation of a new family of MDs.
Methods:
An extension of the previously defined GDIs is presented, and for this purpose, the concept of Higher
Order Derivatives and Mixed Derivatives is introduced. These novel approaches to obtaining 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) of fragmenting the molecular structures. The result of applying
the higher order and mixed GDIs over any molecular structure allows 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 KeysFinder
Framework in TOMOCOMD-CARDD system.
Results:
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 compared to previously 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.
Conclusion:
It can be suggested that the higher-order and mixed GDIs, appear as a promissory tool in
QSAR/QSPRs, similarity/dissimilarity analysis and virtual screening studies.
Cebyšev inequalities for co-ordinated \(QC\)-convex and \((s,QC)\)-convex
