title: Discrete Derivatives for Atom-Pairs as a Novel Graph-Theoretical Invariant for Generating New Molecular Descriptors: Orthogonality, Interpretation and QSARs/QSPRs on Benchmark Databases.
abstract: This report presents a new mathematical method based on the concept of the derivative of a molecular graph (G) with respect to a given event (S) to codify chemical structure information. The derivate over each pair of atoms in the molecule is defined as ∂G/∂S(vi  , vj )=(fi -2fij +fj )/fij , where fi (or fj ) and fij are the individual frequency of atom i (or j) and the reciprocal frequency of the atoms i and j, respectively. These frequencies characterize the participation intensity of atom pairs in S. Here, the event space is composed of molecular sub-graphs which participate in the formation of the G skeleton that could be complete (representing all possible connected sub-graphs) or comprised of sub-graphs of certain orders or types or combinations of these. The atom level graph derivative index, Δi , is expressed as a linear combination of all atom pair derivatives that include the atomic nuclei i. Global [total or local (group or atom-type)] indices are obtained by applying the so called invariants over a vector of Δi values. The novel MDs are validated using a data set of 28 alkyl-alcohols and other benchmark data sets proposed by the International Academy of Mathematical Chemistry. Also, the boiling point for the alcohols, the adrenergic blocking activity of N,N-dimethyl-2-halo-phenethylamines and physicochemical properties of polychlorinated biphenyls and octanes are modeled. These models exhibit satisfactory predictive power compared with other 0-3D indices implemented successfully by other researchers. In addition, tendencies of the proposed indices are investigated using examples of various types of molecular structures, including chain-lengthening, branching, heteroatoms-content, and multiple bonds. On the other hand, the relation of atom-based derivative indices with (17) O NMR of a series of ethers and carbonyls reflects that the new MDs encode electronic, topological and steric information. Linear independence between the graph derivative indices and other 0-3D MDs is demonstrated by using principal component analysis on a dataset of 41 heterogeneous molecules. It is concluded that the graph derivative indices are independent indices containing important structural information to be used in QSPR/QSAR and drug design studies, and permit obtaining easier, more interpretable and robust mathematical models than the majority of those reported in the literature.
authors.affiliation: Unit of Computer-Aided Molecular "Biosilico" Discovery and Bioinformatic Research (CAMD-BIR Unit), Facultad de Química y Farmacia,. Universidad Central "Marta Abreu" de Las Villas, Carretera a Camajuani Km 5 1/2, Santa Clara, 54830, Villa Clara, Cuba. fax: 963543156; phone: 963543156.
authors.forename: Oscar
authors.initials: O
authors.lastname: Martínez-Santiago
authors.affiliation: Unit of Computer-Aided Molecular "Biosilico" Discovery and Bioinformatic Research (CAMD-BIR Unit), Facultad de Química y Farmacia,. Universidad Central "Marta Abreu" de Las Villas, Carretera a Camajuani Km 5 1/2, Santa Clara, 54830, Villa Clara, Cuba. fax: 963543156; phone: 963543156.
authors.forename: Reisel
authors.initials: R
authors.lastname: Millán-Cabrera
authors.affiliation: Unit of Computer-Aided Molecular "Biosilico" Discovery and Bioinformatic Research (CAMD-BIR Unit), Facultad de Química y Farmacia,. Universidad Central "Marta Abreu" de Las Villas, Carretera a Camajuani Km 5 1/2, Santa Clara, 54830, Villa Clara, Cuba. fax: 963543156; phone: 963543156. ymarrero77@yahoo.es, ymponce@gmail.com.|Doctorado en Toxicología Ambiental, Facultad de Química Farmacéutica, Universidad de Cartagena, Cartagena de Indias, Bolívar, Colombia. ymarrero77@yahoo.es, ymponce@gmail.com.
authors.forename: Yovani
authors.initials: Y
authors.lastname: Marrero-Ponce
authors.affiliation: Unit of Computer-Aided Molecular "Biosilico" Discovery and Bioinformatic Research (CAMD-BIR Unit), Facultad de Química y Farmacia,. Universidad Central "Marta Abreu" de Las Villas, Carretera a Camajuani Km 5 1/2, Santa Clara, 54830, Villa Clara, Cuba. fax: 963543156; phone: 963543156.
authors.forename: Stephen J
authors.initials: SJ
authors.lastname: Barigye
authors.affiliation: Unit of Computer-Aided Molecular "Biosilico" Discovery and Bioinformatic Research (CAMD-BIR Unit), Facultad de Química y Farmacia,. Universidad Central "Marta Abreu" de Las Villas, Carretera a Camajuani Km 5 1/2, Santa Clara, 54830, Villa Clara, Cuba. fax: 963543156; phone: 963543156.|Department of Computer Sciences, Faculty of Informatics, Camaguey University, Camaguey City, 74650 Camaguey Cuba.
authors.forename: Yoan
authors.initials: Y
authors.lastname: Martínez-López
authors.affiliation: Institut Universitari de Ciència Molecular, Universitat de València, Edifici d'Instituts de Paterna, Polígon la Coma s/n, E-46071 Valencia, Spain.
authors.forename: Francisco
authors.initials: F
authors.lastname: Torrens
authors.affiliation: Unidad de Investigación de Diseño de Fármacos y Conectividad Molecular, Departamento de Química Física, Facultad de Farmacia, Universitat de València, Spain.
authors.forename: Facundo
authors.initials: F
authors.lastname: Pérez-Giménez
keywords: DIVATI; Event; Frequency matrix; Generalized incidence matrix; Genetic algorithm; Invariant; Molecular descriptors; QSPR; Sub-graph; TOMOCOMD-CARDD