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Item Derivatives in discrete mathematics: a novel graph-theoretical invariant for generating new 2/3D molecular descriptors. I. Theory and QSPR application(Journal of computer-aided molecular design, 2012) Marrero-Ponce, Yovani; Santiago, Oscar Martínez; Barigye, Stephen J.; Torrens, FranciscoShow more In this report, we present a new mathematical approach for describing chemical structures of organic molecules at atomic-molecular level, proposing for the first time the use of the concept of the derivative (∂ ) of a molecular graph (MG) with respect to a given event (E), to obtain a new family of molecular descriptors (MDs). With this purpose, a new matrix representation of the MG, which generalizes graph’s theory’s traditional incidence matrix, is introduced. This matrix, denominated the generalized incidence matrix, Q, arises from the Boolean representation of molecular sub-graphs that participate in the formation of the graph molecular skeleton MG and could be complete (representing all possible connected sub-graphs) or constitute sub-graphs of determined orders or types as well as a combination of these. The Q matrix is a non-quadratic and unsymmetrical in nature, its columns (n) and rows (m) are conditions (letters) and collection of conditions (words) with which the event occurs. This non-quadratic and unsymmetrical matrix is transformed, by algebraic manipulation, to a quadratic and symmetric matrix known as relations frequency matrix, F, which characterizes the participation intensity of the conditions (letters) in the events (words). With F, we calculate the derivative over a pair of atomic nuclei. The local index for the atomic nuclei i, Δ i , can therefore be obtained as a linear combination of all the pair derivatives of the atomic nuclei i with all the rest of the j′s atomic nuclei. Here, we also define new strategies that generalize the present form of obtaining global or local (group or atom-type) invariants from atomic contributions (local vertex invariants, LOVIs). In respect to this, metric (norms), means and statistical invariants are introduced. These invariants are applied to a vector whose components are the values Δ i for the atomic nuclei of the molecule or its fragments. Moreover, with the purpose of differentiating among different atoms, an atomic weighting scheme (atom-type labels) is used in the formation of the matrix Q or in LOVIs state. The obtained indices were utilized to describe the partition coefficient (Log P) and the reactivity index (Log K) of the 34 derivatives of 2-furylethylenes. In all the cases, our MDs showed better statistical results than those previously obtained using some of the most used families of MDs in chemometric practice. Therefore, it has been demonstrated to that the proposed MDs are useful in molecular design and permit obtaining easier and robust mathematical models than the majority of those reported in the literature. All this range of mentioned possibilities open “the doors” to the creation of a new family of MDs, using the graph derivative, and avail a new tool for QSAR/QSPR and molecular diversity/similarity studies.Show more Item Discrete Derivatives for Atom-Pairs as a Novel Graph Theoretical Invariant for Generating New Molecular Descriptors: Orthogonality, Interpretation and QSARs/ QSPRs on Benchmark Databases(Molecular Informatics, 2014) Martínez-Santiago, Oscar; Marrero-Ponce, Yovani; Barigye, Stephen J.; Torrens, Francisco; Pérez-Giménez, FacundoShow more 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 17O 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.Show more Item Relations Frequency Hypermatrices in Mutual, Conditional and Joint Entropy-Based Information Indices(Journal of Computational Chemistry, 2013) Barigye, Stephen J.; Marrero-Ponce, Yovani; Martı´nez-Lopez, Yoan; Torrens, FranciscoShow more Graph-theoretic matrix representations constitute the most popular and significant source of topological molecular descriptors (MDs). Recently, we have introduced a novel matrix representation, named the duplex relations frequency matrix, F, derived from the generalization of an incidence matrix whose row entries are connected subgraphs of a given molecular graph G. Using this matrix, a series of information indices (IFIs) were proposed. In this report, an extension of F is presented, introducing for the first time the concept of a hypermatrix in graph-theoretic chemistry. The hypermatrix representation explores the n-tuple participation frequencies of vertices in a set of connected subgraphs of G. In this study we, however, focus on triple and quadruple participation frequencies, generating triple and quadruple relations frequency matrices, respectively. The introduction of hypermatrices allows us to redefine the recently proposed MDs, that is, the mutual, conditional, and joint entropy-based IFIs, in a generalized way. These IFIs are implemented in GT-STAF (acronym for Graph Theoretical Thermodynamic STAte Functions), a new module of the TOMOCOMD-CARDD program. Information theoretic-based variability analysis of the proposed IFIs suggests that the use of hypermatrices enhances the entropy and, hence, the variability of the previously proposed IFIs, especially the conditional and mutual entropy based IFIs. The predictive capacity of the proposed IFIs was evaluated by the analysis of the regression models, obtained for physico-chemical properties the partition coefficient (Log P) and the specific rate constant (Log K) of 34 derivatives of 2-furylethylene. The statistical parameters, for the best models obtained for these properties, were compared to those reported in the literature depicting better performance. This result suggests that the use of the hypermatrix-based approach, in the redefinition of the previously proposed IFIs, avails yet other valuable tools beneficial in QSPR studies and diversity analysis.Show more