Bond‐extended stochastic and nonstochastic bilinear indices. I. QSPR/QSAR applications to the description of properties/activities of small‐medium size organic compounds
Authors: Marrero‐Ponce Yovani , Martínez Eugenio R. , Casañola‐Martín Gerardo M. , Pérez‐Giménez Facundo , Díaz Yunaimy Echevería , Garcia‐Domenech Ramon , Rodriguez Brogues José E. 
Published: November 2009

Bond‐extended stochastic and nonstochastic bilinear indices are introduced in this article as novel bond‐level molecular descriptors (MDs). These novel totals (whole‐molecule) MDs are based on bilinear maps (forms) similar to use defined in linear algebra. The proposed nonstochastic indices try to match molecular structure provided by the molecular topology by using the kth Edge(Bond)‐Adjacency Matrix (Ek, designed here as a nonstochastic E matrix). The stochastic parameters are computed by using the kth stochastic edge‐adjacency matrix, ESk, as matrix operators of bilinear transformations. This new edge (bond)‐adjacency relationship can be obtained directly from Ek and can be considered like a new matrix‐transformation strategic to obtain new relations for a molecular graph. In both set of MDs, chemical information is codified by using different pair combinations of atomic weightings (in this case four atomic‐labels: atomic mass, polarizability, van der Waals volume, and electronegativity). In addition, a local‐fragment (bond‐type) formalism was also developed. The kth bond‐type bilinear indices are calculated by summing the kth bond bilinear indices of all bonds of the same bond type in the molecules. The new set of MDs can be easily and quickly calculated in our in house software TOMOCOMD‐CARDD (topological molecular computational design computer‐aided‐rational‐drug design). The reported application and utilization of these MDs for predictive capability correlations of structure with physicochemical and pharmacological properties are reviewed. Three benchmark datasets have been used to evaluate the QSPR/QSAR behavior of the new bond‐level TOMOCOMD‐CARDD MDs. We developed the QSPR models to describe several physicochemical properties of octane isomers (First Case) and, to analyze of the boiling point of 28 alkyl‐alcohols (Second Case) and to examine of the specific rate constant (log k), the partition coefficient (log P), as well as the antibacterial activity of 34 derivatives of 2‐furylethylenes (Third Case). For these three rounds, the quantitative models found are significant from a statistical point of view and permit a clear interpretation of the studied properties in terms of the structural features of molecules. A leave‐one‐out cross‐validation procedure revealed that the regression models had a good predictability. The comparison with other approaches reveals good performance of the method proposed. Therefore, it is clearly demonstrated that this suitability is higher than that shown by other 2D/3D well‐known sets of MDs. The approach described here appears to be a very promising structural invariant, useful for QSPR/QSAR studies and shown to provide an excellent alternative or guides for discovery and optimization of new lead compounds, reducing the time and cost of the traditional procedure. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2011