Mathematical descriptor

Basak, S. C., Mills, D., Gute, B. D. Predicting bioactivity and toxicity of chemicals from mathematical descriptors A chemical-cum-biochemical approach. In Advances in Quantum Chemistry, Klein, D. J., Brandas, E., Eds., Elsevier, Amsterdam, 2004, in press. 

In the present work, we will use a relatively low level of theory to derive 32 weakly correlated molecular descriptors, each based on the subdivision and classification of the molecular surface area according to three fundamental properties contribution to ClogP, molar refractivity, and atomic partial charge. The resulting collection will be shown to have applicability in QSAR, QSPR, and compound classification. Moreover, the derived 32 descriptors linearly encode most of the information of a collection of traditional mathematical descriptors used in QSAR and QSPR. 

Many mathematical descriptors for sorption isotherms have been proposed. One of the more famous is that of Brunauer et al. (1938), the B.E.T. isotherm, which is based on the concept of a measurable amount of monomolecular layer (vicinal) water for a particular food. Wolf et al. (1985) compiled 2201 references on sorption isotherm data for foods. An example of the type, detail, and accuracy of sorption isotherm data available in the literature is presented in Table 3.2. 

Describing a fact, property, or a situation this leads us to the topic of Mathematical Descriptors that are able to represent such information in a computer program. 

Properties Cases are indexed by properties — typically one- or multidimensional experimental or mathematical descriptors. Descriptors allow a specific definition of the case according to the relevant features of a case. 

Throughout the realm of molecular modeling, the concept of molecular shape arises over and over in one form or another. Just what do scientists mean by a molecule s shape, and how can one use three-dimensional shape in modeling. In Chapter 5, Professor Gustavo A. Arteca examines these issues and delineates the hierarchical levels of molecular shape and shape descriptors. He explains molecular shape in terms of mathematical descriptors of nuclear geometry, connectivity, and molecular surfaces. Of special note are his comments on shape dynamics of flexible molecules and descriptors of relative shape. 

The Wiener index for isopentane is 18. The Wiener index is used as a mathematical descriptor used in structure - property relationships. There are many other related descriptors for molecular graphs, including shape index, chirality index, Szeged index [12]. A comparison of the boiling points of alkanes against the different indices has been shown in the literature [13]. 

The development and application of molecular shape descriptors is an active area in computational chemistry and biology. The main goal of our work is to develop mathematical descriptors that can determine whether two molecules have comparable shapes. In this chapter we present a series of molecular shape descriptors developed oti the basis of molecular vdW space. The molecules are treated in the hard sphere approximation, as a body composed from a collection of atomic fused spheres. Each sphere is centered in the corresponding nucleus and it is characterized by its Cartesian coordinates and by its vdW radius, r. These molecular vdW shape descriptors depend only on the internal structure of the molecule, being invariants to any translation and rotation movement. Consequently, they may inform us that two molecules have comparable shapes, but since they carry no information about the absolute orientation or position of the molecule, they are not useful for computing molecular superposition. 

Basak SC, Mills D, Gute BD (2006) Predicting Bioactivity and Toxicity of Chemicals from Mathematical Descriptors A Chemical-cum-Biochemical Approach. In Klein DJ, Brandas E (eds) Advances in Quantum Chemistry, Elsevier, in press... 

As previously mentioned, the acronym QSAR stands for the quantitative structure-activity relationship. However, there may be some ambiguity associated with the attribute quantitative. It does not necessarily follows that results expressed or having numerical representation are necessarily quantitative. Qualitative results can equally be numerically represented. Strictly speaking, we define and view QSAR models as quantitative only when the numerically expressed models allow meaningful interpretation of the numerical results obtained for the structure-activity relationship within the basic concepts of the particular model. This means that the physicochemical models should allow quantitative interpretation of the numerical physicochemical descriptors used and that the structure-mathematical models should allow quantitative interpretation of the numerical structure-mathematical descriptors used. We will use the symbol qsar and QSAR as the abbreviation for qualitative structure-activity relationship. Such are the relationships that are non-numerical and the relationships that may be numerical but the variables used are interrelated and thus do not allow unique interpretation of the MRA equations. Because all molecular descriptors hitherto used in QSAR, whether they are based on physicochemical properties, quantum mechanical calculations, or molecular graphs, are all interrelated, it follows that all such hitherto reported results, without further elaboration, remain essentially qualitative, being qsar rather than QSAR. 

Structure-property-activity studies. The other is the pair of shape indices of Randic [28], proposed in 2001, which is one of very few indices that have shown very good regressions with several physicochemical properties of octane isomers. There are not two but at least two dozen that would be worth mentioning here, but this is a book about solved and unsolved problems in structural chemistry and not a book about topological indices. As one can see, both above-mentioned indices came after a good 25 years of intensive search for simple novel mathematical descriptors for QSAR, which resulted in several hundred reports on new indices of various complexities. In contrast, the Estrada index and the proposed shape indices of Randic are conceptual and computationally unusually simple and elegant. 

From that point of view, new mathematical descriptors of recent times are more likely than not to make many ad hoc descriptors of the past less relevant for QSAR than they appear today. 

The two matrix representations of the protein segments, the amino acid adjacency matrix and the decagonal isometries matrix, are derived from the sequence information alone. As has been demonstrated, mathematical descriptors, dependent on the sequence information alone, have successfully revealed the underlying characteristics and patterns of given sequences. Their numerical nature also makes them easier to incorporate into a mathematical model. In addition, as has been well illustrated in chemical graph theory, when considering characterization of molecules, one can... 

The book discusses a number of important problems in chemistry that have not been fully understood or fully appreciated, such as the notion of aromaticity and conjugated circuits, the generalized Htickel 4n + 2 Rule, and the nature of quantitative structure-property-activity relationships (QSARs), which have resulted in only partially solved problems and approximated solutions that are inadequate. It also describes advantages of mathematical descriptors in QSAR, including their use in screening eombinatorial libraries to search for structures with high similarity to the target compounds. 

The alternatives to mathematical descriptors derived from molecular graphs or molecular geometry are the traditional QSAR (quantitative structure-activity relationship) descriptors and quantum chemically computed parameters. The former include the partition coefficient for oil/water (often octanol/water) (log P), the Hammet sigma value (electronic parameter that measures the electron withdrawal from and the electron release to the aromatic ring by a substituent, the Taft s parameters for the electronic effects of substituents in aliphatic compounds (a ), and a steric parameter for the proximity of substituents on reaction sites (Es)- Also selected molecular properties, such as molar refractivity (MR), polarizability (a), molecular weight (MW), and density (d), have been used. 

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