Kier and Hall

Correlations of Kier and Hall Kier and Hall [20] have studied relationships between dj° and various MCIs. For example, they report the following equation for alkanes (C5-C9)  

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Many of the descriptors which can be calculated from the 2D structure rely upon the molecular graph representation because of the need for rapid calculations. Kier and Hall have developed a large number of topological indices, each of which characterises the molecular structure as a single number [Hall and Kier 1991]. Every non-hydrogen atom ir the molecule is characterised by two delta values, the simple delta Si and the valence delta SJ ... 

Weiner index Randic indices Kier and Hall indices Inforanation content Connectivity index Balaban index... 

Most of the 2D QSAR methods are based on graph theoretic indices, which have been extensively studied by Randic [29] and Kier and Hall [30,31]. Although these structural indices represent different aspects of molecular structures, their physicochemical meaning is unclear. Successful applications of these topological indices combined with multiple linear regression (MLR) analysis are summarized in Ref. 31. On the other hand, parameters derived from various experiments through chemometric methods have also been used in the study of peptide QSAR, where partial least square (PLS) [32] analysis has been employed [33]. 

Kier and Hall developed an interesting concept termed molecular connectivity, in which the molecular-connectivity index, x, was defined as... 

Kier and Hall used the valence 5 index from Eq. (3) to define the family of molecular connectivity indices "Xt [13-15] ... 

Kier and Hall noticed that the quantity (S -S) jn, where n is the principal quantum number and 5 is computed with Eq. (2), correlates with the Mulliken-Jaffe electronegativities [19, 20]. This correlation suggested an application of the valence delta index to the computation of the electronic state of an atom. The index (5 -5)/n defines the Kier-Hall electronegativity KHE and it is used also to define the hydrogen E-state (HE-state) index. 

Another set of particularly useful 2D-based topological descriptors are the so-called electrotopological state index (E-state) descriptors developed by Kier and Hall [36],... 

Molecular connectivity indices (MCI) or % (Kier and Hall 1976, Andren et al. 1987, Nirmalakhandan and... 

A system of molecular connectivity indices was developed and extensively exploited by Kier and Hall [102-104,113], Hall and Kier [108,109,115,120,125] and Kier [107]. The zero-order (°y) and second-order (2y) molecular connectivity indices are the closest members to the l index described above. The... 

Liu et al. [27] introduced a new lipoaffinity descriptor to account for the effect of molecular hydrophobicity on BBB penetration. The descriptor was defined based on Kier and Hall s atom-type electrotopological state indices and therefore its evaluation is fast, requires no molecular 3D information, and... 

Traditional 2D-QSAR descriptors are generally considered to be the characteristics of a molecule, as a chemist would perceive the molecules. The molecules are described by their physical properties, subdivided surface area (86), atom counts and bonds, Kier and Hall connectivity and kappa shape indices... 

Calculated descriptors have generally fallen into two broad categories those that seek to model an experimentally determined or physical descriptor (such as ClogP or CpKJ and those that are purely mathematical [such as the Kier and Hall connectivity indices (4)]. Not surprisingly, the latter category has been heavily populated over the years, so much so that QSAR/QSPR practitioners have had to rely on model validation procedures (such as leave-k-out cross-validation) to avoid models built upon chance correlation. Of course, such procedures are far less critical when very few descriptors are used (such as with the Hansch, Leo, and Abraham descriptors) it can even be argued that they are unnecessary. 

An example is the method proposed by Kier and Hall, which usually considers only the C atoms and excludes the H atoms from consideration. Take the four isomers of pentane below, which are connected very differently. Each molecule can be described by a connection number 5 for each of the carbon atoms, 1 for a primary carbon, 2 for a secondary carbon, etc. 

Kier and Hall proposed a series of connectivity indexes. The zeroth index sums the reciprocal of the square roots of the connection numbers of each individual atom. 

The model of a molecule derived from the circle of research (Fig. 4) must be critically evaluated as to whether it contains the information associated with the emergent behaviour of the molecule. In other words, the question must be asked whether information from the atom level exceeds the logical depth necessary to understand the emergent properties of molecules as they relate to drug research (Kier and Hall, 1992). 

The second variant of QSAR is the use of actual structural descriptors, such as molecular orbital indices or topological codes, to define numerically the structure of a molecule and to find linear relationships with numerical biological data (Kier and Hall, 1976, 1992). 

Correlations of Needham, Wei, and Seybold Similar to the correlation of Kier and Hall, the correlation [21] uses MCIs as independent variables. The model has been derived for alkanes (C2-C9) ... 

Molecular Connectivity-AHvb Relationship Kier and Hall [14] derived the following relationship for alkanes (C2-C16) ... 

MCI-Boiling Point Relationships Kier and Hall [15], using connectivity indices, reported the following fit for alkanes (C5-C9) ... 

Molecular Connectivity- Kow Relationships Kier and Hall [34] have analyzed the correlation between K0w and various MCIs for hydrocarbons and monofunctional alcohols, ethers, ketones, acids, esters, and amines. Analogous relationships have been studied by Finizio et al. [35] for substituted s-triazines and by Govers et al. [36] for thioureas. 

X Randic branching index extension of the Randic approach by Kier and Hall [19]... 

Nirmalakhandan and Speece (1988) developed a correlation for log KAW using connectivity indices x (which have been developed, exploited, and discussed by Kier and Hall [1986]) and the polarizability . The standard error was 0.445 log units. 

Kier and Hall (1976) and Hall et al. (1975) have pioneered the use of the connectivity index as a descriptor of molecular structure. It is an expression of the sum of the degrees of connectedness of each atom in a molecule. Indices can be calculated to various degrees or orders, thus encoding increasing information about the structure. Although the index has been used with success in a number of applications, it is not entirely clear on theoretical grounds why this is so. It appears that the index generally expresses molar volume or area. 

First order MCIs, most commonly used in estimating Koc, can be hand calculated, although for large molecules the use of one of the computer programs previously mentioned greatly simplifies the task. A brief description of the procedure to calculate MCIs appears here. For a more detailed discussion of the procedure for calculating MCIs, see Kier and Hall (1986). 

Total positive partial charge Dipole moment from partial charges Kier and Hall molecular shape indices Solvent-accessible surface area... 

The manner in which chemical structure is described depends upon the concept applied. A chemical can variously be considered as a microscopic ensemble of nuclei and electrons and so may be described with energy functions. Alternatively, it can be regarded as a macroscopic collection of molecules and characterized with physicochemical properties. The term chemical structure is also related to the molecular formula (i.e., the atoms of which the molecule is composed, and the way in which those atoms are connected). According to Kier and Hall (2001), the structure is the count of each atom, identified as its element, along with a description, enumeration, or characterization... 

Based on the premise that a useful description of molecular structure represents the molecule in a surrounding of other structures, Kier and Hall developed the encounter concept to provide a theoretical basis for the interpretation of the molecular connectivity indices (Kier and Hall, 2000 2001). According to this concept, the measurement of a physical (or biological) property reflects the collective influence of the encounters between each molecule and other molecules in its direct environment. In the light of this theoretical concept, the interpretation of the value is that it encodes the relative accessibility of a bond to encounter another bond in another molecule, the resultant encounter which may lead to an intermolecular interaction. The term accessibility is defined as the topological and electronic availability of one bond to engage in some interaction with another bond. 

Bahnick and Doucette (1988) devised a polarity term, Au/J, calculated as the difference between a modified %v (calculated assuming that polar atoms have been replaced by carbon) and the normal U/J (i.e., 1st order valence molecular connectivity). Note that this is not the same as A1 devised by Kier and Hall (1986). 

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