Kier and Hall index

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

The Wiener index [86] can be expressed in terms of the distance matrix [87] and equals the half-sum of all distance matrix entries. Randi(5 [88] and Kier and Hall indices of order 0-3 [89] are calculated from coordination numbers of atoms or from values of atomic connectivity. The Kier shape index (order 1-3) [90] depends on the number of skeletal atoms, molecular branching, and the ratio of the atomic radius and the radius of the carbon atom in the sp hybridization state. The Kier flexibility index [90] is derived from the Kier shape index. The Balaban index depends on the row sums of the entries of the distance matrix and the cyclomatic number [92,93]. The information content index and its derivatives (order 0-2) are based on the Shannon information theory [95]. Modifications of the information content index are structural information content, complementary information content, and bond information content [96],... 

We believe that much more efficient descriptors can be discovered and this is a focus of our work. Until recently QSAR analyses have used relatively simple molecular descriptors based on substituent constants (e.g., Hammett constants, n, or molar refractivities), physicochemical properties (e.g., partition coefficients), or topological indices (e.g., Randic, Weiner, or Kier and Hall indices). Recently we and others have developed several new information rich, computationally efficient representations. The most interesting of these are the molecular eigenvalue indices. 

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 ... 

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

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. 

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). 

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

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. 

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

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. 

To derive these equations, log P (hydrophobic parameter), MR (molar refrac-tivity index), and MV (molar volume) were calculated using software freely available on the internet (wwwlogP.com, www.daylight.com). The first-order valence molecular connectivity index of substituents was calculated as suggested by Kier and Hall [46,47]. In these equations, is cross-vahdated obtained by the leave-one-out jackknife procedure. Its value higher than 0.6 defines the good predictive ability of the equation. The different indicator variables in these equations were defined as follows. 

Many different approaches to QSAR have been developed since Hansch s seminal work. These include both 2D (two-dimensional) and 3D (three-dimensional) QSAR methods. Most of the 2D QSAR methods employ graph theoretic indices to characterize molecular structures, which have been extensively studied by Radio, Kier and Hall. Similarly, ADAPT system employs topo-... 

These are truly structural descriptors because they are based only on the two-dimensional representation of a chemical structure. The most widely known descriptors are those that were originally proposed by Randic (173) and extensively developed by Kier and Hall (27). The strength of this approach is that the required information is embedded in the hydrogen-suppressed framework and thus no experimental measurements are needed to define molecular connectivity indices. For each bond the Ck term is calculated. The summation of these terms then leads to the derivation of X, the molecular connectivity index for the molecule. 

Differential connectivity indices are defined as differences between connectivity indices "x d -> valence connectivity indices [Hall and Kier, 1986 Kier and Hall, 1991] ... 

Kier and Hall defined [Kier and Hall, 1986 Kier and Hall, 1977b] a general scheme based on the Randic index to calculate also zero-order and higher-order descriptors, thus obtaining connectivity indices of m th order, usually known as Kier-Hall connectivity indices. They are calculated by the following ... 

Analogously, atom-type HE-state indices were proposed as molecular descriptors calculated by summing hydrogen electrotopological states of all atoms of the same atom type [Kier and Hall, 1999]. 

Six different weighting schemes are proposed (1) the unweighted case u (yvi = 1 i —, n, where A is the number of atoms for each compound), (2) atomic mass m, (3) the - van der Waals volume v, (4) the Sanderson - atomic electronegativity e, (5) the - atomic polarizability p and (6) the - electrotopological state indices of Kier and Hall 5. All the weights are also scaled with respect to the carbon atom, and their values are shown in Table W-5 moreover, as all the weights must be positive, the electrotopological indices are scaled thus ... 

Another set of topological descriptors is the electrotopological state indices (E-state indices) developed by Kier and Hall [26, 27]. These descriptors are based on the topological state of a particular atom with corrections for electronic interactions due to other atoms in the structure. This methodology originally devised for nonhydrogen atoms only has been extended to also include E-state indices for hydrogen atoms [28] and to also include atom-type E-state indices, for... 

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