Autocorrelation vectors

Let us start with a classic example. We had a dataset of 31 steroids. The spatial autocorrelation vector (more about autocorrelation vectors can be found in Chapter 8) stood as the set of molecular descriptors. The task was to model the Corticosteroid Ringing Globulin (CBG) affinity of the steroids. A feed-forward multilayer neural network trained with the back-propagation learning rule was employed as the learning method. The dataset itself was available in electronic form. More details can be found in Ref. [2]. 

Z eb index, Wiener index. Balaban J index, connectivity indices chi (x), kappa (k) shape indices, molecular walk counts, BCUT descriptors, 2D autocorrelation vector... 

In order to transform the information fi om the structural diagram into a representation with a fixed number of components, an autocorrelation function can be used [8], In Eq. (19) a(d) is the component of the autocorrelation vector for the topological distance d. The number of atoms in the molecule is given by N. 

A range of physicochemical properties such as partial atomic charges [9] or measures of the polarizabihty [10] can be calculated, for example with the program package PETRA [11]. The topological autocorrelation vector is invariant with respect to translation, rotation, and the conformer of the molecule considered. An alignment of molecules is not necessary for the calculation of their autocorrelation vectors. 

In the calculation of a 3D autocorrelation vector the spatial distance is used as given by Eq. (20). 

Here, the component of the autocorrelation vector a for the distance interval between the boundaries dj (lower) and (upper) is the sum of the products of property p for atoms i and j, respectively, having a Euclidian distance d within this interval. 

In contrast to the topological autocorrelation vector, it is possible to distinguish between different conformations of a molecule using 3D autocorrelation vectors. 

The calculation of autocorrelation vectors of surface properties [25] is similar (Eq. (21), with the distance d XiXj) between two points and Xj on the molecular surface within the interval between d[ and d a certain property p, e.g., the electrostatic potential (ESP) at a point on the molecular surface and the number of distance intervals 1). 

The component of the autocorrelation vector for a certain distance interval between the boundaries 4 and du is the sum of the products of the property p x,) at a point Xi on the molecular surface with the same property p Xj) at a point Xj within a certain distance d Xj,Xj) normalized by the number of distance intervals 1. All pairs of points on the surface are considered only once. 

The 3D autocorrelation vector of the three xylene isomers in Figure 8-4 differ only with respect to the component relating to the two methyl groups. For o-xylene it is... 

Figure 8-4. Comparison of 3D autocorrelation vectors of o-, m-, and j-xylene (without hydrogen atoms) Atomic property p = 1.

When the resolution of the autocorrelation vector is decreased, some signals, e.g., those for the methyl groups in m- and / -xylene, may collapse. In such a case, one cannot distinguish between these two isomers. 

The representation of molecules by molecular surface properties was introduced in Section 2.10. Different properties such as the electrostatic potential, hydrogen bonding potential, or hydrophobicity potential can be mapped to this surface and seiwe for shape analysis [44] or the calculation of surface autocorrelation vectors (refer to Section 8.4.2). 

For each property, an autocorrelation vector of length 17 was calculated. This yielded a descriptor vector which contained 136 (8 x 17) individual descriptors. 

Through the application of the GA to the topological autocorrelation vector, the descriptor could be reduced from 136 to nine descriptors and the quality of projection was increased by 9.4%. Thus, besides the improvement in efficiency, an improvement in quality could be obtained. 

Autocorrelation of Topological Structure Moreau and Broto [24,25] have suggested the autocorrelation vector of a molecular graph as the source for molecular descriptors. This method assumes that each atom i in the graph is uniquely associated with a numeric quantity, qit such as the atomic number, atomic mass, (v)(, (ds), Sv, or electronegativity. The intrinsic atom values of the electrotopological state [26] and the atomic Rd and log Kow parameters [27,28] are other potential atomic descriptors suitable to construct autocorrelation vectors. Generally, the fcth element of the autocorrelation vector is defined as... 

Sadowski etal. [49] have described the use of 3D autocorrelation vectors that are based on the electrostatic potential measured on the molecular surface of a molecule. The electrostatic potential was measured over 12 different distances giving 12 autocorrelation coefficients per molecule. The vectors were calculated for the molecules in two different combinatorial libraries a xanthene library and a cubane library. The compounds were then used to train a Kohonen network. The network was successfully able to separate the libraries. 

Bauknecht, H., Zell, A., Bayer, H., Levi, P, Wagener, M., Sadowski, J. and Gasteiger, J. (1996). Locating Biologically Active Compounds in Medium-Sized Heterogeneous Datasets by Topological Autocorrelation Vectors Dopamine and Benzodiazepine Agonists. J.Chem.Inf.Com-putScL, 36,1205-1213. 

Zakarya, D., Tiyal, F. and Chastrette, M. (1993b). Use of the Multifunctional Autocorrelation Method to Estimate Molar Volumes of Alkanes and Oxygenated Compounds Comparison Between Components of Autocorrelation Vectors and Topological Indexes. J.Phys.Org.Chem., 6,574-582. 

The following section gives an overview of feature trees and 2D autocorrelation vectors, the two most important graph-based topological descriptors used for virtual screening. 

In contrast to the topological autocorrelation vectors in the 3D autocorrelation vector, the spatial distance between atoms is used for calculation. Flence, using 3D autocorrelation vectors, it is possible to distinguish between different conformations of a molecule. The calculation of autocorrelation vectors of surface properties is similar to equation (10.2) ... 

Topological autocorrelation vectors are calculated from the two-dimensional structure of a molecule that can be expressed as molecular graph. One of the original... 

Autocorrelation Vectors (topological autocorrelation vectors, autocorrelation of a topological Structure, ATS) are molecular descriptors for a property distribution along the topological structure. 

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