Pairwise-distance descriptors

C, E E Hodgkin and Richards W G 1993. The Utilisation of Gaussian Functions for the Rapid nation of Molecular Similarity. Journal of Chemical Information and Computer Science 32 188-192. C and I D Kuntz 1995, Investigating the Extension of Pairwise Distance Pharmacophore sures to Triplet-based Descriptors, Journal of Computer-Aided Molecular Design 9 373-379. 

Good, A.G. and Kuntz, I.D. Investigating the extension of pairwise distance pharmacophore measures to triplet-based descriptors./. Comput.-Aided Mol. Des. 1995, 9, 373-379. 

In order to apply the SA protocol, one of the keys is to design a mathematical function that adequately measures the diversity of a subset of selected molecules. Because each molecule is represented by molecular descriptors, geometrically it is mapped to a point in a multidimensional space. The distance between two points, such as Euclidean distance, Tanimoto distance, and Mahalanobis distance, then measures the dissimilarity between any two molecules. Thus, the diversity function to be designed should be based on all pairwise distances between molecules in the subset. One of the functions is as follows ... 

The descriptors used for pairwise distance measurements can be continuous, as in a physicochemical property, or binary e.g., the presence or absence of a specific substructure). For continuous chemical spaces, nearly all metrics are based on the generalized Minkowski metric given in (1), where % represents the Mi feature of the ith molecule, k is the total number of features, and r the order of the metric. 

Good, A.C. and Kuntz, I.D. (1995). Investigating the Extension of Pairwise Distance Pharmacophore Measures to Triplet-Based Descriptors. J.Comput.Aid.Molec.Des., 9, 373-379. 

Fig. 1. Clustering versus partitioning. In cluster analysis, compounds (gray dots) are grouped together based on the calculation of pairwise intermolecular distances in chemical space. By contrast, partitioning methods subdivide chemical space into sections into which compounds fall based on their calculated descriptor coordinates.

As discussed in Subheading 1., the primary design criterion is often based on either similarity or diversity. Quantifying these measures requires that the compounds are represented by numerical descriptors that enable pairwise molecular similarities or distances to be calculated or that allow the definition of a multidimensional property space in which the molecules can be placed. 

When molecules are represented by high-dimensional descriptors such as 2D fingerprints or several hundred topological indices, then the diversity of a library of compounds is usually calculated using a function based on the pairwise (dis)similarities of the molecules. Pairwise similarity can be quantified using a similarity or distance coefficient. The Tanimoto coefficient is most often used with binary fingerprints and is given by the formula below ... 

Distance-based metrics quantify the diversity of a set of compounds as a function of their pairwise (dis)similarities in a descriptor space. It is important to mention that distance coefficients are analogous to distances in multidimensional geometric space, although they are usually not equivalent to such distances. For a distance coefficient to be described as a metric, it must possess the following four properties (1) Distance values must be nonzero and the distance from an object to itself must be zero. (2) Distance values must be symmetric. (3) Distance values must obey the triangular inequality. (4) Distances between nonidentical objects must be greater than zero. A coefficient containing only the first three properties is dubbed a pseudometric, and one without the third property is a nonmetric. 

Figure 13.8. Comparison of pairwise biological distances versus descriptor differences for 138 angiotensinconverting enzyme (ACE) inhibitors (upper panel) as example for neighborhood plots (a) CoMFA ster-ic fields (b) 2D fingerprints (c) molecular weight.

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