Topological similarity measures

Apparently, the concept of similarity plays an important role in the chemistry of functional groups. Motivated by the recent revival of interest in molecular similarity [7-39], we shall present a systematic approach towards a quantum chemical description of functional groups. There are two main components of the approach described in this report. The first component is shape-similarity, based on the topological shape groups and topological similarity measures of molecular electron densities[2,19-34], whereas the second component is the Density Domain approach to chemical bonding [4]. The topological Density Domain is a natural basis for a quantum... 

The algebraic and differential topological similarity measures required much simpler mathematical and computational apparatus than the direct comparisons of the original, complex quantum mechanical objects. 

Mezey, P.G. (1995) Density domain bonding topology and molecular similarity measures., In Topics in Current Chemistry, Vol. 173, Molecular Similarity, Sen, K. (Ed.), Springer-Verlag, Heidelberg. 

Mezey, P.G. Density Domain Bonding Topology and Molecular Similarity Measures. 173, 63-83 (1995). 

The concept of chemistry space pervades, either explicitly or implicitly, much of the literature in chemoinformatics. As is discussed in Subheading 3., chemistry spaces are induced by various similarity measures. The different similarity measures do not, however, give rise to topologically equivalent chemistry spaces—nearest-neighbor relations are generally not preserved among chemistry spaces induced by different similarity measures. The consequences of this are manifold. An especially egregious consequence is that the results of similarity searches based on different similarity measures can differ substantially. And there is no easy solution to this problem. 

Molecular similarity analysis has developed substantially over the years, especially as digital computers became faster, more compact, and widely available to scientists. Handling large sets of molecules is generally not a problem. The main problem confronting MSA is the problem of the lack of topological invariance of the chemistry spaces induced by the various similarity measures. Unfortunately, this problem may be fundamentally related to the inherent subjectivity of similarity and thus cannot be addressed in any simple manner. 

In a subsequent study, we examined the influence of seven similarity indices on the enrichment of actives using the topological CATS descriptor and the 12 COBRA datasets [31]. In particular, we evaluated to what extent different similarity measures complement each other in terms of the retrieved active compounds. Retrospective screening experiments were carried out with seven similarity measures Manhattan distance, Euclidian distance, Tanimoto coefficient, Soergel distance, Dice coefficient, cosine coefficient, and spherical distance. Apart from the GPCR dataset, considerable enrichments were achieved. Enrichment factors for the same datasets but different similarity measures differed only slightly. For most of the datasets the Manhattan and the Soergel distance... 

Density Domain Bonding Topology and Molecular Similarity Measures... 

This chapter focuses on step 3. For step 1, descriptors may include property values, biological properties, topological indexes, and structural fragments. The performance of these descriptors and forms of representation have been analyzed by Brown and Brown and Martin. Similarity searching for step 2 has been discussed by Downs and Willett characteristics of various similarity measures have been discussed by Barnard, Downs, and Willett. " For step 4, little has been published specifically about visualization and analysis of results for chemical data sets. Flowever, most publications that focus on implementing systems that utilize clustering do provide details of how the results were displayed or analyzed. 

P. G. Mezey, Density Domain Bonding Topology and Molecular Similarity Measures, in Topics in Current Chemistry Molecular Similarity (K. Sen, ed.), Vol. 173, pp. 63-83. Springer-Verlag, Berlin, 1995. 

We have seen that a simple list of Betti numbers of the shape groups can serve as a numerical shape code for a partitioned molecular surface. Some of the alternative topological shape de.scriptors of molecular surfaces, such as the shape matrices s(a,b) and shape graphs g(a,b), can also serve as 3D topological shape codes 143,109,110,158,199]. In Chapter 6, several examples of shape codes are described and used as numerical shape similarity measures. 

Pictures of high resolution appear crisp, whereas pictures of low resolution appear fuzzy. A decrease of resolution is accompanied by an increase of fuzziness. Consequently, similarity measures based on the minimum level of resolution required to distinguish objects can be formulated in terms of the maximum level of fuzziness at which the objects are distinguishable. Similarity can be regarded as fuzzy equivalence. This principle provides an alternative mathematical basis for using the methods of topological resolution [262] in similarity analysis the theory of fuzzy sets [382-385]. 

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