Shape group analysis

Walker, P. D., Maggiora, G. M., Johnson, M. A., Petke, J. D., and Mezey, P. G. (1995) Shape group-analysis of molecular similarity—Shape similarity of 6-membered aromatic ring-systems../. Chem. Inf. Comput. Sci. 35, 568-578. 

Skinny molecular range, [af, a< ) af is defined above, whereas is the maximum threshold at and below which all locally nonconvex domains on the surface of density domains are simply connected. In simpler terms, in the skinny molecular range all nuclei are found within a single density domain, but there are formal "neck regions on the surface of density domains. In the terminology of shape group analysis [2], rings of D) type can be found on the surface of density domains. 

The results of the shape group analysis can be summarized using the Betti numbers. The distribution of various values of Betti numbers bPp(a,b) as a... 

The Shape Group analysis can be carried out for all these regional electron densities using both the non-interacting and the interacting RIDCO formalisms. 

For example, consider a given electron density threshold and reference curvature parameter pair (a,b), and assume that the associated truncated contour surface falls into seven pieces. Furthermore, assume that a shape group analysis gives the sequence... 

The main tool for a systematic, topological shape and similarity analysis of molecules is the shape group analysis of molecular electron density clouds [13-25]. The shape group methods are not restricted to molecular electron densities however, in the present context, we shall phrase our brief review of these techniques in terms of electron densities. 

In terms of a shape group analysis on these densities and the computed shape similarities, the following shape similarities, s(pA, Pa(ab>) and s(pB, Pb(ab)), appear to have special importance. 

Shape Group Analysis of Surfaces and Related Techniques... 

The key notion in shape group analysis is evaluation of topological invariants not only for the original surface, but also for a family of surfaces derived by using geometrical properties of the initial surface. Conceptually, this is the same approach discussed in the preceding section for the derivation of a family of knot polynomials from a given molecular space curve. 

G. A. Arteca and P. G. Mezey, Chem. Phys., 161, 1 (1992). Deformation of Electron Densities in Static External Fields Shape Group Analysis for Small Molecules. 

An important concern is the efficient detection of local shape changes introduced by chemical changes in remote locations of a molecule. One simple approach [20] applied a truncation method, compatible with the truncation process already used within the shape group methods for molecular shape analysis [41-44]. 

For both types of FIDCO surfaces, the usual Shape Group method [2] of electron density shape analysis is applicable. The additional formal domain boundaries AD i(Ga b(3)) and AD i(GA(B)(a)) introduce one additional index -1, which can be treated the same way as relative curvature indices. The one-dimensional homology groups obtained by truncations using all possible index combinations are the shape groups of FIDCO surfaces. The (a,b)-parameter maps and shape codes are generated the same way as for complete molecules [2],... 

For a shape complementarity analysis of functional groups we shall follow the shape complementarity approach described for molecules in ref. [2],... 

This generalization of the Centrally Inverted Map Method (CIMM) of molecular shape complementarity analysis [2] to FIDCOs of functional groups replaces the problem of complementarity evaluation with a conceptually and computationally simpler similarity evaluation. 

The standard Shape Group Method is applicable for the analysis of the entire series of non-interacting RIDCOs, for a whole range of density thresholds a, with the provision of an additional domain type representing the connection of region R to the rest of the molecule within the actual RM system. This additional domain type D i is defined as... 

The Shape Group Method (SGM) for the Analysis of Molecular Shapes... 

The nondifferentiability of these surfaces at the seams of interpenetrating spheres as well as the local nondifferentiability of solvent accessible surfaces or union surfaces, are a technical disadvantage. Local nondifferentiability limits the application of the shape group methods in their original form that requires second derivatives for curvature analysis. For example, at every point r of a VDWS where two or more atomic spheres interpenetrate one another, the surface is not smooth and is not differentiable. For such nondifferentiable molecular surfaces, alternative shape descriptors and shape codes have been introduced. 

This Centrally Inverted Map Method (CIMM) of molecular shape complementarity analysis allows one to use the techniques of similarity measures. In fact, the problem of shape complementarity is converted into a problem of similarity between the original (a,b) parameter map of shape groups HP (a,b) of molecule M] and the centrally inverted (a,b) parameter map of the complementary HP2-ii(a,b) shape groups of molecule M2. 

In multiple shape comparisons, efficient, algorithmic shape analysis methods are of particular importance. The shape group and shape code methods provide a framework for such analysis however, the input information they require, such as the 3D electron densities or electrostatic potentials often involve time consuming calculations. This is the case for large molecules or molecular systems, important in drug design and molecular engineering applications. Efficient calculation and representation of these molecular functions is of special importance in such cases. 

Isopotential contours of the composite nuclear potentials (NUPCO s, see Chapter 4), provide an inexpensive, approximate shape representation that can be computed easily even for very large molecules. Although NUPCO s only approximate the MIDCO s of molecules, the family of NUPCO s of a molecule describes an important molecular property that has a major effect on the actual molecular shape. Consequently, NUPCO s can be used for direct comparisons between molecules, and similar NUPCO s are likely to be associated with similar molecular shapes. All the shape analysis techniques originally developed for MIDCO s are equally applicable to NUPCO s. The shape groups, the (a,b) parameter maps [where a is the nuclear potential threshold of a NUPCO G(a)], the shape matrices, shape codes, and the shape globe invariance maps of NUPCO s of molecules can serve as inexpensive methods for the detection and evaluation of a particular aspect of molecular similarity. 

The QShAR (Quantitative Shape-Activity Relations) method, combined with the integrated main and side effect modeling of bioactive molecules, forms the conceptual basis of the approaches described in this chapter. The density scalable FSGH method for a simple representation of molecular bodies, in combination with the Shape Group Method and various other shape code approaches for quantitative shape analysis, as well as the multiple shape ranking methods for integrated main and side effect analysis, are the components of a computational implementation of the basic concepts. 

The integrated main effect and multiple side-effect analysis by multiple shape ranking of bioactive molecules, based on the density scalable FSGH and Shape Group Methods, are suitable for shape analysis of large molecules. 

Most symmorphy groups hp are rather complicated and their direct use for molecular shape characterization and shape similarity analysis is not a trivial task. Some simplifications are possible using a technique based on the Brouwer fixed point theorem, as described in reference [43]. 

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