Shape similarity measure

Shape Similarity Measures of Functional Groups Based on Local Shapes... 

Shape similarity between two molecules or molecular regions A and B can be expressed using the following shape-similarity measure ... 

SHAPE SIMILARITY MEASURES AND DISSIMILARITY MEASURES IN THE STUDY OF HOST-GUEST INTERACTIONS... 

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. 

By contrast, relative shape analysis and relative shape descriptors can change for each molecule, depending on the other molecule used for comparison. For n molecules there are n(n-l)/2 molecule pairs, hence n(n-l)/2 families of relative shape descriptors of the given type. Consequently, in the study of shape similarities in large molecular families, the quadratic dependence of the number of relative shape descriptors on the number of molecules is a disadvantage and the use of relative shape descriptors is often impractical. Shape similarity measures based on relative shape descriptors are called similarity measures of the second kind. 

A general family of shape similarity measures of the second kind is based on the principle of rescaling the size of one object until it fits within the other object. The degree of scaling required gives a similarity measure. 

For the simpler case of molecule pair interactions, such as the interactions between two reactants, local shape complementarity is of importance. The basic principle of local shape similarity measures is also applicable for the construction of local shape complementarity measures. 

Tlie main assumption of this approach is that the shape of the molecule is closely related to the shape of the - binding site cavity and, as a consequence, to the biological activity. Therefore, a shape reference compound is chosen which represents the binding site cavity, and the similarity (or commonality) measured between the reference shape and the shape of other compounds is used to determine the biological activity of these compounds. As well as the shape similarity measures, other molecular descriptors such as those in - Hansch analysis can be used to evaluate the biological response. The MSA model is thus defined as ... 

Mezey, P.G. (1992). Shape-Similarity Measures for Molecular Bodies A Three-Dimensional Topological Approach to Quantitative Shape-Activity Relations. J.Chem.lnf.Comput.Sci., 32, 650-656. 

Hence while the topological shape analysis and shape similarity measures do not truly represent a Non-Quantum Similarity approach, nevertheless, in practical terms, they offer a powerful alternative. 

If, for the ranges [0.001,0.1 a.u.] for the density threshold values a and [—1.0,1.0] for the reference curvature b, a rather natural 41 x 21 grid is used, justified by the logarithmic scales, then the elements of the 41x21 matrix can also be stored as an integer vector C of 861 components. Using this alternative, the shape similarity measure x(A,B) can also be expressed as... 

The nonvisual shape similarity measures of molecules as well as molecular fragments, using the numerical shape code method, provide the basis for a shape complementarity measure. A simple transformation of the local shape codes generates a representation that is suitable for a direct evaluation of local shape complementarity. 

The so-called distance matrix error is a shape similarity measure based on rms deviations.220-222 jn this case, the distance between two configurations a and b is defined as ... 

With respect to a reference shape T, the T-hull of a molecular body M is computed as the intersection of all rotated and translated versions of T, which contains M. T-hulls provide shape similarity measures for molecules and for their solvent accessible surfaces. T-hulls are also the basis for a shape quantization . ... 

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