The degree of similarity of three-dimensional bodies: Application to molecular shape analysis

Similarity of shape features of three-dimensional bodies is of importance in many fields. Computational methods that are suitable to provide numerical measures for such similarities are expected to find applications in a wide variety of areas. Whereas relative measures based on direct pair comparisons are useful, nevertheless, methods that involve absolute shape descriptors are expected to be more universally applicable. The general “grade of similarity” concept proposed in this study is based on such absolute shape descriptors of three-dimensional bodies. The study of similarity of the three-dimensional shapes of molecules as represented, for example, by their electronic charge distributions, or electrostatic potentials, or simply by their fused spheres Van der Waals surfaces, is an important component of modem drug design. A family of topological methods, the shape group methods (SGM), have been proposed recently for the study of the shapes of formal molecular bodies, evaluating and comparing numerical shape codes for the non-visual comparison of molecules by the computer. In this contribution a new, and conceptually simpler numerical measure of shape similarity is proposed, applicable for the computer evaluation of similarity of arbitrary three-dimensional objects of closed surfaces. The technique is suggested for the non-visual, numerical evaluation of shape similarity of formal molecular bodies and contour surfaces.

[1]  F. Harary,et al.  Graphical shapes: Seeing graphs of chemical curves and molecular surfaces , 1988 .

[2]  Paul G. Mezey,et al.  The shape of molecular charge distributions: Group theory without symmetry , 1987 .

[3]  Paul G. Mezey,et al.  Shape group studies of molecular similarity: Shape groups and shape graphs of molecular contour surfaces , 1988 .

[4]  Paul G. Mezey,et al.  Global and local relative convexity and oriented relative convexity; application to molecular shapes in external fields , 1988 .

[5]  Frank Harary,et al.  Similarity and complexity of the shapes of square-cell configurations , 1991 .

[6]  Gabor A. Somorjai,et al.  Modern concepts in surface science and heterogeneous catalysis , 1990 .

[7]  Gustavo A. Arteca,et al.  Shape group theory of van der Waals surfaces , 1989 .

[8]  P. Mezey.  A global approach to molecular chirality , 1991 .

[9]  Gustavo A. Arteca,et al.  Molecular similarity and molecular shape changes along reaction paths: a topological analysis and consequences on the Hammond postulate , 1989 .

[10]  J. Thorpe,et al.  Lecture Notes on Elementary Topology and Geometry. , 1967 .

[11]  Gustavo A. Arteca,et al.  Molecular conformations and molecular shape: A discrete characterization of continua of van der Waals surfaces , 1988 .

[12]  R. Ho.  Algebraic Topology , 2022 .

[13]  P. Mezey,et al.  Discrete characterization of cross-sections of molecular surfaces , 1989 .

[14]  Comparison of potential energy maps and molecular shape invariance maps for two-dimensional conformational problems , 1990 .

[15]  Frank Harary,et al.  Graph Theory , 2016 .

[16]  Frank Harary,et al.  Spanning subgraphs of a hypercube iii: meshes , 1988 .

[17]  P. Mezey.  New developments in molecular chirality , 1991 .

[18]  Paul G. Mezey,et al.  Potential Energy Hypersurfaces , 1987 .