Some notes on Randić-Razinger's approach to characterization of molecular shapes

Journal of Chemical Information and Computer Sciences

Volumn 37, Issue 5, 1997, Pages 900-912

  Zefirov, Nikolai S ^a   Tratch, Serge S ^a  

^a MOSCOW STATE UNIVERSITY   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0001364262     PISSN: 00952338     EISSN: None     Source Type: Journal    
DOI: 10.1021/ci9700182     Document Type: Article

References (54)

1. Randić, M.; Razinger, M. Molecular Topographic Indices. J. Chem. Inf. Comput. Sci. 1995, 35, 140-147.
2. Randić, M. Molecular Shape Profiles. J. Chem. Inf. Comput. Sci. 1995, 35, 373-382.
3. Randić, M.; Razinger, M. On Characterization of Molecular Shapes. J. Chem. Inf. Comput. Sci. 1995, 35, 594-606.
4. Randić, M.; Razinger, M. Molecular Shapes and Chirality. J. Chem. Inf. Comput. Sci. 1996, 36, 429-441.
5. Mezey, P. G. Shape in Chemistry: An Introduction to Molecular Shape and Topology; VCH Publ.: New York, 1993. The broader list of references on results of P. G. Mezey's group may be found in refs 2-4.
6. Strictly speaking, neglecting hydrogens and substituents means that molecular periphery (and hence molecular shape, its codes, invariants, etc.) does not characterize real planar molecules but relates to planar embeddings of corresponding hydrogen-depleted, or skeleton, or framework graphs. In the case of polybenzenoidal hydrocarbons, these three sorts of graphs coincide but in the general case they can be differentiated, see sections 7 and 9 in ref 18a.
7. Balaban, A. T. Chemical Graphs. Part 12. Configuration of Annulenes. Tetrahedron 1971, 27, 6115-6131.
8. 9d may serve as examples.
9. (a) Balaban, A. T.; Harary, F. Chemical Graphs. Part 5. Enumeration and Proposed Nomenclature of Benzenoid Cata-Condensed Polycyclic Aromatic Hydrocarbons. Tetrahedron 1968, 24, 2505-2516.
10. (b) Mgel'skaya, E. V. Graphs of Polycyclic Compounds. In Problems of Algorithmic Analysis for Structural Information (Computational Systems, 119); Novosibirsk, 1987; pp 71-90 [in Russian].
11. (c) Knop, J. V.; Szymanski, K.; Jeričević, Z.; Trinajstić, N. Computer Enumeration and Generation of Benzenoid Hydrocarbons and Identification of Bay Regions. J. Comput. Chem. 1983, 4, 23-32.
12. (d) Dobrynin, A. A. An Effective Generation Algorithm for Graphs of Unbranched Hexagonal Systems. In Mathematical Problems of Chemical Informatics (Computational Systems, 130); Novosibirsk, 1989; pp 3-38 [in Russian].
13. d, one or several reflection lines correspond; just these "mirror lines" must be absent in two-dimensionally chiral planar molecules.
14. ij denotes graph-theoretical distance between atoms i and j in a molecular graph of order n.
15. (b) Hosoya, H. Topological Index. A Newly Proposed Quantity Characterizing the Topological Nature of Structure Isomers of Saturated Hydrocarbons. Bull. Chem. Soc. Jpn. 1971, 44, 2332-2339.
16. (c) Wiener, H. Structure Determination of Paraffin Boiling Points. J. Am. Chem. Soc. 1947, 69, 17-20.
17. 5 are based on a somewhat fuzzy concept of molecular surface which, in turn, significantly depends on the assumed threshold for electron density. For this reason, "continuous" approaches are less properly adapted for organic chemists which traditionally operate with molecular graphs and their embeddings in 2D or 3D space.
18. In more rigorous mathematical terminology, this polygonal figure is a convex hull of the planar n-point system. Similarly, in the case of nonplanar n-point system in 3D space, its convex hull is represented by m-point (4 ≤ m ≤ n) convex polyhedral figure.
19. 3 of planar as well as spatial molecular shapes.
20. ij are real distances between hydrogen atoms divided by 1.54 (i.e., by the length of normal C-C distance). All H⋯H distances in the staggered ethane rotamer are, in turn, calculated from coordinate values represented in T. Clark's Handbook of Computational Chemistry; John Wiley & Sons,Inc.; New York, 1985.
21. (a) Tratch, S. S. Mathematical Models in Stereochemistry. I. Combinatorial Characteristics of Composition, Connectivity, and Configuration of Organic Molecules. Zh. Organ. Khim. 1995, 31, 1320-1351 [in Russian].
22. (b) Tratch, S. S.; Zefirov, N. S. Combinatorial Models and Algorithms in Chemistry. Ladder of Combinatorial Objects and Its Application to Formalization of Structural Problems of Organic Chemistry. In Principles of Symmetry and Systemology in Chemistry; Stepanov, N. F., Ed.; Moscow State University Press: Moscow, 1987; pp 54-86 [in Russian].
23. (a) Tratch, S. S. Combinatorial Approach to Stereochemical Problems: A Brief Survey. Abstracts of the Sixth International Conference on Mathematical Chemistry; Pitlochry, Scotland, July 10-14, 1995.
24. (b) Tratch, S. S. Combinatorial Objects in Chemistry: Formal Models, Enumeration Techniques, and Generation Algorithms. Abstracts of the Eleventh International Course & Conference on the Interfaces Among Mathematics, Chemistry, and Computer Sciences; Dubrovnik, Croatia, June 24-29, 1996.
25. (a) Tratch, S. S.; Zefirov, N. S. Algebraic Chirality Criteria and Their Application to Chirality Classification in Rigid Molecular Systems. J. Chem. Inf. Comput. Sci. 1996, 36, 448-464.
26. p). Rep. Mol. Theory 1990, 1, 149-163.
27. (a) Zefirov, N. S.; Kozhushkov, S. I.; Kuznetsova, T. S.; Kokoreva, O. V.; Lukin, K. A.; Ugrak, B. I.; Tratch, S. S. Triangulanes: Stereoisomerism and General Method of Synthesis. J. Am. Chem. Soc. 1990, 112, 7702-7707.
28. (b) Tratch, S. S. Logical-Combinatorial Approaches to Design Problems for Organic Structures, Reactions, and Configurations (Doctoral Dissertation Thesis); Moscow, 1993; Vol. 2, pp 56-144 [in Russian].
29. (c) Tratch, S. S.; Devdariani, R. O.; Zefirov, N. S. Combinatorial Models and Algorithms in Chemistry. Configuration-Topological Analogs of Wiener Index. Zh. Organ. Khim. 1990, 26, 921-932 [in Russian].
30. Linear or 1D point configurations are not considered in this paper because relative dispositions of points in 1D space can be trivially characterized by ordered sequences of point numbers.
31. Drozd, V. N.; Zefirov, N. S.; Sokolov, V. I.; Stankevitch, I. V. Topological Definition of the Notion "Stereochemical Configuration". Zh. Organ. Khimii 1979, 15, 1785-1793 [in Russian].
32. All nondirected graphs and multigraphs can also be described by symmetric functions χ from the sets of ordered pairs [1, 2], [2, 1], [1, 3], [3, 1], ... into the set of allowed edge multiplicities; the symmetry property means that χ([i, j]) = χ([j, i]) for all pairs of vertices i and j, i≠ j.
33. 24b was the first in mathematical chemistry who has treated graphs in this way.
34. (a) Faradgev, I. A. Constructive Enumeration of Combinatorial Objects. In Algorithmic Investigations in Combinatorics; Faradgev, I. A., Ed.; Nauka Press: Moscow, 1978; pp 3-11 [in Russian].
35. (b) Kerber, A. On Graphs and Their Enumeration. MATCH 1975, 1, 5-10.
36. 26a states "On the lowest level of description only the composition of molecules with respect to certain types of fragments is recorded. The next level specifies molecular constitution, that is, the bonding connectivity among the fragments." and "On the next level, opening the door to what is called stereo-chemistry, molecules are recognized to live in 3-dimensional space. On this level the description records qualitative geometrical features...". The very similar (and also qualitative) ideas were discussed in refs 26b,c.
37. (a) Hässelbarth, W. A. Generalisation of the Polya/de Bruijn Enumeration Theory and Its Application to "Chemical Combinatorics". MATCH 1986, 20, 241-250.
38. (b) Turro, N. J. Geometric and Topological Thinking in Organic Chemistry. Agnew. Chem., Int. Ed. Engl. 1986, 25, 882-901.
39. (c) Maggiora, G. M.; Johnson, M. A. Introduction to Similarity in Chemistry. In Concepts and Applications of Molecular Similarity; Johnson, M. A.; Maggiora, G. M., Eds.; John Wiley & Sons, Inc.: New York, 1985; pp 1-13.
40. 28b for ordered quadruples related to four-atomic chains.
41. (a) Dreiding, A. S.; Wirth, K. The Multiplex. A Classification of Finite Ordered Point Sets in Oriented d-Dimensional Spaces. MATCH 1980, 8, 341-352.
42. (b) Johnson, M.; Tsai, C.; Nicholson, V. Chemical Stereographs: An Extension of Chemical Graphs for Representing Stereochemical and Conformational Structures. J. Math. Chem. 1991, 7, 3-38.
43. In a more rigorous formulation, all even permutations of i, j, k retain the sign of ψ, while all odd permutations of point numbers convert -1 into +1 and vice versa, just the same situation is observed for ordered pairs [i, j] (related to 1D configurons {i, j) and for ordered quadruples [i, j, k, l] (related to 3D configurons {i, j, k, l}). This fact shows that in all cases ψ is an alternating function, and this is the main difference between functions χ and ψ (χ may be regarded as a symmetric function, cf. note 22).
44. 16a, 18b to be defined by reversed sign of the determinant which corresponds to 3 × 3 (or 4 × 4) matrix with columns related to points i, j, k (and l). The first rows of the both matrices consist of units while the second, third (and fourth) rows are formed out of x-, y- (and z-) coordinates of the point system under consideration.
45. In this and other papers, we use the lexicographic order of triples and quadruples. According to this order (in the case n = 5), the values of ψ in Figure 8b correspond to the following list of 10 triples: [1, 2, 3], [1, 2, 4], [1, 2, 5], [1, 3, 4], [1, 3, 5], [1, 4, 5], [2, 3, 4], [2, 3, 5], [2, 4, 5], [3, 4, 5]. The "reversed" lexicographic order results however in more effective solution of some computational problems and hence is well-adapted for computer representation of functions ψ. According to this order, the same 10 triples form the following sequence: [1, 2, 3], [1, 2, 4], [1, 3, 4], [2, 3, 4], [1, 2, 5], [1, 3, 5], [2, 3, 5], [1, 4, 5], [2, 4, 5], [3, 4, 5].
46. Although to any disposition of numbered points in 2D or 3D space the unique function ψ evidently corresponds, the reverse statement is not true due to existence of geometrically nonrealizable point configurations. For this reason, the general realizability criterion for functions ψ is a crucial point in computer-assisted solution of several generation problems related to different kinds of configurations. At present, we can only assert that to any realizable n-point 2D or 3D configuration, the convex hulls for all m-point, m = 4, 5, ..., n, subconfigurations must necessarily exist.
47. Slightly different definitions of subgraphs and partial (sub)graphs can be found in mathematical literature. In this paper, we follow the terminology having been used in N. Christophide's handbook (Graph Theory: An Algorithmic Approach; Academic Press: New York, 1985).
48. Randić, M. Graphical Enumeration of Conformations of Chains. Int. J. Quant. Chem., Quant. Biol. Symp. 1980, 7, 187-197.
49. (b) Balaban A. T.; Harary, F. Chemical Graphs. V. Enumeration and Proposed Nomenclature of Benzenoid Cata-Condensed Polycyclic Aromatic Hydrocarbons. Tetrahedron 1968, 24, 2505-2516.
50. The codes formed out of zeros and units (or out of integers 0, 1, 2 in ternary case) can be treated as binary (or ternary) numbers to which the decimal equivalents uniquely correspond. On the other hand, the codes consisting of -1 and +1 (and 0 in ternary case) more clearly reflect relative orientations of points in corresponding 2D or 3D configurons. The choice between coding rules is not, however, important because any of possible coding systems can easily be converted into another one.
51. m. The "circulation independent" canonicalization rules seem, however, to be more general in the sense that they do not depend on the choice between two possible "walking directions" and hence can be applied to all 2D as well as 3D ring configurations. Note that in 3D space, clockwise and anticlockwise directions cannot, in principle, be differentiated.
52. We differentiate between geometrical, graphical, and chemical realizability of ("complete" or partial) 2D and 3D configurations. The point (or labeled, or graph, or molecular) configuration is geometrically nonrealizable if there exist no dispositions of points (or graph vertices, or atoms) to which the prescribed values of ψ correspond. The geometrically realizable graph (or molecular) configuration is graphically nonrealizable if in any of its embeddings at least two graph edges necessarily intersect. The less precisely defined notion of chemical realizability relates to graphically realizable molecular configurations; this kind of realizability evidently depends on allowed values of interatomic distances and bond and dihedral angles which can, in turn, be affected by numerous other factors.
53. Tratch, S. S. On Molecular Shapes and Ring 2D Configurations. Abstracts of the Eleventh International Course & Conference on the Interfaces Among Mathematics, Chemistry, and Computer Sciences; Dubrovnik, Croatia, June 24-29, 1996.
54. (b) Tratch, S. S.; Zefirov, N. S. Mathematical Models of Ring Configurations and Their Application to Enumeration and Generation Problems. Manuscript in preparation.