The argent program system: A second-generation tool aimed at combinatorial search for new types of organic reactions. 2. Mathematical models in ARGENT-1

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Volumn , Issue 46, 2002, Pages 275-301

  Tratch, Serge S ^a   Molchanova, Marina S ^b   Zefirov, Nikolai S ^a  

^a MOSCOW STATE UNIVERSITY   (Russian Federation)

^b N D ZELINSKY INSTITUTE OF ORGANIC CHEMISTRY   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

ALGORITHM; ARTICLE; CHEMICAL REACTION; COMPUTER PROGRAM; EXPERIMENTAL MODEL; MATHEMATICAL ANALYSIS;

EID: 0036459268     PISSN: 03406253     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (58)

1. Zefirov, N.S.; Tratch, S.S.; Molchanova, M.S. Previous paper in this issue.
2. 3c by one of the authors as the lecture course at the 11th Dubrovnik International Conference.
3. (a) Balasubramanian, K. Theor. Chim. Acta 1988, 74, 111-122.
4. (b) Faradgev, I.A. In Algorithmic Investigations in Combinatorics; Nauka Press: Moscow, 1978; pp 3-11.
5. (c) Tratch, S.S. In MATH/CHEM/COMP'96, Book of Abstracts; Inter-University Centre: Dubrovnik, 1996.
6. (a) Masinter, L.M.; Sridharan, N.S.; et al. J. Am. Chem. Soc. 1974, 96, 7714-7723.
7. (b) Masinter, L.M.; Sridharan, N.S.; et al. Ibid.; pp 7702-7714.
8. Tratch, S.S.; Zefirov, N.S. J. Chem. Inf. Comput. Sci. 1998, 38, 331-348.
9. (a) Harary, F. Graph Theory; Addison-Wesley: Reading, MA, 1969.
10. (b) Harary, F.; Palmer, E.M. Graphical Enumeration; Academic Press: New York, 1973.
11. 2h; the latter point symmetry group consists of all 8 operations in the 3D space that convert the planar embedding of the graph into itself.
12. Permutations from the automorphism group or other groups are commonly represented in the cyclic notation (e.g., see Fig. 1a). Cycles are disjoint subsets of the set of permuted elements; their union is the whole set. In any particular cycle of some permutation, each element but the last one moves to the next element of this cycle, and the last element moves to the first one. For example, the permutation (1)(2)(3, 4)(5)(6) of the vertex set means that vertex 3 moves to vertex 4 and vice versa; these two vertices form a cycle of length 2. A cycle of length 1 indicates that the corresponding vertex (1, 2, 5, or 6 in this example) remains unmoved. Cycles of length 1 are often omitted from the notation for the sake of brevity.
13. 9b on the theory and applications of such structures was recently published.
14. (b) Kerber, A. Algebraic Combinatorics Via Finite Group Actions. BI-Wiss.-Verl.: Mannheim, 1991.
15. 6a
16. 6 that the length of orbit Y for any permutation group A - in other words, the number of elements in the corresponding equivalence class - is equal to the index of the subgroup A(y) that stabilizes (moves to itself) some element y ∈ Y: Y = A/A(y). That is why the number of labeled graphs in each equivalence class considered here is equal to the order of the "large" group (which corresponds to some starting graph) divided by the order of the "small" automorphism group (i.e., the group of the resultant vertex- or edge-labeled graph).
17. (a) Some algebraists (e.g., see ref 12b) distinguish between usual permutation groups and actions of abstract groups on appropriate sets; the latter notion is applied in situations where two or more copies of the same group permute elements of the same set.
18. (b) Kaluzhnin, L.A.; Sushtchanski, V.I.; Ustimenko, V.A. Kibernetika 1982, 83-94.
19. 13b we found that three main types of degeneracy are theoretically possible: (1) the degeneracy that appears due to the presence of (-)-automorphisms in the expanded groups of edge-labeled graphs (regular degeneracy); (2) the degeneracy that is caused by some symmetries of graph G but cannot be associated with any (-)-automorphism (semiregular degeneracy); and (3) the degeneracy that is completely independent of the symmetry properties of the topology identifying graph G (irregular degeneracy). All results discussed in this paper, as well as all degenerate chemical interconversions (isomerizations and intermolecular processes) actually investigated by organic chemists, refer only to regular degeneracy; that is why the adjective "regular" is not explicitly used in the text. The present version of the ARGENT-1 program makes it possible to perform specialized search for chemically feasible examples of two other, still unprecedented types of degenerate interconversions. Further, we are planning to publish a separate series of papers on the theoretical investigation of degeneracy and the actual results obtained.
20. (b) Tratch, S.S.; Molchanova, M.S.; Zefirov, N.S. In Molecular Modeling (Proc. 2nd All-Russian Conf.); Moscow, 2001; p 18.
21. This intuitively clear fact may be briefly explained as follows: (+)-automorphisms of any labeled graph are its isomorphisms onto itself, and (-)-automorphisms are its isomorphisms onto its unique antipode (in other words, onto a labeled graph with all paired labels substituted by opposite ones). Thus, the number of (-)-automorphisms is either zero (if the antipode labeling is non-isomorphic to the original one) or coincides with the number of (+)-automorphisms (in the other case).
22. 15b This conclusion stems from the fact that (-)-automorphisms (responsible for the regular degeneracy) may be considered as analogs of improper symmetry operations. The corresponding combinatorial chirality criteria and their applications to classification of chiral molecules are discussed in refs 15c,d.
23. (b) Tratch, S.S.; Zefirov, N.S. In Molecular Modeling (Proc. 1st All-Russian Conf.); Moscow, 1998; p U2.
24. (c) Tratch, S.S. Zh. Org. Khimii 1995, 31, 1320-1351.
25. (d) Tratch, S.S.; Zefirov, N.S. J. Chem. Inf. Comput. Sci. 1996, 36, 448-464.
26. 16b The representation of this process depends on the choice of the resonance structure for the cyanomethide anion: the reaction equation of Fig. 4f and that of Chart 11f in ref 16c can both be used for describing this interesting process.
27. (b) Zefirov, N.S.; Chapovskaya, N.K.; et al. Zh. Org. Khimii 1975, 11, 1981.
28. (c) Tratch, S.S.; Zefirov, N.S. J. Chem. Inf. Comput. Sci. 1998, 38, 349-366.
29. X of all functions (or mappings) f = X → Y from set X into set Y.
30. (b) Harary, F.; Palmer, E. J. Combin. Theory 1966, 1, 157-173.
31. 2.
32. These seven labels are regarded as the "main" atom labels in ARGENT-1 because they cover the whole range of maximal valences from one to seven; surely, the user may extend this set if needed.
33. (a) Tratch, S.S.; Zefirov, N.S. In Principles of Symmetry and Systemology in Chemistry; Moscow University Press: Moscow, 1987; pp 54-86.
34. (b) Tratch, S.S. Doctoral Dissertation; Moscow, 1993; Vol. 2, pp 56-144.
35. (a) Balaban, A.T. Rev. Roum. Chim. 1986, 31, 795-810.
36. (b) Tratch, S.S.; Lomova, O.A.; et al. J. Chem. Inf. Comput. Sci. 1992, 32, 130-139.
37. One-stage labeling problems are considered here only for the sake of brevity; surely, both starting graphs in Fig. 6a can themselves be produced via a two-stage procedure (skeleton graphs corresponding to benzene isomers and their diazaanalogs can be constructed at the first and second labeling stages, respectively). Note that the extremely strained and probably very unstable diazaprismane structure was chosen here only for convenience, so that the symmetry of both graphs in Fig. 6a would be characterized by the same permutation group.
38. 20b,24b); some other applications are briefly mentioned in refs 24c,d.
39. (a) Tratch, S.S.; Devdariani, R.O.; Zefirov, N.S. Zh. Org. Khimii 1990, 26, 921-932.
40. (b) Zefirov, N.S.; Kozhushkov, S.I.; et al. J. Am. Chem. Soc. 1990, 112, 7702-7707.
41. (c) Tratch, S.S. In MATH/CHEM/COMP'96, Book of Abstracts; Inter-University Centre: Dubrovnik, 1996.
42. (d) Zefirov, N.S.; Tratch, S.S. J. Chem. Inf. Comput. Sci. 1997, 37, 900-912.
43. (a) Lunn, A.C.; Senior, J.K. J. Phys. Chem. 1929, 33, 1027-1079.
44. (b) Pólya, G. Acta Math. 1937, 68, 145-254.
45. (c) Iliev, V.V. MATCH 1999, 40, 153-186.
46. Although the permutations presented in Fig. 6a actually permute free valences rather then vertices of both skeleton graphs, they uniquely correspond to vertex automorphisms of the graphs under consideration. For the correspondence between graph symmetries and spatial symmetry operations pertaining to planar embeddings of the graphs in the 3D space, see note 7.
47. More complicated labeling problems (e.g., involving one or more unpaired and an even number of paired labels) are based on similar mathematical models; the examples are to be explicitly considered in one of the forthcoming publications in this series. It is important that "saturation" of free valences by appropriate substituents means just the same as replacement of H in the pyrazine or diazaprismane structure by some unpaired or paired label. Note that H itself, if present, must be considered as one of unpaired labels.
48. 28d,e consideration of both theories lies outside the scope of this paper.
49. (b) Häselbarth, W.; Ruch, E. Isr. J. Chem. 1976/77, 15, 112-115.
50. (c) Ruch, E.; Klein, D.J. Theor. Chim. Acta 1983, 63, 447-472.
51. (d) Fujita, S. J. Math. Chem. 1990, 5, 121-156.
52. (e) Fujita, S. Symmetry and Combinatorial Enumeration in Chemistry. Springer-Verl.: Berlin, 1991.
53. 3)O-) homological modules have not yet been analyzed in literature.
54. (b) Kerber, A. MATCH 1975, 1, 5-10.
55. (c) Tratch, S.S. Unpublished results.
56. (a) Tratch, S.S.; Molchanova, M.S.; Zefirov, N.S. Contribution 3 of this series. To be submitted.
57. (b) Tratch, S.S.; Molchanova, M.S.; Zefirov, N.S. Contribution 4 of this series. To be submitted.
58. (c) Molchanova, M.S.; Tratch, S.S.; Zefirov, N.S. Contributions 5 and 6. In preparation.