Skip to main content
SpringerLink
Log in

Combinatorics of combinatorial chemistry

  • Published:
Journal of Mathematical Chemistry Aims and scope Submit manuscript

Abstract

The hot topic among medicinal chemists today is a novel technique for chemical synthesis in drug research called combinatorial chemistry, where usually a core structure and some building‐block molecules are given and all combinatorially possible combinations are produced. The resulting set of compounds (called a library) can afterwards be systematically screened for a desired biological activity. In this paper we discuss the applications of the mathematical discipline of combinatorics to this process, especially an algorithm for the exhaustive and redundancy‐free generation of a combinatorial library as well as equations for the enumeration of library sizes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S.C. Basak, Use of molecular complexity indices in predictive pharmacology and toxicology: a QSAR approach, Med. Sci. Res. 15 (1987) 605–609.

    CAS  Google Scholar 

  2. S.C. Basak, S. Bertelsen and G.D. Grunwald, Application of graph-theoretical parameters in quantifying molecular similarity and structure-activity studies, J. Chem. Inf. Comput. Sci. 34 (1994) 270–276.

    Article  CAS  Google Scholar 

  3. C. Benecke, R. Grund, R. Hohberger, A. Kerber, R. Laue and T. Wieland, MOLGEN+, a generator of connectivity isomers and stereoisomers for molecular structure elucidation, Anal. Chim. Act. 314 (1995) 141–147.

    Article  CAS  Google Scholar 

  4. C. Benecke, R. Grund, R. Hohberger, A. Kerber, R. Laue and T. Wieland, Chemical isomerism, a challenge for algebraic combinatorics and for computer science, in: Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, eds. C. Cohen, M. Giusti and T. Mora, Lect. Not. Comput. Sci. (Springer, New York, 1995).

    Google Scholar 

  5. N.L. Biggs, K.E. Lloyd and R.J. Wilson, Graph Theory 1736–1936 (Clarendon Press, 1977).

  6. T. Carell, E.A. Wintner, A.J. Sutherland, J. Rebek, Jr., Y.M. Dunayevskiy and P. Vouros, New promise in combinatorial chemistry: synthesis, characterization, and screening of small-molecule libraries in solution, Chem. & Biol. 2 (1995) 171–183.

    Article  CAS  Google Scholar 

  7. R.E. Carhart, D.H. Smith, H. Brown and C. Djerassi, An approach to computer-assisted elucidation of molecular structure, J. Am. Chem. Soc. 97 (1975) 5755–5762.

    Article  CAS  Google Scholar 

  8. R.E. Carhart, D.H. Smith, N.A. Gray, J.G. Nourse and C. Djerassi, GENOA: a computer program for structure elucidation utilizing overlapping and alternative substructures, J. Org. Chem. 46 (1981) 1708–1718.

    Article  CAS  Google Scholar 

  9. A. Cayley, On the mathematical theory of isomers, Philosoph. Magaz. 74(4) (1874) 444–446.

    Google Scholar 

  10. J. Dugundji and I.K. Ugi, An algebraic model of contitutional chemistry as a basis for chemical computer programs, Top. Curr. Chem. 39 (1973) 19–64.

    CAS  Google Scholar 

  11. K. Funatsu, N. Miyabayaski and S. Sasaki, Further development of structure generation in the automated structure elucidation system CHEMICS, J. Chem. Inf. Comput. Sci. 28 (1988) 18–28.

    Article  CAS  Google Scholar 

  12. M.A. Gallop, R.W. Barrett, W.J. Dower, S.P.A. Fodor and E.M. Gordon, Applications of combinatorial technologies to drug discovery. 1. Background and peptide combinatorial libraries, J. Med. Chem. 37 (1994) 1233–1251.

    Article  CAS  Google Scholar 

  13. E.M. Gordon, R.W. Barrett, W.J. Dower, S.P.A. Fodor and M.A. Gallop, Applications of combinatorial technologies to drug discovery. 2. Combinatorial organic synthesis, library screening strategies, and future directions, J. Med. Chem. 37 (1994) 1385–1401.

    Article  CAS  Google Scholar 

  14. R. Grund, A. Kerber and R. Laue, MOLGEN, ein Computeralgebra-System für die Konstruktion molekularer Graphen, MATCH 27 (1992) 87–131.

    Google Scholar 

  15. R. Grund, A. Kerber and R. Laue, Construction of discrete structures, especially isomers, Discr. Appl. Math. 67 (1996) 115–126.

    Article  Google Scholar 

  16. T. Grüner, R. Laue and M. Meringer, Applications for group actions applied to graph generation, in: DIMACS Series in Discrete Mathematics and Theoretical Computer Science, eds. L. Finkelstein, and C. Kantor (Providence, RI, 1995) (in press).

  17. A. Kerber, Algebraic Combinatorics Via Finite Group Actions (BI-Wissenschaftsverlag, Mannheim, Wien, Zürich, 1991).

    Google Scholar 

  18. A. Kerber, ed., Generatoren für molekulare Graphen, Nomenklaturfragen, MATCH 27 (1992).

  19. L.B. Kier and L.H. Hall, Molecular Connectivity in Structure-Activity Analysis (Research Studies Press, Chichester, 1986).

    Google Scholar 

  20. H. Kubinyi, ed., 3D QSAR in Drug Design. Theory, Methods and Applications (ESCOM Science Publishers, Leiden, 1993).

  21. H. Kubinyi, Der Schlüssel zum Schloß. II. Hansch-Analyse, 3D-QSAR und De novo-esign, Pharmazie in unserer Zeit 23(5) (1994) 281–290.

    CAS  Google Scholar 

  22. R. Laue, Construction of combinatorial objects – a tutorial, Bayreuther Mathem. Schr. 43 (1993) 53–96.

    Google Scholar 

  23. E.J. Martin, J.M. Blaney, M.A. Siani, D.C. Spellmeyer, A.K. Wong and W.H. Moos, Measuring diversity: experimental design of combinatorial libraries for drug discovery, J. Med. Chem. 38 (1995) 1431–1436.

    Article  CAS  Google Scholar 

  24. L.M. Masinter, N.S. Sridharan, J. Lederberg and D.H. Smith, Applications of artificial intelligence for chemical inference. XII. Exhaustive generation of cyclic and acyclic isomers, J. Am. Chem. Soc. 96 (1974) 7702–7714.

    Article  CAS  Google Scholar 

  25. M.R. Pavia, The chemical generation of molecular diversity, Network Science (August 1995).

  26. G. Pólya, Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen, Acta mathematica 68 (1937) 145–253.

    Google Scholar 

  27. G. Pólya and R.C. Read, Combinatorial Enumeration of Groups, Graphs, and Chemical Compounds (Springer, Berlin, 1987).

    Google Scholar 

  28. R.C. Read, Every-one a winner, Ann. Discr. Math. 2 (1978) 107–120.

    Article  Google Scholar 

  29. H.J. Redfield, The theory of group-reduced distributions, Am. J. Math. 49 (1927) 433-455.

    Google Scholar 

  30. C.A. Shelley, Heuristic approach for displaying chemical structures, J. Chem. Inf. Comput. Sci. 23 (1983) 61–65.

    Article  CAS  Google Scholar 

  31. R.P. Sheridan and S.K. Kearsley, Using a genetic algorithm to suggest combinatorial libraries, J. Chem. Inf. Comput. Sci. 35 (1995) 310–320.

    Article  CAS  Google Scholar 

  32. I. Ugi, Fast and permanent changes in preparative and pharmaceutical chemistry through multicomponent reactions and their ‘libraries’, Proc. Eston. Acad. Sci. Chem. 44 (1995) 237–273.

    CAS  Google Scholar 

  33. T. Wieland, Konstruktionsalgorithmen bei molekularen Graphen und deren Anwendung, Ph.D. thesis, University of Bayreuth (1996).

  34. T. Wieland, The use of structure generators in predictive pharmacology and toxicology, Arzneim.-Forsch./Drug Res. 46(I) (1996) 223–227.

    CAS  Google Scholar 

  35. T. Wieland, Mathematical simulations in combinatorial chemistry, MATCH 34 (1996) 179–206.

    CAS  Google Scholar 

  36. T. Wieland, A. Kerber and R. Laue, Principles of the generation of constitutional and configurational isomers, J. Chem. Inf. Comput. Sci. 36 (1996) 413–419.

    Article  CAS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Lehrstuhl II für Mathematik, University of Bayreuth, 95440, Bayreuth, Germany

    Thomas Wieland

Authors
  1. Thomas Wieland
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wieland, T. Combinatorics of combinatorial chemistry. Journal of Mathematical Chemistry 21, 141–157 (1997). https://doi.org/10.1023/A:1019166201637

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1019166201637

Keywords

Search

Navigation