, Volume 13, Issue 1–2, pp 77–105 | Cite as

DNA physical mapping and alternating Eulerian cycles in colored graphs

  • P. A. Pevzner


Small-scale DNA physical mapping (such as the Double Digest Problem or DDP) is an important and difficult problem in computational molecular biology. When enzyme sites are modeled by a random process, the number of solutions to DDP is known to increase exponentially as the length of DNA increases. However, the overwhelming majority of solutions are very similar and can be transformed into each other by simple transformations. Recently, Schmitt and Waterman [SW] introduced equivalence classes on the set of DDP solutions and raised an open problem to completely characterize equivalent physical maps.

We study the combinatorics of multiple solutions and the cassette transformations of Schmitt and Waterman. We demonstrate that the solutions to DDP are closely associated with alternating Eulerian cycles in colored graphs and study order transformations of alternating cycles. We prove that every two alternating Eulerian cycles in a bicolored graph can be transformed into each other by means of order transformations. Using this result we obtain a complete characterization of equivalent physical maps in the Schmitt-Waterman problem. It also allows us to prove Ukkonen's conjecture on word transformations preservingq-gram composition.

Key words

Graph theory DNA mapping DNA sequencing 


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  1. [AK]
    Abrham, J., and Kotzig, A. Transformations of Eulerian tours.Ann. Discrete Math.,8 (1980), 65–69.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [AY]
    Allison, L., and Yee, C. N. Restriction site mapping is in separation theory.CABIOS,4 (1988), 97–101.Google Scholar
  3. [B]
    Bellon, B. Construction of restriction maps.CABIOS,4 (1988), 111–115.Google Scholar
  4. [BMPS]
    Benkouar, A., Manoussakis, Y. G., Paschos, V. T., and Saad R. On the complexity of some Hamiltonian and Eulerian problems in edge-coloured complete graphs. In W. L. Hsu and R. C. T. Lee (eds.),ISA '91 Algorithms. Proceedings of the 2nd International Symposium on Algorithms, Taipei, December 1991. Lecture Notes in Computer Science, Vol. 557. Springer-Verlag, Berlin, 1991, pp. 190–198.Google Scholar
  5. [DK]
    Dix, T. I., and Kieronska, D. H. Errors between sites in restriction site mapping.CABIOS,4 (1988), 117–123.Google Scholar
  6. [DB]
    Durand, R., and Bregegere, F. An efficient program to construct restriction maps from experimental data with realistic error levels.Nucleic Acids Res.,12 (1984), 703–716.CrossRefGoogle Scholar
  7. [E]
    Ebert, J. Computing Eulerian trails.Inform. Process. Lett.,28 (1988), 93–97.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [FSR]
    Fitch, W. M., Smith, T. F., and Ralph, W. W. Mapping the order of DNA restriction fragments,Gene,22 (1983), 19–29.CrossRefGoogle Scholar
  9. [FF]
    Ford, I. R., and Fulkerson, D. R.Flows in Networks. Princeton University Press, Princeton, NJ, 1962.zbMATHGoogle Scholar
  10. [GW]
    Goldstein, L., and Waterman, M. S. Mapping DNA by stochastic relaxation.Adv. in Appl. Math.,8 (1987), 194–207.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [GM1]
    Grigorjev, A. V., and Mironov, A. A. Mapping DNA by stochastic relaxation: a new approach to fragment sizes.CABIOS,6 (1990), 107–111.Google Scholar
  12. [GM2]
    Grigorjev, A. V., and Mironov, A. A. Mapping DNA by stochastic relaxation: a schedule for optimal annealing.J. DNA Mapping and Sequencing,1 (1991), 221–226.Google Scholar
  13. [Ha]
    Hall, M., Jr.,Combinatorial Theory. Toronto, 1967.Google Scholar
  14. [HAY]
    Ho, S. T. S., Allison, L., and Yee, C. N. Restriction site mapping for three or more enzymes.CABIOS,6 (1990), 195–204.Google Scholar
  15. [Hoy]
    Hoyle, P. Use of commercial software on IBM personal computers. In M. J. Bishop and C. J. Rawlings (eds),Nucleic Acids and Protein Sequence Analysis: Practical Approaches. IRL Press, Oxford, 1987, pp. 47–82.Google Scholar
  16. [Ko]
    Kotzig A. Moves without forbidden transitions in a graph.Mat. casopis,18 (1968), 76–80.MathSciNetGoogle Scholar
  17. [Kr]
    Krawczak, M. Algorithms for the restriction site mapping of DNA molecules.Proc. Nat. Acad. Sci. USA,85 (1988), 7298–7301.CrossRefMathSciNetGoogle Scholar
  18. [MAB+]
    Mironov, A. A., Alexandrov, N. N., Bogodarova, N. Yu., Grigorjev, A., Lebedev, V. F., Lunovskaya, L. V., Pevzner, P. A., and Truchan M. E. DNASUN: A Package of Computer Programs for Biotechnology Laboratory (submitted).Google Scholar
  19. [NN]
    Newberg, L., and Naor, D. A lower bound on the number of solutions to the probed partial digest problem.Adv. in Appl. Math.,14 (1993), 172–185.zbMATHCrossRefMathSciNetGoogle Scholar
  20. [NMS]
    Nolan, G. P., Maina, C. V., and Szalay, A. A. Plasmid mapping computer program.Nucleic Acids Res.,12 (1984), 717–729.CrossRefGoogle Scholar
  21. [Pea]
    Pearson, W. Automatic construction of restriction site maps.Nucleic Acids Res.,10 (1982), 217–227.CrossRefGoogle Scholar
  22. [Pev1]
    Pevzner, P. A. Graphs of restrictions and DNA physical mapping.Biopolymers and Cell,5 (1988), 233–237 (in Russian).Google Scholar
  23. [Pev2]
    Pevzner, P. A. ι-tuple DNA sequencing: a computer analysis.J. Biom. Struct. Dyn.,7 (1989), 63–73.Google Scholar
  24. [Pev3]
    Pevzner, P. A. DNA physical mapping. In M. D. Frank-Kamenetzky (ed.),Computer Analysis of Genetic Texts. Nauka, Moscow, 1990, pp. 154–188 (in Russian).Google Scholar
  25. [Pev4]
    Pevzner, P. A.DNA Physical Mapping, Flows in Networks and Minimum Cycles Mean in Graphs. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Vol. 8, 1992, pp. 99–112.Google Scholar
  26. [Pev5]
    Pevzner, P. A. (1994) MAPSUN: a DNA physical mapping computer algorithm (in preparation).Google Scholar
  27. [PM]
    Pevzner, P. A., and Mironov, A. A. An efficient method for physical mapping of DNA molecules.Molek. Biol,21 (1987), 788–796.Google Scholar
  28. [PDO]
    Polner, G., Dorgai, L., and Orosz, L. PMAP, PMAPS: DNA physical map construction programs.Nucleic Acids Res.,12 (1984), 227–236.CrossRefGoogle Scholar
  29. [SW]
    Schmitt, W., and Waterman, M. Multiple solutions of DNA restriction mapping problem.Adv. in Appl. Math.,12 (1991), 412–427.zbMATHCrossRefMathSciNetGoogle Scholar
  30. [S]
    Stefik, M. Inferring DNA structure from segmentation data.Artificial Intelligence,11 (1978), 85–114.CrossRefGoogle Scholar
  31. [TDMH]
    Tuffery, P., Dessen, P., Mugnier, C., and Hazout, S. Restriction map construction using a “complete sentences compatibility” algorithm.CABIOS,4 (1988), 103–110.Google Scholar
  32. [U]
    Ukkonen, E. Approximate string matching withq-grams and maximal matches.Theoret. Comput. Sci.,92 (1992), 191–211.zbMATHCrossRefMathSciNetGoogle Scholar
  33. [WG]
    Waterman, M. S., and Griggs, J. R. Interval graphs and maps of DNA.Bull. Math. Biol.,48 (1986), 189–195.zbMATHMathSciNetGoogle Scholar
  34. [Y]
    Yap, R. H. C. Restriction site mapping in CLP(ℛ).Proceedings of the 8th International Conference on Logic Programming, MIT Press, Cambridge, MA, 1991, pp. 521–534.Google Scholar
  35. [ZFL]
    Zehetner, G., Frischauf, A., and Lehrach, H. Approaches to restriction map determination. In M. J. Bishop and C. J. Rawlings (eds.),Nucleic Acids and Protein Sequences Analysis, Practical Approaches. IRL Press, Oxford, 1987, pp. 147–164.Google Scholar
  36. [ZL]
    Zehetner, G., and Lehrach, H. A computer program package for restriction map analysis and manipulation,Nucleic Acids Res.,14 (1986), 335–349.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1995

Authors and Affiliations

  • P. A. Pevzner
    • 1
  1. 1.Computer Science DepartmentThe Pennsylvania State UniversityUniversity ParkUSA

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