The Discretizable Molecular Distance Geometry Problem seems Easier on Proteins

  • Leo Liberti
  • Carlile Lavor
  • Antonio Mucherino


Distance geometry methods are used to turn a set of interatomic distances given by Nuclear Magnetic Resonance (NMR) experiments into a consistent molecular conformation. In a set of papers (see the survey [8]) we proposed a Branch-and-Prune (BP) algorithm for computing the set X of all incongruent embeddings of a given protein backbone. Although BP has a worst-case exponential running time in general, we always noticed a linear-like behaviour in computational experiments. In this chapter we provide a theoretical explanation to our observations. We show that the BP is fixed-parameter tractable on protein-like graphs and empirically show that the parameter is constant on a set of proteins from the Protein Data Bank.


Branch-and-Prune Symmetry Distance geometry Fixed-parameter tractable Protein conformation 



The authors wish to thank the Brazilian research agencies FAPESP and CNPq and the French research agency CNRS and École Polytechnique for financial support.


  1. 1.
    Berman, H. Westbrook, J., Feng, Z., Gilliland, G., Bhat, T., Weissig, H., Shindyalov, I., Bourne, P.: The protein data bank. Nucleic Acid Res. 28, 235–242 (2000)Google Scholar
  2. 2.
    Connelly, R.: Generic global rigidity. Discrete Comput. Geom. 33, 549–563 (2005)Google Scholar
  3. 3.
    Crippen, G., Havel, T.: Distance Geometry and Molecular Conformation. Wiley, New York (1988)Google Scholar
  4. 4.
    Dong, Q., Wu, Z.: A geometric build-up algorithm for solving the molecular distance geometry problem with sparse distance data. J. Global Optim. 26, 321–333 (2003)Google Scholar
  5. 5.
    Eren, T., Goldenberg, D., Whiteley, W., Yang, Y., Morse, A., Anderson, B., Belhumeur, P.: Rigidity, computation, and randomization in network localization. In: IEEE Infocom Proceedings, 2673–2684 (2004)Google Scholar
  6. 6.
    Graver, J.E., Servatius, B., Servatius, H.: Combinatorial Rigidity. Graduate Studies in Math., AMS (1993)Google Scholar
  7. 7.
    Lavor, C. Lee, J., John, A.L.S., Liberti, L., Mucherino, A., Sviridenko, M.: Discretization orders for distance geometry problems. Optim. Lett. 6, 783–796 (2012)Google Scholar
  8. 8.
    Lavor, C., Liberti, L., Maculan, N., Mucherino, A.: Recent advances on the discretizable molecular distance geometry problem. Eur. J. Oper. Res. 219, 698–706 (2012)Google Scholar
  9. 9.
    Lavor, C., Liberti, L. Maculan, N. Mucherino, A.: The discretizable molecular distance geometry problem. Comput. Optim. Appl. 52, 115–146 (2012)Google Scholar
  10. 10.
    Lavor, C., Liberti, L., Mucherino, A.: On the solution of molecular distance geometry problems with interval data. In: IEEE Conference Proceedings, International Workshop on Computational Proteomics (IWCP10), International Conference on Bioinformatics and Biomedicine (BIBM10), Hong Kong, 77–82 (2010)Google Scholar
  11. 11.
    Lavor, C., Mucherino, A., Liberti, L., Maculan, N.: On the computation of protein backbones by using artificial backbones of hydrogens. J. Global Optim. 50, 329–344 (2011)Google Scholar
  12. 12.
    Liberti, L., Lavor, C.: On a relationship between graph realizability and distance matrix completion. In: Kostoglou, V., Arabatzis, G., Karamitopoulos, L. (eds.) Proceedings of BALCOR, vol. I, pp. 2–9. Hellenic OR Society, Thessaloniki (2011)Google Scholar
  13. 13.
    Liberti, L., Lavor, C., Maculan, N.: A branch-and-prune algorithm for the molecular distance geometry problem. Int. Trans. Oper. Res. 15, 1–17 (2008)Google Scholar
  14. 14.
    Liberti, L., Lavor, C., Mucherino, A., Maculan, N.: Molecular distance geometry methods: from continuous to discrete. Int. Trans. Oper. Res. 18, 33–51 (2010)Google Scholar
  15. 15.
    Liberti, L., Masson, B., Lavor, C., Lee, J., Mucherino, A.: On the number of solutions of the discretizable molecular distance geometry problem, Tech. Rep. 1010.1834v1[cs.DM], arXiv (2010)Google Scholar
  16. 16.
    Liberti, L., Masson, B., Lee, J., Lavor, C., Mucherino, A.: On the number of solutions of the discretizable molecular distance geometry problem, Lecture Notes in Computer Science. In: Wang, W., Zhu, X., Du, D-Z. (eds.) Proceedings of the 5th Annual International Conference on Combinatorial Optimization and Applications (COCOA11), Zhangjiajie, China, vol.6831, pp. 322–342 (2011)Google Scholar
  17. 17.
    Moré, J., Wu, Z.: Global continuation for distance geometry problems. SIAM J. Optim. 7(3), 814–846 (1997)Google Scholar
  18. 18.
    Mucherino, A., Lavor, C., Liberti, L.: The discretizable distance geometry problem, to appear in Optimization Letters (DOI:10.1007/s11590-011-0358-3).Google Scholar
  19. 19.
    Mucherino, A., Liberti, L., Lavor, C.: MD-jeep: an implementation of a branch and prune algorithm for distance geometry problems, Lectures Notes in Computer Science. In: Fukuda, K., et al. (eds.) Proceedings of the Third International Congress on Mathematical Software (ICMS10), Kobe, Japan, vol. 6327, pp. 186–197 (2010)Google Scholar
  20. 20.
    Saxe, J.: Embeddability of weighted graphs in k-space is strongly NP-hard. In: Proceedings of \(1{7}^{th}\) Allerton Conference in Communications, Control and Computing, pp. 480–489 (1979)Google Scholar
  21. 21.
    Schlick, T.: Molecular Modelling and Simulation: An Interdisciplinary Guide. Springer, New York (2002)Google Scholar

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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.LIX, École PolytechniquePalaiseauFrance
  2. 2.Department of Applied MathsUniversity of CampinasCampinas-SPBrazil
  3. 3.IRISAUniversity of Rennes 1RennesFrance

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