Abstract
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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
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)
Connelly, R.: Generic global rigidity. Discrete Comput. Geom. 33, 549–563 (2005)
Crippen, G., Havel, T.: Distance Geometry and Molecular Conformation. Wiley, New York (1988)
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)
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)
Graver, J.E., Servatius, B., Servatius, H.: Combinatorial Rigidity. Graduate Studies in Math., AMS (1993)
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)
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)
Lavor, C., Liberti, L. Maculan, N. Mucherino, A.: The discretizable molecular distance geometry problem. Comput. Optim. Appl. 52, 115–146 (2012)
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)
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)
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)
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)
Liberti, L., Lavor, C., Mucherino, A., Maculan, N.: Molecular distance geometry methods: from continuous to discrete. Int. Trans. Oper. Res. 18, 33–51 (2010)
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)
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)
Moré, J., Wu, Z.: Global continuation for distance geometry problems. SIAM J. Optim. 7(3), 814–846 (1997)
Mucherino, A., Lavor, C., Liberti, L.: The discretizable distance geometry problem, to appear in Optimization Letters (DOI:10.1007/s11590-011-0358-3).
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)
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)
Schlick, T.: Molecular Modelling and Simulation: An Interdisciplinary Guide. Springer, New York (2002)
Acknowledgements
The authors wish to thank the Brazilian research agencies FAPESP and CNPq and the French research agency CNRS and École Polytechnique for financial support.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Science+Business Media New York
About this chapter
Cite this chapter
Liberti, L., Lavor, C., Mucherino, A. (2013). The Discretizable Molecular Distance Geometry Problem seems Easier on Proteins. In: Mucherino, A., Lavor, C., Liberti, L., Maculan, N. (eds) Distance Geometry. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-5128-0_3
Download citation
DOI: https://doi.org/10.1007/978-1-4614-5128-0_3
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-5127-3
Online ISBN: 978-1-4614-5128-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)