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Optimization Letters

, Volume 6, Issue 8, pp 1671–1686 | Cite as

The discretizable distance geometry problem

  • A. Mucherino
  • C. Lavor
  • L. Liberti
Original Paper

Abstract

We introduce the discretizable distance geometry problem in \({\mathbb{R}^3}\) (DDGP3), which consists in a subclass of instances of the Distance Geometry Problem for which an embedding in \({\mathbb{R}^3}\) can be found by means of a discrete search. We show that the DDGP3 is a generalization of the discretizable molecular distance geometry problem (DMDGP), and we discuss the main differences between the two problems. We prove that the DDGP3 is NP-hard and we extend the Branch & Prune (BP) algorithm, previously used for the DMDGP, for solving instances of the DDGP3. Protein graphs may or may not be in DMDGP and/or DDGP3 depending on vertex orders and edge density. We show experimentally that as distance thresholds decrease, PDB protein graphs which fail to be in the DMDGP still belong to DDGP3, which means that they can still be solved using a discrete search.

Keywords

Distance geometry DDGP3 DMDGP Combinatorial reformulations Branch and prune 

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References

  1. 1.
    Anderson, E., Bai, Z., Dongarra, J., Greenbaum, A., McKenney, A., Du Croz, J., Hammerling, S., Demmel, J., Bischof, C., Sorensen, D.: LAPACK: a Portable Linear Algebra Library for High-Performance Computers. In: Supercomputing ’90: Proceedings of the 1990 ACM/IEEE conference on Supercomputing, pp. 2–11. IEEE Computer Society Press, New York (1990)Google Scholar
  2. 2.
    Berman H.M., Westbrook J., Feng Z., Gilliland G., Bhat T.N., Weissig H., Shindyalov I.N., Bourne P.E.: The protein data bank. Nucleic Acids Res. 28, 235–242 (2000)CrossRefGoogle Scholar
  3. 3.
    Carvalho R.S., Lavor C., Protti F.: Extending the Geometric Buildup Algorithm for the Molecular Distance Geometry Problem. Inf. Process. Lett. 108, 234–237 (2008)CrossRefGoogle Scholar
  4. 4.
    Coope I.D.: Reliable Computation of the Points of Intersection of n Spheres in n-space. ANZIAM J. 42, 461–477 (2000)MathSciNetGoogle Scholar
  5. 5.
    Crippen G.M., Havel T.F.: Distance Geometry and Molecular Conformation. John Wiley & Sons, New York (1988)MATHGoogle Scholar
  6. 6.
    Eren, T., Goldenberg, D.K., Whiteley, W., Yang, Y.R., Morse, A.S., Anderson, B.D.O., Belhumeur, P.N.: Rigidity, Computation, and Randomization in Network Localization. In: IEEE Infocom Proceedings, pp. 2673–2684 (2004)Google Scholar
  7. 7.
    Havel T.F.: Distance Geometry. In: Grant, D.M., Harris, R.K. (eds) Encyclopedia of Nuclear Magnetic Resonance, pp. 1701–1710. Wiley, New York (1995)Google Scholar
  8. 8.
    Huang H.-X., Liang Z-A., Pardalos P.M.: Some Properties for the Euclidean Distance Matrix and Positive Semidefinite Matrix Completion Problem. J. Global Optim. 25(1), 3–21 (2003)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Krislock, N.: Semidefinite Facial Reduction for Low-Rank Euclidean Distance Matrix Completion, PhD thesis, University of Waterloo, Waterloo (2010)Google Scholar
  10. 10.
    Lavor, C., Lee, J., Lee-St. John, A., Liberti, L., Mucherino, A., Sviridenko, M.: Discretization orders for distance geometry problems. Optim. Lett. (2011, in press)Google Scholar
  11. 11.
    Lavor, C., Liberti, L., Maculan, N.: Discretizable molecular distance geometry problem, Tech. Rep. q-bio.BM/0608012, arXiv (2006)Google Scholar
  12. 12.
    Lavor C., Liberti L., Maculan N.: Molecular distance geometry problem. In: Floudas, C., Pardalos, P. (eds) Encyclopedia of Optimization, pp. 2305–2311. Springer, New York (2009)Google Scholar
  13. 13.
    Lavor, C., Liberti, L., Maculan, N., Mucherino, A.: The discretizable molecular distance geometry problem. Comput. Optim. Appl. (2011, in press)Google Scholar
  14. 14.
    Lavor, C., Liberti, L., Maculan, N., Mucherino, A.: Recent advances on the discretizable molecular distance geometry problem. Eur. J. Oper. Res. (2011, in press)Google Scholar
  15. 15.
    Lavor, C., Mucherino, A., Liberti, L., Maculan, N.: Computing Artificial Backbones of Hydrogen Atoms in order to Discover Protein Backbones. In: IEEE Conference Proceedings, International Multiconference on Computer Science and Information Technology (IMCSIT09), Workshop on Computational Optimization (WCO09), Mragowo, Poland, pp. 751–756 (2009)Google Scholar
  16. 16.
    Lavor, C., Mucherino, A., Liberti, L., Maculan, N.: An artificial backbone of hydrogens for finding the conformation of protein molecules. In: Proceedings of the Computational Structural Bioinformatics Workshop (CSBW09), Washington D.C., USA, pp. 152–155 (2009)Google Scholar
  17. 17.
    Lavor C., Mucherino A., Liberti L., Maculan N.: On the computation of protein backbones by using artificial backbones of hydrogens. J. Global Optim. 50(2), 329–344 (2011)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Lavor C., Mucherino A., Liberti L., Maculan N.: Discrete approaches for solving molecular distance geometry problems using NMR data. Int. J. Comput. Biosci. 1(1), 88–94 (2010)Google Scholar
  19. 19.
    Liberti L., Lavor C., Maculan N.: A Branch-and-Prune algorithm for the molecular distance geometry problem. Int. Trans. Oper. Res. 15(1), 1–17 (2008)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Liberti, L., Lavor, C., Mucherino, A.: An exponential algorithm for the discretizable molecular distance geometry problem is polynomial on proteins. In: Proceedings of the 7th International Symposium on Bioinformatics Research and Applications (ISBRA11), Changsha, China (2011)Google Scholar
  21. 21.
    Liberti L., Lavor C., Mucherino A., Maculan N.: Molecular distance geometry methods: from continuous to discrete. Int. Trans. Oper. Res. 18(1), 33–51 (2010)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Liu X., Pardalos P.M. et al.: A Tabu based pattern search method for the distance geometry problem. In: Giannessi, F. (eds) New Trends in Mathematical Programming, pp. 223–234. Kluwer Academic Publishers, Dordrecht (1998)Google Scholar
  23. 23.
    Moré J.J., Wu Z.: Global continuation for distance geometry problems. SIAM J. Optim. 7, 814–836 (1997)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Moré J.J., Wu Z.: Distance geometry optimization for protein structures. J. Global Optim. 15, 219–223 (1999)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Mucherino, A., Lavor, C.: The Branch and Prune algorithm for the molecular distance geometry problem with inexact distances. In: Proceedings of World Academy of Science, Engineering and Technology (WASET), International Conference on Bioinformatics and Biomedicine (ICBB09), Venice, Italy, pp. 349–353 (2009)Google Scholar
  26. 26.
    Mucherino, A., Liberti, L., Lavor, C., Maculan, N.: Comparisons between an exact and a MetaHeuristic algorithm for the molecular distance geometry problem. In: ACM Conference Proceedings, Genetic and Evolutionary Computation Conference (GECCO09), Montréal, Canada, pp. 333–340 (2009)Google Scholar
  27. 27.
    Mucherino A., Lavor C., Liberti L., Maculan N.: On the definition of artificial backbones for the discretizable molecular distance geometry problem. Mathematica Balkanica 23(3–4), 289–302 (2009)MathSciNetMATHGoogle Scholar
  28. 28.
    Mucherino, A., Lavor, C., Liberti, L., Maculan, N.: Strategies for solving distance geometry problems with inexact distances by discrete approaches. In: Proceedings of Toulouse Global Optimization 2010 (TOGO10), Toulouse, France, pp. 93–96 (2010)Google Scholar
  29. 29.
    Mucherino, A., Lavor, C., Malliavin, T., Liberti, L., Nilges, M., Maculan, M.: Influence of pruning devices on the solution of molecular distance geometry problems. In: Pardalos, P.M., Rebennack, S. (eds.) Proceedings of the 10th International Symposium on Experimental Algorithms (SEA11), Crete, Greece. Lecture Notes in Computer Science, vol. 6630, pp. 206–217 (2011)Google Scholar
  30. 30.
    Mucherino, A., Liberti, L., Lavor, C.: MD-jeep: an Implementation of a Branch & Prune Algorithm for Distance Geometry Problems. In: Fukuda, K., et al. (eds.) Proceedings of the Third International Congress on Mathematical Software (ICMS10), Kobe, Japan. Lectures Notes in Computer Science, vol. 6327, pp. 186–197 (2010)Google Scholar
  31. 31.
    Pardalos, P.M., Shalloway, D., Xue, G. (eds.) (1996) Global Minimization of Nonconvex Energy Functions: Molecular Conformation and Protein Folding. AMS, DIMACSGoogle Scholar
  32. 32.
    Saxe, J.B.: Embeddability of Weighted Graphs in k-space is Strongly NP-hard. In: Proceedings of 17th Allerton Conference in Communications, Control, and Computing, Monticello, IL, pp. 480–489 (1979)Google Scholar
  33. 33.
    So M.-C., Ye Y.: Theory of semidefinite programming for sensor network localization. Math. Program. 109, 367–384 (2007)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Wu D., Wu Z.: An updated geometric build-up algorithm for solving the molecular distance geometry problem with sparse distance Data. J. Global Optim. 37, 661–673 (2007)MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Wu D., Wu Z., Yuan Y.: Rigid versus unique determination of protein structures with geometric buildup. Optim. Lett. 2, 319–331 (2008)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.CERFACSToulouseFrance
  2. 2.Department of Applied Mathematics (IMECC-UNICAMP)State University of CampinasCampinasBrazil
  3. 3.LIX, École PolytechniquePalaiseauFrance

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