Advertisement

Protein Structure by Semidefinite Facial Reduction

  • Babak Alipanahi
  • Nathan Krislock
  • Ali Ghodsi
  • Henry Wolkowicz
  • Logan Donaldson
  • Ming Li
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7262)

Abstract

All practical contemporary protein NMR structure determination methods use molecular dynamics coupled with a simulated annealing schedule. The objective of these methods is to minimize the error of deviating from the NOE distance constraints. However, this objective function is highly nonconvex and, consequently, difficult to optimize. Euclidean distance geometry methods based on semidefinite programming (SDP) provide a natural formulation for this problem. However, complexity of SDP solvers and ambiguous distance constraints are major challenges to this approach. The contribution of this paper is to provide a new SDP formulation of this problem that overcomes these two issues for the first time. We model the protein as a set of intersecting two- and three-dimensional cliques, then we adapt and extend a technique called semidefinite facial reduction to reduce the SDP problem size to approximately one quarter of the size of the original problem. The reduced SDP problem can not only be solved approximately 100 times faster, but is also resistant to numerical problems from having erroneous and inexact distance bounds.

Keywords

Molecular structural biology nuclear magnetic resonance 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alipanahi, B., Gao, X., Karakoc, E., Donaldson, L., Li, M.: PICKY: a novel SVD-based NMR spectra peak picking method. Bioinformatics 25(12), i268–i275 (2009)CrossRefGoogle Scholar
  2. 2.
    Alipanahi, B., Gao, X., Karakoc, E., Li, S., Balbach, F., Feng, G., Donaldson, L., Li, M.: Error tolerant NMR backbone resonance assignment and automated structure generation. Journal of Bioinformatics and Computational Biology 0(1), 1–26 (2011)Google Scholar
  3. 3.
    Alipanahi, B., Krislock, N., Ghodsi, A.: Manifold learning by semidefinite facial reduction (2011) (unpublished manuscript) (in preparation)Google Scholar
  4. 4.
    Alipanahi, B.: New Approaches to Protein NMR Automation. Ph.D. thesis, University of Waterloo (2011)Google Scholar
  5. 5.
    Biswas, P., Toh, K.C., Ye, Y.: A distributed SDP approach for large-scale noisy anchor-free graph realization with applications to molecular conformation. SIAM J. Sci. Comput. 30, 1251–1277 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Braun, W., Bösch, C., Brown, L.R., Go, N., Wüthrich, K.: Combined use of proton-proton overhauser enhancements and a distance geometry algorithm for determination of polypeptide conformations. application to micelle-bound glucagon. Biochimica et Biophysica Acta 667(2), 377–396 (1981)Google Scholar
  7. 7.
    Braun, W., Go, N.: Calculation of protein conformations by proton-proton distance constraints. a new efficient algorithm. Journal of Molecular Biology 186(3), 611–626 (1985)CrossRefGoogle Scholar
  8. 8.
    Brünger, A.T.: X-PLOR Version 3.1: A System for X-ray Crystallography and NMR. Yale University Press (1993)Google Scholar
  9. 9.
    Chen, V.B., Arendall, W.B., Headd, J.J., Keedy, D.A., Immormino, R.M., Kapral, G.J., Murray, L.W., Richardson, J.S., Richardson, D.C.: MolProbity: all-atom structure validation for macromolecular crystallography. Acta Crystallographica. Section D, Biological Crystallography 66(pt.1), 12–21 (2010)CrossRefGoogle Scholar
  10. 10.
    Doreleijers, J.F., Mading, S., Maziuk, D., Sojourner, K., Yin, L., Zhu, J., Markley, J.L., Ulrich, E.L.: BioMagResBank database with sets of experimental NMR constraints corresponding to the structures of over 1400 biomolecules deposited in the protein data bank. Journal of Biomolecular NMR 26(2), 139–146 (2003)CrossRefGoogle Scholar
  11. 11.
    Doreleijers, J.F., Nederveen, A.J., Vranken, W., Lin, J., Bonvin, A.M., Kaptein, R., Markley, J.L., Ulrich, E.L.: BioMagResBank databases DOCR and FRED containing converted and filtered sets of experimental NMR restraints and coordinates from over 500 protein PDB structures. Journal of Biomolecular NMR 32(1), 1–12 (2005)CrossRefGoogle Scholar
  12. 12.
    Güntert, P.: Structure calculation of biological macromolecules from NMR data. Quarterly Reviews of Biophysics 31(2), 145–237 (1998)CrossRefGoogle Scholar
  13. 13.
    Güntert, P.: Automated NMR structure calculation with CYANA. Methods in Molecular Biology 278, 353–378 (2004)Google Scholar
  14. 14.
    Güntert, P., Mumenthaler, C., Wüthrich, K.: Torsion angle dynamics for NMR structure calculation with the new program DYANA. Journal of Molecular Biology 273, 283–298 (1997)CrossRefGoogle Scholar
  15. 15.
    Havel, T.F., Wüthrich, K.: A Distance Geometry Program for Determining the Structures of Small Proteins and Other Macromolecules From Nuclear Magnetic Resonance Measurements of Intramolecular H-H Proxmities in Solution. Bulletin of Mathematical Biology 46(4), 673–698 (1984)zbMATHGoogle Scholar
  16. 16.
    Krislock, N.: Semidefinite Facial Reduction for Low-Rank Euclidean Distance Matrix Completion. Ph.D. thesis, University of Waterloo (2010)Google Scholar
  17. 17.
    Krislock, N., Wolkowicz, H.: Explicit sensor network localization using semidefinite representations and facial reductions. SIAM J. Optimiz. 20, 2679–2708 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Kuszewski, J., Gronenborn, A.M., Clore, G.M.: Improving the quality of NMR and crystallographic protein structures by means of a conformational database potential derived from structure databases. Protein Sci. 5(6), 1067–1080 (1996)CrossRefGoogle Scholar
  19. 19.
    Kuszewski, J., Gronenborn, A.M., Clore, G.M.: Improvements and extensions in the conformational database potential for the refinement of NMR and x-ray structures of proteins and nucleic acids. Journal of Magnetic Resonance 125(1), 171–177 (1997)CrossRefGoogle Scholar
  20. 20.
    Leung, N.H.Z., Toh, K.C.: An SDP-based divide-and-conquer algorithm for large-scale noisy anchor-free graph realization. SIAM J. Sci. Comput. 31, 4351–4372 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Lewis, A., Overton, M.: Nonsmooth optimization via BFGS. Submitted to SIAM J. Optimiz. (2009)Google Scholar
  22. 22.
    Linge, J.P., Habeck, M., Rieping, W., Nilges, M.: ARIA: automated NOE assignment and NMR structure calculation. Bioinformatics 19(2), 315–316 (2003)CrossRefGoogle Scholar
  23. 23.
    Moré, J.J., Wu, Z.: Global continuation for distance geometry problems. SIAM J. Optimiz. 7, 814–836 (1997)zbMATHCrossRefGoogle Scholar
  24. 24.
    Nilges, M., Clore, G.M., Gronenborn, A.M.: Determination of three-dimensional structures of proteins from interproton distance data by hybrid distance geometry-dynamical simulated annealing calculations. FEBS Letters 229(2), 317–324 (1988)CrossRefGoogle Scholar
  25. 25.
    Raman, S., Lange, O.F., Rossi, P., Tyka, M., Wang, X., Aramini, J., Liu, G., Ramelot, T.A., Eletsky, A., Szyperski, T., Kennedy, M.A., Prestegard, J., Montelione, G.T., Baker, D.: NMR structure determination for larger proteins using Backbone-Only data. Science 327(5968), 1014–1018 (2010)CrossRefGoogle Scholar
  26. 26.
    Ramana, M.V., Tunçel, L., Wolkowicz, H.: Strong duality for semidefinite programming. SIAM J. Optimiz. 7(3), 641–662 (1997)zbMATHCrossRefGoogle Scholar
  27. 27.
    Schoenberg, I.J.: Remarks to Maurice Fréchet’s article Sur la définition axiomatique d’une classe d’espace distanciés vectoriellement applicable sur l’espace de Hilbert. Ann. of Math. 36(3), 724–732 (1935)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Schwieters, C., Kuszewski, J., Tjandra, N., Clore, G.: The Xplor-NIH NMR molecular structure determination package. Journal of Magnetic Resonance 160, 65–73 (2003)CrossRefGoogle Scholar
  29. 29.
    Shen, Y., Lange, O., Delaglio, F., Rossi, P., Aramini, J.M., Liu, G., Eletsky, A., Wu, Y., Singarapu, K.K., Lemak, A., Ignatchenko, A., Arrowsmith, C.H., Szyperski, T., Montelione, G.T., Baker, D., Bax, A.: Consistent blind protein structure generation from NMR chemical shift data. Proceedings of the National Academy of Sciences of the United States of America 105(12), 4685–4690 (2008)CrossRefGoogle Scholar
  30. 30.
    Tütüncü, R., Toh, K., Todd, M.: Solving semidefinite-quadratic-linear programs using SDPT3. Math. Program. 95(2, ser. B), 189–217 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Vandenberghe, L., Boyd, S.: Semidefinite programming. SIAM Review 38(1), 49–95 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Wei, H., Wolkowicz, H.: Generating and measuring instances of hard semidefinite programs. Mathematical Programming 125, 31–45 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Weinberger, K.Q., Saul, L.K.: Unsupervised learning of image manifolds by semidefinite programming. In: IEEE Computer Society Conference on Computer Vision and Pattern Recognition, vol. 2, pp. 988–995 (2004)Google Scholar
  34. 34.
    Williams, G.A., Dugan, J.M., Altman, R.B.: Constrained global optimization for estimating molecular structure from atomic distances. Journal of Computational Biology 8(5), 523–547 (2001)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Babak Alipanahi
    • 1
  • Nathan Krislock
    • 2
  • Ali Ghodsi
    • 3
  • Henry Wolkowicz
    • 4
  • Logan Donaldson
    • 5
  • Ming Li
    • 1
  1. 1.David R. Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada
  2. 2.INRIA Grenoble Rhône-AlpesFrance
  3. 3.Department of Statistics and Actuarial ScienceUniversity of WaterlooWaterlooCanada
  4. 4.Department of Combinatorics and OptimizationUniversity of WaterlooWaterlooCanada
  5. 5.Department of BiologyYork UniversityTorontoCanada

Personalised recommendations