A Geometric Arrangement Algorithm for Structure Determination of Symmetric Protein Homo-oligomers from NOEs and RDCs

  • Jeffrey W. Martin
  • Anthony K. Yan
  • Chris Bailey-Kellogg
  • Pei Zhou
  • Bruce R. Donald
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6577)


Nuclear magnetic resonance (NMR) spectroscopy is a primary tool to perform structural studies of proteins in the physiologically-relevant solution-state. Restraints on distances between pairs of nuclei in the protein, derived from the nuclear Overhauser effect (NOE) for example, provide information about the structure of the protein in its folded state. NMR studies of symmetric protein homo-oligomers present a unique challenge. Current techniques can determine whether an NOE restrains a pair of protons across different subunits or within a single subunit, but are unable to determine in which subunits the restrained protons lie. Consequently, it is difficult to assign NOEs to particular pairs of subunits with certainty, thus hindering the structural analysis of the oligomeric state. Hence, computational approaches are needed to address this subunit ambiguity. We reduce the structure determination of protein homo-oligomers with cyclic symmetry to computing geometric arrangements of unions of annuli in a plane. Our algorithm, disco, runs in expected O(n 2) time, where n is the number of distance restraints, and is guaranteed to report the exact set of oligomer structures consistent with ambiguously-assigned inter-subunit distance restraints and orientational restraints from residual dipolar couplings (RDCs). Since the symmetry axis of an oligomeric complex must be parallel to an eigenvector of the alignment tensor of RDCs, we can represent each distance restraint as a union of annuli in a plane encoding the configuration space of the symmetry axis. Oligomeric protein structures with the best restraint satisfaction correspond to faces of the arrangement contained in the greatest number of unions of annuli. We demonstrate our method using two symmetric protein complexes: the trimeric E. coli Diacylglycerol Kinase (DAGK), whose distance restraints possess at least two possible subunit assignments each; and a dimeric mutant of the immunoglobulin-binding domain B1 of streptococcal protein G (GB1) using ambiguous NOEs. In both cases, disco computes oligomer structures with high accuracy.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Chen, C.Y., Georgiev, I., Anderson, A.C., Donald, B.R.: Computational structure-based redesign of enzyme activity. Proceedings of the National Academy of Sciences 106, 3764–3769 (2009)CrossRefGoogle Scholar
  2. 2.
    Frey, K.M., Georgiev, I., Donald, B.R., Anderson, A.C.: Predicting resistance mutations using protein design algorithms. Proceedings of the National Academy of Sciences 107, 13707–13712 (2010)CrossRefGoogle Scholar
  3. 3.
    Goodsell, D.S., Olson, A.J.: Structural symmetry and protein function. Annual Review of Biophysics and Biomolecular Structure 29(1), 105–105 (2000), doi:10.1146/annurev.biophys.29.1.105CrossRefGoogle Scholar
  4. 4.
    Levy, E.D., Erba, E.B., Robinson, C.V., Teichmann, S.A.: Assembly reflects evolution of protein complexes. Nature 453, 1262–1265 (2008), 10.1038/nature06942CrossRefGoogle Scholar
  5. 5.
    White, S.H.: The progress of membrane protein structure determination. Protein Science 13, 1948–1949 (2004)CrossRefGoogle Scholar
  6. 6.
    Potluri, S., Yan, A.K., Donald, B.R., Bailey-Kellogg, C.: A complete algorithm to resolve ambiguity for intersubunit NOE assignment in structure determination of symmetric homo-oligomers. Protein Science 16, 69–81 (2007)CrossRefGoogle Scholar
  7. 7.
    Ikura, M., Bax, A.: Isotope-filtered 2D NMR of a protein-peptide complex: study of a skeletal muscle myosin light chain kinase fragment bound to calmodulin. Journal of the American Chemical Society 114, 2433–2440 (1992)CrossRefGoogle Scholar
  8. 8.
    Saxe, J.: Embeddability of weighted graphs in k-space is strongly NP-hard. In: Proceedings of the 17th Allerton Conference in Communications, Control, and Computing, pp. 480–489 (1979)Google Scholar
  9. 9.
    Wang, L., Mettu, R.R., Donald, B.R.: A polynomial-time algorithm for de novo protein backbone structure determination from nuclear magnetic resonance data. Journal of Computational Biology 13, 1267–1288 (2006)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Potluri, S., Yan, A.K., Chou, J.J., Donald, B.R., Bailey-Kellogg, C.: Structure determination of symmetric homo-oligomers by a complete search of symmetry configuration space, using NMR restraints and van der Waals packing. Proteins: Structure, Function, and Bioinformatics 65, 203–219 (2006)CrossRefGoogle Scholar
  11. 11.
    Wang, X., Bansal, S., Jiang, M., Prestegard, J.H.: RDC-assisted modeling of symmetric protein homo-oligomers. Protein Science 17, 899–907 (2008)CrossRefGoogle Scholar
  12. 12.
    Wang, C.S.E., Lozano-Pérez, T., Tidor, B.: AmbiPack: A systematic algorithm for packing of macromolecular structures with ambiguous distance constraints. Proteins: Structure, Function, and Genetics 32, 26–42 (1998)CrossRefGoogle Scholar
  13. 13.
    Nilges, M.: A calculation strategy for the structure determination of symmetric demers by 1H NMR. Proteins: Structure, Function, and Genetics 17, 297–309 (1993)CrossRefGoogle Scholar
  14. 14.
    Kovacs, H., O’Donoghue, S.I., Hoppe, H.J., Comfort, D., Reid, K.B.M., Campbell, I.D., Nilges, M.: Solution structure of the coiled-coil trimerization domain from lung surfactant protein D. Journal of Biomolecular NMR 24, 89–102 (2002), 10.1023/A:1020980006628CrossRefGoogle Scholar
  15. 15.
    O’Donoghue, S.I., Chang, X., Abseher, R., Nilges, M., Led, J.J.: Unraveling the symmetry ambiguity in a hexamer: Calculation of the R6 human insulin structure. Journal of Biomolecular NMR 16(2), 93–108 (2000), 10.1023/A:1008323819099CrossRefGoogle Scholar
  16. 16.
    Bardiaux, B., Bernard, A., Rieping, W., Habeck, M., Malliavin, T.E., Nilges, M.: Influence of different assignment conditions on the determination of symmetric homodimeric structures with ARIA. Proteins: Structure, Function, and Bioinformatics 75, 569–585 (2009)CrossRefGoogle Scholar
  17. 17.
    Van Horn, W.D., Kim, H.J., Ellis, C.D., Hadziselimovic, A., Sulistijo, E.S., Karra, M.D., Tian, C., Sönnichsen, F.D., Sanders, C.R.: Solution nuclear magnetic resonance structure of membrane-integral diacylglycerol kinase. Science 324, 1726–1729 (2009)CrossRefGoogle Scholar
  18. 18.
    Byeon, I.J.L., Louis, J.M., Gronenborn, A.M.: A protein contortionist: Core mutations of GB1 that induce dimerization and domain swapping. Journal of Molecular Biology 333, 141–152 (2003)CrossRefGoogle Scholar
  19. 19.
    Donald, B.R., Martin, J.: Automated NMR assignment and protein structure determination using sparse dipolar coupling constraints. Progress in Nuclear Magnetic Resonance Spectroscopy 55, 101–127 (2009)CrossRefGoogle Scholar
  20. 20.
    Al-Hashimi, H.M., Bolon, P.J., Prestegard, J.H.: Molecular symmetry as an aid to geometry determination in ligand protein complexes. Journal of Magnetic Resonance 142, 153–158 (2000)CrossRefGoogle Scholar
  21. 21.
    Bewley, C.A., Clore, G.M.: Determination of the relative orientation of the two halves of the domain-swapped dimer of cyanovirin-N in solution using dipolar couplings and rigid body minimization. Journal of the American Chemical Society 122, 6009–6016 (2000)CrossRefGoogle Scholar
  22. 22.
    Oxenoid, K., Chou, J.J.: The structure of phospholamban pentamer reveals a channel-like architecture in membranes. Proceedings of the National Academy of Sciences of the United States of America 102, 10870–10875 (2005)CrossRefGoogle Scholar
  23. 23.
    Schnell, J.R., Chou, J.J.: Structure and mechanism of the M2 proton channel of influenza A virus. Nature 451(7178), 591–595 (2008), 10.1038/nature06531CrossRefGoogle Scholar
  24. 24.
    Wang, J., Pielak, R.M., McClintock, M.A., Chou, J.J.: Solution structure and functional analysis of the influenza B proton channel. Nat. Struct. Mol. Biol. 16, 1267–1271 (2009), 10.1038/nsmb.1707CrossRefGoogle Scholar
  25. 25.
    Losonczi, J.A., Andrec, M., Fischer, M.W.F., Prestegard, J.H.: Order matrix analysis of residual dipolar couplings using singular value decomposition. Journal of Magnetic Resonance 138, 334–342 (1999)CrossRefGoogle Scholar
  26. 26.
    Lozano-Perez, T.: Automatic planning of manipulator transfer movements. IEEE Transactions on Systems, Man and Cybernetics 11, 681–698 (1981)CrossRefGoogle Scholar
  27. 27.
    Martin, J.W., Yan, A.K., Bailey-Kellogg, C., Zhou, P., Donald, B.R.: Supplementary information: A geometric arrangement algorithm for structure determination of symmetric protein homo-oligomers from NOEs and RDCs (2011),
  28. 28.
    Halperin, D.: Arrangements. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, 2nd edn., pp. 529–562. CRC Press, Inc., Boca Raton (1997)Google Scholar
  29. 29.
    Hanniel, I., Halperin, D.: Two-dimensional arrangements in CGAL and adaptive point location for parametric curves. In: Näher, S., Wagner, D. (eds.) WAE 2000. LNCS, vol. 1982, pp. 171–182. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  30. 30.
    Schwieters, C.D., Kuszewski, J.J., Tjandra, N., Clore, G.M.: The Xplor-NIH NMR molecular structure determination package. Journal of Magnetic Resonance 160, 65–73 (2003)CrossRefGoogle Scholar
  31. 31.
    Berman, H.M., Westbrook, J., Feng, Z., Gilliland, G., Bhat, T.N., Weissig, H., Shindyalov, I.N., Bourne, P.E.: The protein data bank. Nucl. Acids Res. 28, 235–242 (2000)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Jeffrey W. Martin
    • 1
  • Anthony K. Yan
    • 1
    • 2
  • Chris Bailey-Kellogg
    • 3
  • Pei Zhou
    • 2
  • Bruce R. Donald
    • 1
    • 2
  1. 1.Department of Computer ScienceDuke UniversityDurhamUSA
  2. 2.Department of BiochemistryDuke University Medical CenterDurhamUSA
  3. 3.Department of Computer ScienceDartmouth CollegeHanoverUSA

Personalised recommendations