Conformational Transitions and Principal Geodesic Analysis on the Positive Semidefinite Matrix Manifold

  • Xiao-Bo Li
  • Forbes J. Burkowski
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8492)


Given an initial and final protein conformation, generating the intermediate conformations provides important insight into the protein’s dynamics. We represent a protein conformation by its Gram matrix, which is a point on the rank 3 positive semidefinite matrix manifold, and show matrices along the geodesic linking an initial and final Gram matrix can be used to generate a feasible pathway for the protein’s structural change. This geodesic is based on a particular quotient geometry. If a protein is known to contain domains or groups of atoms that act as rigid clusters, facial reduction can be used to decrease the size of the Gram matrices before calculating the geodesic. The geodesic between two conformations is only one path a protein’s Gram matrix can follow; principal geodesic analysis (PGA) is one possible strategy to find other geodesics.


cliques rigid clusters elastic network interpolation Euclidean distance matrix Gram matrix positive semidefinite facial reduction geodesic Riemannian manifold principal geodesic analysis 


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  1. 1.
    Absil, P.A., Mahony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton (2008)MATHGoogle Scholar
  2. 2.
    Alfakih, A., Khandani, A., Wolkowicz, H.: Solving euclidean distance matrix completion problems via semidefinite programming. Computational Optimization and Applications 12(1-3), 13–30 (1999)MATHMathSciNetGoogle Scholar
  3. 3.
    Alipanahi, B.: New Approaches to Protein NMR Automation. Ph.D. thesis, University of Waterloo (2011)Google Scholar
  4. 4.
    Alipanahi, B., Krislock, N., Ghodsi, A., Wolkowicz, H., Donaldson, L., Li, M.: Determining protein structures from noesy distance constraints by semidefinite programming. Journal of Computational Biology 20(4), 296–310 (2013)CrossRefMathSciNetGoogle 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 Journal on Scientific Computing 30(3), 1251–1277 (2008)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Bonnabel, S., Sepulchre, R.: Riemannian metric and geometric mean for positive semidefinite matrices of fixed rank. SIAM J. Matrix Anal. Appl. 31(3), 1055–1070 (2009), CrossRefMathSciNetGoogle Scholar
  7. 7.
    Burkowski, F.J.: Structural Bioinformatics: An Algorithmic Approach. Chapman & Hall/CRC (2009)Google Scholar
  8. 8.
    Edelman, A., Arias, T.A., Smith, S.T.: The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. Appl. 20(2), 303–353 (1998)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Fletcher, P.T., Joshi, S.: Principal geodesic analysis on symmetric spaces: Statistics of diffusion tensors. In: Sonka, M., Kakadiaris, I.A., Kybic, J. (eds.) CVAMIA-MMBIA 2004. LNCS, vol. 3117, pp. 87–98. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  10. 10.
    Fletcher, P.T., Joshi, S.: Riemannian geometry for the statistical analysis of diffusion tensor data. Signal Processing 87(2), 250–262 (2007)CrossRefMATHGoogle Scholar
  11. 11.
    Gerstein, M., Krebs, W.: A database of macromolecular motions. Nucleic Acids Res. 26, 4280–4290 (1998)CrossRefGoogle Scholar
  12. 12.
    Goh, A.: Riemannian manifold clustering and dimensionality reduction for vision-based analysis. In: Machine Learning for Vision-Based Motion Analysis, pp. 27–53. Springer (2011)Google Scholar
  13. 13.
    Goh, A., Vidal, R.: Clustering and dimensionality reduction on Riemannian manifolds. In: IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2008, pp. 1–7. IEEE (2008)Google Scholar
  14. 14.
    Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. Johns Hopkins University Press, Baltimore (1996)MATHGoogle Scholar
  15. 15.
    Journée, M., Bach, F., Absil, P.A., Sepulchre, R.: Low-rank optimization on the cone of positive semidefinite matrices. SIAM J. Optim. 20(5), 2327–2351 (2010)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Kim, M.K.: Elastic Network Models of Biomolecular Structure and Dynamics. Ph.D. thesis, The Johns Hopkins University (2004)Google Scholar
  17. 17.
    Kim, M.K., Jernigan, R.L., Chirikjian, G.S.: Efficient generation of feasible pathways for protein conformational transitions. Biophysical Journal 83(3), 1620–1630 (2002)CrossRefGoogle Scholar
  18. 18.
    Kim, M.K., Jernigan, R.L., Chirikjian, G.S.: Rigid-cluster models of conformational transitions in macromolecular machines and assemblies. Biophysical Journal 89(1), 43–55 (2005)CrossRefGoogle Scholar
  19. 19.
    Kleywegt, G.J., Jones, T.A.: Phi/psi-chology: Ramachandran revisited. Structure 4, 1395–1400 (1996)CrossRefGoogle Scholar
  20. 20.
    Krislock, N.: Semidefinite facial reduction for Low-Rank Euclidean Distance Matrix Completion. Ph.D. thesis, School of Computer Science, University of Waterloo (2010)Google Scholar
  21. 21.
    Krislock, N., Wolkowicz, H.: Explicit sensor network localization using semidefinite representations and facial reductions. SIAM Journal on Optimization 20(5), 2679–2708 (2010)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Meyer, G.: Geometric optimization algorithms for linear regression on fixed-rank matrices. Ph.D. thesis, University of Liège (2011)Google Scholar
  23. 23.
    Mishra, B., Meyer, G., Sepulchre, R.: Low-rank optimization for distance matrix completion. In: 2011 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC), Orlando, FL, USA, December 12-15, pp. 4455–4460 (2011)Google Scholar
  24. 24.
    Mishra, B., Meyer, G., Bach, F., Sepulchre, R.: Low-rank optimization with trace norm penalty. arXiv preprint arXiv:1112.2318 (2011)Google Scholar
  25. 25.
    Morris, A.L., MacArthur, M.W., Hutchinson, E.G., Thornton, J.M.: Stereochemical quality of protein structure coordinates. Proteins: Structure, Function, and Bioinformatics 12(4), 345–364 (1992)CrossRefGoogle Scholar
  26. 26.
    Pettersen, E.F., Goddard, T.D., Huang, C.C., Couch, G.S., Greenblatt, D.M., Meng, E.C., Ferrin, T.E.: UCSF Chimera–a visualization system for exploratory research and analysis. J. Comp. Chem. 25(13), 1605–1612 (2004)CrossRefGoogle Scholar
  27. 27.
    Teodoro, M.L., Phillips Jr., G.N., Kavraki, L.E.: Understanding protein flexibility through dimensionality reduction. Journal of Computational Biology 10(3-4), 617–634 (2003)CrossRefGoogle Scholar
  28. 28.
    Vandereycken, B.: Riemannian and multilevel optimization for rank-constrained matrix problems. Ph.D. thesis, Department of Computer Science, KU Leuven (2010)Google Scholar
  29. 29.
    Vonrhein, C., Schlauderer, G.J., Schulz, G.E.: Movie of the structural changes during a catalytic cycle of nucleoside monophosphate kinases. Structure 3, 483–490 (1995)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Xiao-Bo Li
    • 1
  • Forbes J. Burkowski
    • 1
  1. 1.University of WaterlooCanada

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