Dirichlet Priors for MAP Inference of Protein Conformation Abundances from SAXS

  • A. Emre Onuk
  • Murat Akcakaya
  • Jaydeep Bardhan
  • Deniz Erdogmus
  • Dana H. Brooks
  • Lee Makowski
Article
  • 97 Downloads

Abstract

Estimation of mixture coefficients of protein conformations in solution find applications in understanding protein behavior. We describe a method for maximum a posteriori (MAP) estimation of the mixture coefficients of ensemble of conformations in a protein mixture solution using measured small angle X-ray scattering (SAXS) intensities. The proposed method builds upon a model for the measurements of crystallographically determined conformations. Assuming that a priori information on the protein mixture is available, and that priori information follows a Dirichlet distribution, we develop a method to estimate the relative abundances with MAP estimator. The Dirichlet distribution depends on concentration parameters which may not be known in practice and thus need to be estimated. To estimate these unknown concentration parameters we developed an expectation-maximization (EM) method. Adenylate kinase (ADK) protein was selected as the test bed due to its known conformations Beckstein et al. (Journal of Molecular Biology, 394(1), 160 1). Known conformations are assumed to form the full vector bases that span the measurement space. In Monte Carlo simulations, mixture coefficient estimation performances of MAP and maximum likelihood (ML) (which assumes a uniform prior on the mixture coefficients) estimators are compared. MAP estimators using known and unknown concentration parameters are also compared in terms of estimation performances. The results show that prior knowledge improves estimation accuracy, but performance is sensitive to perturbations in the Dirichlet distribution’s concentration parameters. Moreover, the estimation method based on EM algorithm shows comparable results to approximately known prior parameters.

Keywords

SAXS intensity Bayesian estimation Expectation-maximization Dirichlet prior ML estimation Adenylate kinase 

References

  1. 1.
    Beckstein, O., Denning, E.J., Perilla, J.R., & Woolf, T.B. (2009). Journal of Molecular Biology, 394 (1), 160.CrossRefGoogle Scholar
  2. 2.
    Putnam, C. D., Hammel, M., Hura, G. L., & Tainer, J. A. (2007). Quarterly Reviews of Biophysics, 40(03), 191.CrossRefGoogle Scholar
  3. 3.
    Konarev, P. V., Volkov, V. V., Sokolova, A. V., Koch, M. H., & Svergun, D. I. (2003). Journal of Applied Crystallography, 36(5), 1277.CrossRefGoogle Scholar
  4. 4.
    Onuk, A. E., Akcakaya, M., Bardhan, J. P., Erdogmus, D., Brooks, D. H., & Makowski, L. (2015). IEEE Transactions on Signal Processing, 63(20), 5383.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Onuk, A.E., Akcakaya, M., Bardhan, J., Erdogmus, D., Brooks, D.H., & Makowski, L. Machine learning for signal processing (MLSP). In 2015 IEEE 25th International Workshop on (IEEE, 2015) (pp. 1–5).Google Scholar
  6. 6.
    Makowski, L., Rodi, D. J., Mandava, S., Minh, D. D., Gore, D. B., & Fischetti, R. F. (2008). Journal of Molecular Biology, 375(2), 529.CrossRefGoogle Scholar
  7. 7.
    Duda, R. O., Hart, P. E., & Stork, D.G. (2012). Pattern classification: Wiley.Google Scholar
  8. 8.
    Stoica, P., & Selen, Y. (2004). Signal processing magazine. IEEE, 21(4), 36.Google Scholar
  9. 9.
    Minka, T. (2000). Estimating a dirichlet distribution.Google Scholar
  10. 10.
    Dempster, A. P., Laird, N. M., & Rubin, D. B. (1977). Journal of the Royal Statistical Society. Series B (methodological), 1–38.Google Scholar
  11. 11.
    Moon, T. K. (1996). Signal processing magazine. IEEE, 13(6), 47.Google Scholar
  12. 12.
    Bishop, C.M. (2006). Pattern recognition and machine learning: Springer.Google Scholar
  13. 13.
    Neumann, J., & Taub, A. (1961). Collected works Vol. V. New York: Pergamon.Google Scholar
  14. 14.
    Chen, Y. (2005). Statistics & Probability Letters, 72(4), 277.Google Scholar
  15. 15.
    Svergun, D., Barberato, C., & Koch, M. (1995). Journal of Applied Crystallography, 28(6), 768.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • A. Emre Onuk
    • 1
  • Murat Akcakaya
    • 2
  • Jaydeep Bardhan
    • 3
  • Deniz Erdogmus
    • 1
  • Dana H. Brooks
    • 1
  • Lee Makowski
    • 4
  1. 1.Electrical and Computer Engineering DepartmentNortheastern UniversityBostonUSA
  2. 2.Electrical and Computer Engineering DepartmentUniversity of PittsburghPittsburghUSA
  3. 3.Mechanical and Industrial EngineeringBostonUSA
  4. 4.Chemistry and Chemical BiologyNortheastern UniversityBostonUSA

Personalised recommendations