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A Euclidean distance matrix model for protein molecular conformation

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Abstract

Protein molecular conformation is an important and challenging problem in biophysics. It is to recover the structure of proteins based on limited information such as noised distances, lower and upper bounds on some distances between atoms. In this paper, based on the recent progress in numerical algorithms for Euclidean distance matrix (EDM) optimization problems, we propose a EDM model for protein molecular conformation. We reformulate the problem as a rank-constrained least squares problem with linear equality constraints, box constraints, as well as a cone constraint. Due to the nonconvexity of the problem, we develop a majorized penalty approach to solve the problem. We apply the accelerated block coordinate descent algorithm proposed in Sun et al. (SIAM J Optim 26(2):1072–1100, 2016) to solve the resulting subproblem. Extensive numerical results demonstrate the efficiency of the proposed model.

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References

  1. Alfakih, A.Y., Wolkowicz, H.: Euclidean Distance Matrices and the Molecular Conformation Problem. Faculty of Mathematics, University of Waterloo, Waterloo (2002)

    MATH  Google Scholar 

  2. 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(3), 1251–1277 (2008)

    Article  MathSciNet  Google Scholar 

  3. Biswas, P., Liang, T.C., Toh, K.C., Ye, Y., Wang, T.C.: Semidefinite programming approaches for sensor network localization with noisy distance measurements. IEEE Trans. Autom. Sci. Eng. 3(4), 360–371 (2006)

    Article  Google Scholar 

  4. Biswas, P., Ye, Y.: A distributed method for solving semidefinite programs arising from ad hoc wireless sensor network localization. In: Hager, W.W., Pardalos, P.M., Huang, S.-J. (eds.) Multiscale Optimization Methods and Applications, pp. 69–84. Springer, Boston (2006)

    Chapter  Google Scholar 

  5. Cao, M.Z., Li, Q.N.: An ordinal weighted EDM model for nonmetric multidimensional scaling: an application to image ranking. Technical report, Beijing Institute of Technology (2018)

  6. Cox, T.F., Cox, M.A.A.: Multidimensional Scaling, 2nd edn. Chapman and hall/CRC, Boca Raton (2000)

    Book  Google Scholar 

  7. Ding, C., Qi, H.D.: Convex Euclidean distance embedding for collaborative position localization with NLOS mitigation. Comput. Optim. Appl. 66(1), 187–218 (2017)

    Article  MathSciNet  Google Scholar 

  8. Ding, C., Qi, H.D.: Convex optimization learning of faithful Euclidean distance representations in nonlinear dimensionality reduction. Math. Program. 164(1–2), 341–381 (2017)

    Article  MathSciNet  Google Scholar 

  9. Dattorro, J.: Convex Optimization and Euclidean Distance Geometry. Meboo publish Google Scholar, Palo Alto (2005)

    MATH  Google Scholar 

  10. Dokmanic, I., Parhizkar, R., Ranieri, J., Vetterli, M.: Euclidean distance matrices: essential theory, algorithms, and applications. IEEE Signal Process. Mag. 32(6), 12–30 (2015)

    Article  Google Scholar 

  11. Dai, Y.J., Yao, Z.Q., Li, Q.N., Xie, D.: Innovative posture sensing method for large engineering manipulators based on nearest Euclidean distance matrix. Technical report, Xiangtan University (2018)

  12. Fang, X.Y., Toh, K.C.: Using a Distributed SDP Approach to Solve Simulated Protein Molecular Conformation Problems. Distance Geometry, pp. 351–376. Springer, New York (2013)

    MATH  Google Scholar 

  13. Gansner, E.R., Hu, Y., North, S.: A maxent-stress model for graph layout. IEEE Trans. Vis. Comput. Gr. 19(6), 927–940 (2013)

    Article  Google Scholar 

  14. Gao, Y.: Structured low rank matrix optimization problems: a penalized approach. PhD thesis, National University of Singapore, August (2010)

  15. Jiang, K.F., Sun, D.F., Toh, K.C.: A partial proximal point algorithm for nuclear norm regularized matrix least squares problems. Math. Program. Comput. 6, 281–325 (2014)

    Article  MathSciNet  Google Scholar 

  16. Huang, H.X., Liang, Z.A., Pardalos, P.M.: Some properties for the Euclidean distance matrix and positive semidefinite matrix completion problems. J. Glob. Optim. 25(1), 3–21 (2003)

    Article  MathSciNet  Google Scholar 

  17. Hoai An, L.T.: Solving large scale molecular distance geometry problems by a smoothing technique via the Gaussian transform and D.C. programming. J. Glob. Optim. 27(4), 375–397 (2003)

    Article  MathSciNet  Google Scholar 

  18. Kelder, D.D., Gabriëlle, M.: Distance geometry and molecular conformation. Trends Pharmacol. Sci. 10(4), 164 (1988)

    Article  Google Scholar 

  19. Lavor, C., Alves, R., Figueiredo, W., Petraglia, A., Maculan, N.: Clifford algebra and the discretizable molecular distance geometry problem. Adv. Appl. Clifford Algebras 25(4), 925–942 (2015)

    Article  MathSciNet  Google Scholar 

  20. Li, Q.N., Li, D.H.: A projected semismooth Newton method for problems of calibrating least squares covariance matrix. Oper. Res. Lett. 39(2), 103–108 (2011)

    Article  MathSciNet  Google Scholar 

  21. 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)

    Article  MathSciNet  Google Scholar 

  22. Li, Q.N., Qi, H.D.: An inexact smoothing Newton method for Euclidean distance matrix optimization under ordinal constraints. J. Comput. Math. 35(4), 467–483 (2017)

    MathSciNet  MATH  Google Scholar 

  23. 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(6), 4351–4372 (2009)

    Article  MathSciNet  Google Scholar 

  24. Liberti, L., Lavor, C., Maculan, N., Mucherino, A.: Euclidean distance geometry and applications. Siam Rev. 56(1), 3–69 (2014)

    Article  MathSciNet  Google Scholar 

  25. Li, X.D., Sun, D.F., Toh, K.C.: QSDPNAL: a two-phase proximal augmented Lagrangian method for convex quadratic semidefinite programming. Math. Program. Comput. 10, 1–41 (2018)

    Article  MathSciNet  Google Scholar 

  26. Li, X.D., Sun, D.F., Toh, K.C.: A block symmetric Gauss–Seidel decomposition theo for convex composite quadratic programming and its applications. Math. Program. 16, 1–24 (2017)

    Google Scholar 

  27. Moré, J.J., Wu, Z.: Distance geometry optimization for protein structures. J. Glob. Optim. 15(3), 219–234 (1999)

    Article  MathSciNet  Google Scholar 

  28. Qi, H.D.: A semismooth Newton method for the nearest Euclidean distance matrix problem. SIAM J. Matrix Anal. Appl. 34(1), 67–93 (2013)

    Article  MathSciNet  Google Scholar 

  29. Qi, H.D., Xiu, N.H., Yuan, X.M.: A lagrangian dual approach to the single-source localization problem. IEEE Trans. Signal Process. 61(15), 3815–3826 (2013)

    Article  MathSciNet  Google Scholar 

  30. Qi, H.D., Yuan, X.M.: Computing the nearest Euclidean distance matrix with low embedding dimensions. Math. program. 147(1–2), 351–389 (2014)

    Article  MathSciNet  Google Scholar 

  31. Schoenberg, I.J.: Remarks to Maurice Fréchet’s article “Sur la définition axiomatique d’une classe d’espaces distanciés vectoriellement applicable sur l’espace de Hilbert”. Ann. Math. 36(3), 724–732 (1935)

    Article  MathSciNet  Google Scholar 

  32. Sun, D.F., Toh, K.C., Yang, L.Q.: An efficient inexact ABCD method for least squares semidefinite programming. SIAM J. Optim. 26(2), 1072–1100 (2016)

    Article  MathSciNet  Google Scholar 

  33. Toh, K.C.: An inexact primal–dual path-following algorithm for convex quadratic SDP. Math. Program. 112(1), 221–254 (2008)

    Article  MathSciNet  Google Scholar 

  34. Toh, K.C.: User guide for QSDP-0—a Matlab software package for convex quadratic semidefinite programming. Technical report, Department of Mathematics, National University of Singapore, Singapore (2010)

  35. Tütüncü, R.H., Toh, K.C., Todd, M.J.: Solving semidefinite-quadratic-linear programs using SDPT3. Math. Program. Ser. B 95(2), 189–217 (2003)

    Article  MathSciNet  Google Scholar 

  36. Wegner, M., Taubert, O., Schug, A., Meyerhenke, H.: Maxent-stress optimization of 3D biomolecular models. arXiv preprint arXiv:1706.06805 (2017)

  37. Young, G., Householder, A.S.: Discussion of a set of points in terms of their mutual distances. Psychometrika 3(1), 19–22 (1938)

    Article  Google Scholar 

  38. Yu, P.P., Li, Q.N.: Ordinal distance metric learning with MDS for image ranking. Asia Pac. J. Oper. Res. 35(1), 1850007 (2018)

    Article  MathSciNet  Google Scholar 

  39. Zou, Z., Bird, R.H., Schnabel, R.B.: A stochastic/perturbation global optimization algorithm for distance geometry problems. J. Glob. Optim. 11(1), 91–105 (1997)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, China

    Fengzhen Zhai

  2. School of Mathematics and Statistics/Beijing Key Laboratory on MCAACI, Beijing Institute of Technology, Beijing, 100081, China

    Qingna Li

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  1. Fengzhen Zhai
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  2. Qingna Li
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Correspondence to Qingna Li.

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This author’s research is supported by National Science Foundation of China (No. 11671036).

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Zhai, F., Li, Q. A Euclidean distance matrix model for protein molecular conformation. J Glob Optim 76, 709–728 (2020). https://doi.org/10.1007/s10898-019-00771-4

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  • DOI: https://doi.org/10.1007/s10898-019-00771-4

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