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.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Alfakih, A.Y., Wolkowicz, H.: Euclidean Distance Matrices and the Molecular Conformation Problem. Faculty of Mathematics, University of Waterloo, Waterloo (2002)
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)
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)
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)
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)
Cox, T.F., Cox, M.A.A.: Multidimensional Scaling, 2nd edn. Chapman and hall/CRC, Boca Raton (2000)
Ding, C., Qi, H.D.: Convex Euclidean distance embedding for collaborative position localization with NLOS mitigation. Comput. Optim. Appl. 66(1), 187–218 (2017)
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)
Dattorro, J.: Convex Optimization and Euclidean Distance Geometry. Meboo publish Google Scholar, Palo Alto (2005)
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)
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)
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)
Gansner, E.R., Hu, Y., North, S.: A maxent-stress model for graph layout. IEEE Trans. Vis. Comput. Gr. 19(6), 927–940 (2013)
Gao, Y.: Structured low rank matrix optimization problems: a penalized approach. PhD thesis, National University of Singapore, August (2010)
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)
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)
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)
Kelder, D.D., Gabriëlle, M.: Distance geometry and molecular conformation. Trends Pharmacol. Sci. 10(4), 164 (1988)
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)
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)
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)
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)
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)
Liberti, L., Lavor, C., Maculan, N., Mucherino, A.: Euclidean distance geometry and applications. Siam Rev. 56(1), 3–69 (2014)
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)
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)
Moré, J.J., Wu, Z.: Distance geometry optimization for protein structures. J. Glob. Optim. 15(3), 219–234 (1999)
Qi, H.D.: A semismooth Newton method for the nearest Euclidean distance matrix problem. SIAM J. Matrix Anal. Appl. 34(1), 67–93 (2013)
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)
Qi, H.D., Yuan, X.M.: Computing the nearest Euclidean distance matrix with low embedding dimensions. Math. program. 147(1–2), 351–389 (2014)
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)
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)
Toh, K.C.: An inexact primal–dual path-following algorithm for convex quadratic SDP. Math. Program. 112(1), 221–254 (2008)
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)
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)
Wegner, M., Taubert, O., Schug, A., Meyerhenke, H.: Maxent-stress optimization of 3D biomolecular models. arXiv preprint arXiv:1706.06805 (2017)
Young, G., Householder, A.S.: Discussion of a set of points in terms of their mutual distances. Psychometrika 3(1), 19–22 (1938)
Yu, P.P., Li, Q.N.: Ordinal distance metric learning with MDS for image ranking. Asia Pac. J. Oper. Res. 35(1), 1850007 (2018)
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)
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This author’s research is supported by National Science Foundation of China (No. 11671036).
About this article
Cite this article
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
- Protein molecular conformation
- Euclidean distance matrix
- Accelerated block coordinate descent method
- Majorized penalty approach