VAMPnets for deep learning of molecular kinetics

Abstract

There is an increasing demand for computing the relevant structures, equilibria, and long-timescale kinetics of biomolecular processes, such as protein-drug binding, from high-throughput molecular dynamics simulations. Current methods employ transformation of simulated coordinates into structural features, dimension reduction, clustering the dimension-reduced data, and estimation of a Markov state model or related model of the interconversion rates between molecular structures. This handcrafted approach demands a substantial amount of modeling expertise, as poor decisions at any step will lead to large modeling errors. Here we employ the variational approach for Markov processes (VAMP) to develop a deep learning framework for molecular kinetics using neural networks, dubbed VAMPnets. A VAMPnet encodes the entire mapping from molecular coordinates to Markov states, thus combining the whole data processing pipeline in a single end-to-end framework. Our method performs equally or better than state-of-the-art Markov modeling methods and provides easily interpretable few-state kinetic models.

Full text

Its licence does not allow us to show the full text. Read it at PubMed Central:

References 84

1. 1.

   Lindorff-Larsen K, Piana S, Dror RO, Shaw DE. How fast-folding proteins fold. Science. 2011;334:517-520.

   Not in our library10.1126/science.1208351 (opens in a new tab)

2. 2.

   Plattner N, Doerr S, Fabritiis GD, Noé F. Complete protein-protein association kinetics in atomic detail revealed by molecular dynamics simulations and Markov modelling. Nat. Chem. 2017;9:1005-1011.

   Not in our library10.1038/nchem.2785 (opens in a new tab)

3. 3.

   Kohlhoff KJ, et al. Cloud-based simulations on google exacycle reveal ligand modulation of gpcr activation pathways. Nat. Chem. 2014;6:15-21.

   Not in our library10.1038/nchem.1821 (opens in a new tab)

4. 4.

   Doerr S, Harvey MJ, Noé F, Fabritiis GD. HTMD: high-throughput molecular dynamics for molecular discovery. J. Chem. Theory Comput. 2016;12:1845-1852.

   Not in our library10.1021/acs.jctc.6b00049 (opens in a new tab)

5. 5.

   Ufimtsev IS, Martinez TJ. Graphical processing units for quantum chemistry. Comp. Sci. Eng. 2008;10:26-34.

   Not in our library10.1109/mcse.2008.148 (opens in a new tab)

6. 6.

   Marx, D. & Hutter, J. in Modern Methods and Algorithms of Quantum Chemistry ofNIC Series Vol. 1, 301–449 Ab initio molecular dynamics: theory and implementation (ed Grotendorst, J.) (John von Neumann Institute for Computing, Jülich, 2000).

   Not in our library

7. 7.

   Schütte C, Fischer A, Huisinga W, Deuflhard P. A direct approach to conformational dynamics based on hybrid monte carlo. J. Comput. Phys. 1999;151:146-168.

   Not in our library10.1006/jcph.1999.6231 (opens in a new tab)

8. 8.

   Prinz JH, et al. Markov models of molecular kinetics: generation and validation. J. Chem. Phys. 2011;134:174105.

   Not in our library10.1063/1.3565032 (opens in a new tab)

9. 9.

   Swope WC, Pitera JW, Suits F. Describing protein folding kinetics by molecular dynamics simulations: 1. Theory J. Phys. Chem. B. 2004;108:6571-6581.

   Not in our library10.1021/jp037421y (opens in a new tab)

10. 10.

    Noé F, Horenko I, Schütte C, Smith JC. Hierarchical analysis of conformational dynamics in biomolecules: transition networks of metastable states. J. Chem. Phys. 2007;126:155102.

    Not in our library10.1063/1.2714539 (opens in a new tab)

11. 11.

    Chodera JD, et al. Automatic discovery of metastable states for the construction of Markov models of macromolecular conformational dynamics. J. Chem. Phys. 2007;126:155101.

    Not in our library10.1063/1.2714538 (opens in a new tab)

12. 12.

    Buchete NV, Hummer G. Coarse master equations for peptide folding dynamics. J. Phys. Chem. B. 2008;112:6057-6069.

    Not in our library10.1021/jp0761665 (opens in a new tab)

13. 13.

    Scherer MK, et al. PyEMMA 2: a software package for estimation, validation and analysis of Markov models. J. Chem. Theory Comput. 2015;11:5525-5542.

    Not in our library10.1021/acs.jctc.5b00743 (opens in a new tab)

14. 14.

    Harrigan MP, et al. Msmbuilder: statistical models for biomolecular dynamics. Biophys. J. 2017;112:10-15.

    Not in our library10.1016/j.bpj.2016.10.042 (opens in a new tab)

15. 15.

    Humphrey W, Dalke A, Schulten K. Vmd—visual molecular dynamics. J. Mol. Graph. 1996;14:33-38.

    Not in our library10.1016/0263-7855(96)00018-5 (opens in a new tab)

16. 16.

    McGibbon RT, et al. Mdtraj: a modern open library for the analysis of molecular dynamics trajectories. Biophys. J. 2015;109:1528-1532.

    Not in our library10.1016/j.bpj.2015.08.015 (opens in a new tab)

17. 17.

    Noé F, Nüske F. A variational approach to modeling slow processes in stochastic dynamical systems. Multiscale Model. Simul. 2013;11:635-655.

    Not in our library10.1137/110858616 (opens in a new tab)

18. 18.

    Nüske F, Keller BG, Pérez-Hernández G, Mey ASJS, Noé F. Variational approach to molecular kinetics. J. Chem. Theory Comput. 2014;10:1739-1752.

    Not in our library10.1021/ct4009156 (opens in a new tab)

19. 19.

    Perez-Hernandez G, Paul F, Giorgino T, D Fabritiis G, Noé F. Identification of slow molecular order parameters for Markov model construction. J. Chem. Phys. 2013;139:015102.

    Not in our library10.1063/1.4811489 (opens in a new tab)

20. 20.

    Schwantes CR, Pande VS. Improvements in Markov state model construction reveal many non-native interactions in the folding of ntl9. J. Chem. Theory Comput. 2013;9:2000-2009.

    Not in our library10.1021/ct300878a (opens in a new tab)

21. 21.

    Molgedey L, Schuster HG. Separation of a mixture of independent signals using time delayed correlations. Phys. Rev. Lett. 1994;72:3634-3637.

    Not in our library10.1103/physrevlett.72.3634 (opens in a new tab)

22. 22.

    Ziehe Andreas, Müller Klaus-Robert. TDSEP — an efficient algorithm for blind separation using time structure. ICANN 98. 1998;675-680.

    Not in our library

23. 23.

    Harmeling S, Ziehe A, Kawanabe M, Müller KR. Kernel-based nonlinear blind source separation. Neural Comput. 2003;15:1089-1124.

    Not in our library10.1162/089976603765202677 (opens in a new tab)

24. 24.

    Mezić I. Spectral properties of dynamical systems, model reduction and decompositions. Nonlinear Dynam. 2005;41:309-325.

    Not in our library10.1007/s11071-005-2824-x (opens in a new tab)

25. 25.

    Schmid, P. J. & Sesterhenn, J. Dynamic mode decomposition of numerical and experimental data. In 61st Annual Meeting of the APS Division of Fluid Dynamics (American Physical Society, 2008).

    Not in our library

26. 26.

    Tu JH, Rowley CW, Luchtenburg DM, Brunton SL, Kutz JN. On dynamic mode decomposition: theory and applications. J. Comput. Dyn. 2014;1:391-421.

    Not in our library10.3934/jcd.2014.1.391 (opens in a new tab)

27. 27.

    Williams MO, Kevrekidis IG, Rowley CW. A data-driven approximation of the koopman operator: extending dynamic mode decomposition. J. Nonlinear Sci. 2015;25:1307-1346.

    Not in our library10.1007/s00332-015-9258-5 (opens in a new tab)

28. 28.

    Wu H, et al. Variational koopman models: slow collective variables and molecular kinetics from short off-equilibrium simulations. J. Chem. Phys. 2017;146:154104.

    Not in our library10.1063/1.4979344 (opens in a new tab)

29. 29.

    Noé F, Clementi C. Collective variables for the study of long-time kinetics from molecular trajectories: theory and methods. Curr. Opin. Struc. Biol. 2017;43:141-147.

    Not in our library10.1016/j.sbi.2017.02.006 (opens in a new tab)

30. 30.

    Klus, S. et al. Data-driven model reduction and transfer operator approximation. Preprint at http://arXiv:1703.10112 (2017).

    Not in our library

31. 31.

    Noé F, Clementi C. Kinetic distance and kinetic maps from molecular dynamics simulation. J. Chem. Theory Comput. 2015;11:5002-5011.

    Not in our library10.1021/acs.jctc.5b00553 (opens in a new tab)

32. 32.

    Noé F, Banisch R, Clementi C. Commute maps: separating slowly-mixing molecular configurations for kinetic modeling. J. Chem. Theory Comput. 2016;12:5620-5630.

    Not in our library10.1021/acs.jctc.6b00762 (opens in a new tab)

33. 33.

    Bowman, G. R., Pande, V. S. & Noé, F. in Advances in Experimental Medicine and Biology Vol. 797 An introduction to Markov state models and their application to long timescale molecular simulation (Springer, Heidelberg, 2014).

    Not in our library

34. 34.

    Husic BE, Pande VS. Ward clustering improves cross-validated Markov state models of protein folding. J. Chem. Theory Comput. 2017;13:963-967.

    Not in our library10.1021/acs.jctc.6b01238 (opens in a new tab)

35. 35.

    Sheong FK, Silva DA, Meng L, Zhao Y, Huang X. Automatic state partitioning for multibody systems (APM): an efficient algorithm for constructing Markov state models to elucidate conformational dynamics of multibody systems. J. Chem. Theory Comput. 2015;11:17-27.

    Not in our library10.1021/ct5007168 (opens in a new tab)

36. 36.

    Wu H, Noé F. Gaussian Markov transition models of molecular kinetics. J. Chem. Phys. 2015;142:084104.

    Not in our library10.1063/1.4913214 (opens in a new tab)

37. 37.

    Weber M, Fackeldey K, Schütte C. Set-free Markov state model building. J. Chem. Phys. 2017;146:124133.

    Not in our library10.1063/1.4978501 (opens in a new tab)

38. 38.

    Bowman GR, Beauchamp KA, Boxer G, Pande VS. Progress and challenges in the automated construction of Markov state models for full protein systems. J. Chem. Phys. 2009;131:124101.

    Not in our library10.1063/1.3216567 (opens in a new tab)

39. 39.

    Trendelkamp-Schroer B, Wu H, Paul F, Noé F. Estimation and uncertainty of reversible Markov models. J. Chem. Phys. 2015;143:174101.

    Not in our library10.1063/1.4934536 (opens in a new tab)

40. 40.

    Kube S, Weber M. A coarse graining method for the identification of transition rates between molecular conformations. J. Chem. Phys. 2007;126:024103.

    Not in our library10.1063/1.2404953 (opens in a new tab)

41. 41.

    Yao Y, et al. Hierarchical nyström methods for constructing Markov state models for conformational dynamics. J. Chem. Phys. 2013;138:174106.

    Not in our library10.1063/1.4802007 (opens in a new tab)

42. 42.

    Fackeldey K, Weber M. Genpcca—Markov state models for non-equilibrium steady states. WIAS Rep. 2017;29:70-80.

    Not in our library

43. 43.

    Gerber S, Horenko I. Toward a direct and scalable identification of reduced models for categorical processes. Proc. Natl Acad. Sci. USA. 2017;114:4863-4868.

    Not in our library10.1073/pnas.1612619114 (opens in a new tab)

44. 44.

    Hummer G, Szabo A. Optimal dimensionality reduction of multistate kinetic and Markov-state models. J. Phys. Chem. B. 2015;119:9029-9037.

    Not in our library10.1021/jp508375q (opens in a new tab)

45. 45.

    Orioli S, Faccioli P. Dimensional reduction of Markov state models from renormalization group theory. J. Chem. Phys. 2016;145:124120.

    Not in our library10.1063/1.4963196 (opens in a new tab)

46. 46.

    Noé F, Wu H, Prinz JH, Plattner N. Projected and hidden Markov models for calculating kinetics and metastable states of complex molecules. J. Chem. Phys. 2013;139:184114.

    Not in our library10.1063/1.4828816 (opens in a new tab)

47. 47.

    Wu, H. & Noé, F. Variational approach for learning Markov processes from time series data. Preprint at http://arXiv:1707.04659 (2017).

    Not in our library

48. 48.

    McGibbon RT, Pande VS. Variational cross-validation of slow dynamical modes in molecular kinetics. J. Chem. Phys. 2015;142:124105.

    Not in our library10.1063/1.4916292 (opens in a new tab)

49. 49.
    Deep learning · 2015

    No notice
50. 50.

    Krizhevsky, A., Sutskever, I. & Hinton, G. E. Imagenet classification with deep convolutional neural networks. In NIPS'12 Proceedings of the 25th International Conference on Neural Information Processing Systems Vol. 1, 1097–1105 (Curran Associates, Inc., 2012).

    Not in our library

51. 51.

    Mnih V, et al. Human-level control through deep reinforcement learning. Nature. 2015;518:529-533.

    Not in our library10.1038/nature14236 (opens in a new tab)

52. 52.

    Perez-Hernandez G, Noé F. Hierarchical time-lagged independent component analysis: computing slow modes and reaction coordinates for large molecular systems. J. Chem. Theory Comput. 2016;12:6118-6129.

    Not in our library10.1021/acs.jctc.6b00738 (opens in a new tab)

53. 53.

    Nüske F, Schneider R, Vitalini F, Noé F. Variational tensor approach for approximating the rare-event kinetics of macromolecular systems. J. Chem. Phys. 2016;144:054105.

    Not in our library10.1063/1.4940774 (opens in a new tab)

54. 54.

    Koopman B. Hamiltonian systems and transformations in hilbert space. Proc. Natl Acad. Sci. USA. 1931;17:315-318.

    Not in our library10.1073/pnas.17.5.315 (opens in a new tab)

55. 55.

    Knoch F, Speck T. Cycle representatives for the coarse-graining of systems driven into a non-equilibrium steady state. New J. Phys. 2015;17:115004.

    Not in our library10.1088/1367-2630/17/11/115004 (opens in a new tab)

56. 56.

    Wang H, Schütte C. Building Markov state models for periodically driven non-equilibrium systems. J. Chem. Theory Comput. 2015;11:1819-1831.

    Not in our library10.1021/ct500997y (opens in a new tab)

57. 57.

    Horenko I, Hartmann C, Schütte C, Noé F. Data-based parameter estimation of generalized multidimensional Langevin processes. Phys. Rev. E. 2007;76:016706.

    Not in our library10.1103/physreve.76.016706 (opens in a new tab)

58. 58.

    Cybenko G. Approximation by superpositions of a sigmoidal function. Math. Control Signals. 1989;2:303-314.

    Not in our library10.1007/bf02551274 (opens in a new tab)

59. 59.

    Eigen, D., Rolfe, J., Fergus, R. & LeCun, Y. Understanding deep architectures using a recursive convolutional network. Preprint at http://arXiv:1312.1847 (2014).

    Not in our library

60. 60.

    Ranzato, M., Poultney, C., Chopra, S. & LeCun, Y. in Advances in Neural Information Processing Systems 18 (ed Platt, J. et al.) Efficient learning of sparse representations with an energy-based model (MIT Press, 2006).

    Not in our library

61. 61.

    Bengio, Y., Lamblin, P., Popovici, D. & Larochelle, H. in Advances in Neural Information Processing Systems 19, Vol. 19, 153 Greedy layer-wise training of deep networks (MIT Press, 2007).

    Not in our library

62. 62.

    Galen, A., Arora, R., Bilmes, J. & Livescu, K. Deep canonical correlation analysis. In ICML'13 Proceedings of the 30th International Conference on International Conference on Machine Learning Vol. 28, III-1247–III-1255 (2013).

    Not in our library

63. 63.

    Röblitz S, Weber M. Fuzzy spectral clustering by PCCA+: application to Markov state models and data classification. Adv. Data Anal. Classif. 2013;7:147-179.

    Not in our library10.1007/s11634-013-0134-6 (opens in a new tab)

64. 64.

    Sarich M, Noé F, Schütte C. On the approximation quality of Markov state models. Multiscale Model. Simul. 2010;8:1154-1177.

    Not in our library10.1137/090764049 (opens in a new tab)

65. 65.

    Noé F, Schütte C, Vanden-Eijnden E, Reich L, Weikl TR. Constructing the full ensemble of folding pathways from short off-equilibrium simulations. Proc. Natl Acad. Sci. USA. 2009;106:19011-19016.

    Not in our library10.1073/pnas.0905466106 (opens in a new tab)

66. 66.

    Hahnloser RLT. On the piecewise analysis of networks of linear threshold neurons. Neural Netw. 1998;11:691-697.

    Not in our library10.1016/s0893-6080(98)00012-4 (opens in a new tab)

67. 67.

    Nair, V. & Hinton, G. E. Rectified linear units improve restricted Boltzmann machines. In Proceedings of the 27th International Conference on Machine Learning (ICML-10) Vol. 27, 807–814 (ACM, New York, 2010).

    Not in our library

68. 68.

    Kingma, D. P. & Ba, J. Adam: a method for stochastic optimization. Preprint at http://arXiv.org:1412.6980 (2014).

    Not in our library

69. 69.

    Nüske, F., Wu, H., Wehmeyer, C., Clementi, C. & Noé, F. Markov state models from short non-equilibrium simulations—analysis and correction of estimation bias. Preprint at http://arXiv:1701.01665 (2017).

    Not in our library

70. 70.

    Wu H, Paul F, Wehmeyer C, Noé F. Multiensemble Markov models of molecular thermodynamics and kinetics. Proc. Natl Acad. Sci. USA. 2016;113:E3221-E3230.

    Not in our library10.1073/pnas.1525092113 (opens in a new tab)

71. 71.

    Wu H, Mey ASJS, Rosta E, Noé F. Statistically optimal analysis of state-discretized trajectory data from multiple thermodynamic states. J. Chem. Phys. 2014;141:214106.

    Not in our library10.1063/1.4902240 (opens in a new tab)

72. 72.

    Chodera JD, Swope WC, Noé F, Prinz JH, Pande VS. Dynamical reweighting: improved estimates of dynamical properties from simulations at multiple temperatures. J. Phys. Chem. 2011;134:244107.

    Not in our library10.1063/1.3592152 (opens in a new tab)

73. 73.

    Prinz JH, et al. Optimal use of data in parallel tempering simulations for the construction of discrete-state Markov models of biomolecular dynamics. J. Chem. Phys. 2011;134:244108.

    Not in our library10.1063/1.3592153 (opens in a new tab)

74. 74.

    Rosta E, Hummer G. Free energies from dynamic weighted histogram analysis using unbiased Markov state model. J. Chem. Theory Comput. 2015;11:276-285.

    Not in our library10.1021/ct500719p (opens in a new tab)

75. 75.

    Mey ASJS, Wu H, Noé F. xTRAM: estimating equilibrium expectations from time-correlated simulation data at multiple thermodynamic states. Phys. Rev. X. 2014;4:041018.

    Not in our library

76. 76.

    Olsson S, Wu H, Paul F, Clementi C, Noé F. Combining experimental and simulation data of molecular processes via augmented Markov models. Proc. Natl Acad. Sci. USA. 2017;114:8265-8270.

    Not in our library10.1073/pnas.1704803114 (opens in a new tab)

77. 77.

    Hinrichs NS, Pande VS. Calculation of the distribution of eigenvalues and eigenvectors in Markovian state models for molecular dynamics. J. Chem. Phys. 2007;126:244101.

    Not in our library10.1063/1.2740261 (opens in a new tab)

78. 78.

    Noé F. Probability distributions of molecular observables computed from Markov models. J. Chem. Phys. 2008;128:244103.

    Not in our library10.1063/1.2916718 (opens in a new tab)

79. 79.

    Chodera JD, Noé F. Probability distributions of molecular observables computed from Markov models. ii: Uncertainties in observables and their time-evolution. J. Chem. Phys. 2010;133:105102.

    Not in our library10.1063/1.3463406 (opens in a new tab)

80. 80.

    LeCun Y, Bottou L, Bengio Y, Haffner P. Gradient-based learning applied to document recognition. Proc. IEEE. 1998;86:2278-2324.

    Not in our library10.1109/5.726791 (opens in a new tab)

81. 81.

    Schütt, K. T. et al. Moleculenet: a continuous-filter convolutional neural network for modeling quantum interactions. Preprint at http://arXiv:1706.08566 (2017).

    Not in our library

82. 82.

    Srivastava N, Hinton G, Krizhevsky A, Sutskever I, Salakhutdinov R. Dropout: a simple way to prevent neural networks from overfitting. J. Mach. Learn Res. 2014;15:1929-1958.

    Not in our library

83. 83.

    Chollet, F. et al. Keras. https://github.com/fchollet/keras (2015).

    Not in our library

84. 84.

    Abadi, M. et al. TensorFlow: large-scale machine learning on heterogeneous systems. Preprint at http://arXiv.org:1603.04467 (2015).

    Not in our library

Cited by papers we hold 12

• Direct generation of protein conformational ensembles via machine learning · 2023
  No notice
• Machine Learning-Guided Protein Engineering · 2023
  No notice
• Tetranucleosome Interactions Drive Chromatin Folding · 2021
  Not checked
• Deep learning the structural determinants of protein biochemical properties by comparing structural ensembles with DiffNets · 2021
  Not checked
• Unsupervised Learning Methods for Molecular Simulation Data · 2021
  No notice
• A Unified Sparse Optimization Framework to Learn Parsimonious Physics-Informed Models From Data · 2020
  Not checked
• Thermodynamics and Kinetics of Drug-Target Binding by Molecular Simulation · 2020
  No notice
• Quantum chemical accuracy from density functional approximations via machine learning · 2020
  No notice
• Data-driven discovery of coordinates and governing equations · 2019
  No notice
• Past–future information bottleneck for sampling molecular reaction coordinate simultaneously with thermodynamics and kinetics · 2019
  No notice
• Deep learning for universal linear embeddings of nonlinear dynamics · 2018
  No notice
• Towards exact molecular dynamics simulations with machine-learned force fields · 2018
  No notice

Questions about this paper

Reads

A person counts once a day per paper. We keep no cookie and no address: a reader is a hash that changes every day. Any month under five reads shows no number.