PyEMMA 2: A Software Package for Estimation, Validation, and Analysis of Markov Models

Journal of Chemical Theory and Computation

Volumn 11, Issue 11, 2015, Pages 5525-5542

  Scherer, Martin K ^a   Trendelkamp Schroer, Benjamin ^a   Paul, Fabian ^a   Pérez Hernández, Guillermo ^a   Hoffmann, Moritz ^a   Plattner, Nuria ^a   Wehmeyer, Christoph ^a   Prinz, Jan Hendrik ^a   Noé, Frank ^a  

^a FREIE UNIVERSITÄT BERLIN   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84946887423     PISSN: 15499618     EISSN: 15499626     Source Type: Journal    
DOI: 10.1021/acs.jctc.5b00743     Document Type: Article

References (134)

1. Shaw, D. E.; Maragakis, P.; Lindorff-Larsen, K.; Piana, S.; Dror, R.; Eastwood, M.; Bank, J.; Jumper, J.; Salmon, J.; Shan, Y.; Wriggers, W. Atomic-Level Characterization of the Structural Dynamics of Proteins Science 2010, 330, 341-346 10.1126/science.1187409
2. Buch, I.; Harvey, M. J.; Giorgino, T.; Anderson, D. P.; De Fabritiis, G. High-throughput all-atom molecular dynamics simulations using distributed computing J. Chem. Inf. Model. 2010, 50, 397 10.1021/ci900455r
3. Shirts, M.; Pande, V. S. Screen Savers of the World Unite! Science 2000, 290, 1903-1904 10.1126/science.290.5498.1903
4. Phillips, J. C.; Braun, R.; Wang, W.; Gumbart, J.; Tajkhorshid, E.; Villa, E.; Chipot, C.; Skeel, R. D.; Kalé, L.; Schulten, K. Scalable molecular dynamics with NAMD J. Comput. Chem. 2005, 26, 1781-1802 10.1002/jcc.20289
5. Pronk, S.; Páll, S.; Schulz, R.; Larsson, P.; Bjelkmar, P.; Apostolov, R.; Shirts, M. R.; Smith, J. C.; Kasson, P. M.; van der Spoel, D.; Hess, B.; Lindahl, E. GROMACS 4.5: a high-throughput and highly parallel open source molecular simulation toolkit Bioinformatics 2013, 29, 845-854 10.1093/bioinformatics/btt055
6. Pronk, S.; Pouya, I.; Lundborg, M.; Rotskoff, G.; Wesén, B.; Kasson, P. M.; Lindahl, E. Molecular Simulation Workflows as Parallel Algorithms: The Execution Engine of Copernicus, a Distributed High-Performance Computing Platform J. Chem. Theory Comput. 2015, 11, 2600-2608 10.1021/acs.jctc.5b00234
7. Harvey, M.; Giupponi, G.; Fabritiis, G. D. ACEMD: Accelerated molecular dynamics simulations in the microseconds timescale J. Chem. Theory Comput. 2009, 5, 1632 10.1021/ct9000685
8. Eastman, P. et al. OpenMM 4: A Reusable, Extensible, Hardware Independent Library for High Performance Molecular Simulation J. Chem. Theory Comput. 2013, 9, 461-469 10.1021/ct300857j
9. Salomon-Ferrer, R.; Götz, A. W.; Poole, D.; Le Grand, S.; Walker, R. C. Routine microsecond molecular dynamics simulations with AMBER-Part II: Particle Mesh Ewald J. Chem. Theory Comput. 2013, 9, 3878-3888 10.1021/ct400314y
10. 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 10.1006/jcph.1999.6231
11. Swope, W. C.; Pitera, J. W.; Suits, F. Describing protein folding kinetics by molecular dynamics simulations: 1. Theory J. Phys. Chem. B 2004, 108, 6571-6581 10.1021/jp037421y
12. Chekmarev, D. S.; Ishida, T.; Levy, R. M. Long-Time Conformational Transitions of Alanine Dipeptide in Aqueous Solution: Continuous and Discrete-State Kinetic Models J. Phys. Chem. B 2004, 108, 19487-19495 10.1021/jp048540w
13. Singhal, N.; Pande, V. S. Error analysis and efficient sampling in Markovian state models for molecular dynamics J. Chem. Phys. 2005, 123, 204909 10.1063/1.2116947
14. Sriraman, S.; Kevrekidis, I. G.; Hummer, G. Coarse Master Equation from Bayesian Analysis of Replica Molecular Dynamics Simulations J. Phys. Chem. B 2005, 109, 6479-6484 10.1021/jp046448u
15. Noé, F.; Horenko, I.; Schütte, C.; Smith, J. C. Hierarchical Analysis of Conformational Dynamics in Biomolecules: Transition Networks of Metastable States J. Chem. Phys. 2007, 126, 155102 10.1063/1.2714539
16. Chodera, J. D.; Singhal, N.; Pande, V. S.; Dill, K. A.; Swope, W. C. Automatic discovery of metastable states for the construction of Markov models of macromolecular conformational dynamics J. Chem. Phys. 2007, 126, 155101 10.1063/1.2714538
17. Noé, F. Probability Distributions of Molecular Observables computed from Markov Models J. Chem. Phys. 2008, 128, 244103 10.1063/1.2916718
18. Buchete, N. V.; Hummer, G. Coarse Master Equations for Peptide Folding Dynamics J. Phys. Chem. B 2008, 112, 6057-6069 10.1021/jp0761665
19. Pan, A. C.; Roux, B. Building Markov state models along pathways to determine free energies and rates of transitions J. Chem. Phys. 2008, 129, 064107 10.1063/1.2959573
20. Noé, F.; Schütte, C.; Vanden-Eijnden, E.; Reich, L.; Weikl, T. R. Constructing the Full Ensemble of Folding Pathways from Short Off-Equilibrium Simulations Proc. Natl. Acad. Sci. U. S. A. 2009, 106, 19011-19016 10.1073/pnas.0905466106
21. Bowman, G. R.; Beauchamp, K. A.; Boxer, G.; Pande, V. S. Progress and challenges in the automated construction of Markov state models for full protein systems J. Chem. Phys. 2009, 131, 124101 10.1063/1.3216567
22. Voelz, V. A.; Bowman, G. R.; Beauchamp, K. A.; Pande, V. S. Molecular Simulation of ab Initio Protein Folding for a Millisecond Folder NTL9 J. Am. Chem. Soc. 2010, 132, 1526-1528 10.1021/ja9090353
23. Bowman, G. R.; Pande, V. S. Protein folded states are kinetic hubs Proc. Natl. Acad. Sci. U. S. A. 2010, 107, 10890-10895 10.1073/pnas.1003962107
24. Voelz, V. A.; Jäger, M.; Zhu, L.; Yao, S.; Bakajin, O.; Weiss, S.; Lapidus, L. J.; Pande, V. S. Markov State Models of Millisecond Folder ACBP Reveals New Views of the Folding Reaction Biophys. J. 2011, 100, 515a 10.1016/j.bpj.2010.12.3015
25. Beauchamp, K. A.; McGibbon, R.; Lin, Y. S.; Pande, V. S. Simple few-state models reveal hidden complexity in protein folding Proc. Natl. Acad. Sci. U. S. A. 2012, 109, 17807-17813 10.1073/pnas.1201810109
26. Sborgi, L.; Verma, A.; Piana, S.; Lindorff-Larsen, K.; Cerminara, M.; Santiveri, C. M.; Shaw, D. E.; de Alba, E.; Muñoz, V. Interaction Networks in Protein Folding via Atomic-Resolution Experiments and Long-Time-Scale Molecular Dynamics Simulations J. Am. Chem. Soc. 2015, 137, 6506-6516 10.1021/jacs.5b02324
27. Held, M.; Metzner, P.; Prinz, J.-H.; Noé, F. Mechanisms of Protein-Ligand Association and Its Modulation by Protein Mutations Biophys. J. 2011, 100, 701-710 10.1016/j.bpj.2010.12.3699
28. Silva, D.-A.; Bowman, G. R.; Sosa-Peinado, A.; Huang, X. A Role for Both Conformational Selection and Induced Fit in Ligand Binding by the LAO Protein PLoS Comput. Biol. 2011, 7, e1002054 10.1371/journal.pcbi.1002054
29. Buch, I.; Giorgino, T.; De Fabritiis, G. Complete reconstruction of an enzyme-inhibitor binding process by molecular dynamics simulations Proc. Natl. Acad. Sci. U. S. A. 2011, 108, 10184-10189 10.1073/pnas.1103547108
30. Gu, S.; Silva, D.-A.; Meng, L.; Yue, A.; Huang, X. Quantitatively Characterizing the Ligand Binding Mechanisms of Choline Binding Protein Using Markov State Model Analysis PLoS Comput. Biol. 2014, 10, e1003767 10.1371/journal.pcbi.1003767
31. Plattner, N.; Noé, F. Protein conformational plasticity and complex ligand binding kinetics explored by atomistic simulations and Markov models Nat. Commun. 2015, 6, 7653 10.1038/ncomms8653
32. Jiang, H.; Sheong, F. K.; Zhu, L.; Gao, X.; Bernauer, J.; Huang, X. Markov State Models Reveal a Two-Step Mechanism of miRNA Loading into the Human Argonaute Protein: Selective Binding followed by Structural Re-arrangement PLoS Comput. Biol. 2015, 11, e1004404 10.1371/journal.pcbi.1004404
33. de Groot, B.; Daura, X.; Mark, A.; Grubmüller, H. Essential Dynamics of Reversible Peptide Folding: Memory-Free Conformational Dynamics Governed by Internal Hydrogen Bonds J. Mol. Biol. 2001, 309, 299-313 10.1006/jmbi.2001.4655
34. Singhal, N.; Snow, C.; Pande, V. S. Path sampling to build better roadmaps: predicting the folding rate and mechanism of a Trp Zipper beta hairpin J. Chem. Phys. 2004, 121, 415-425 10.1063/1.1738647
35. Chodera, J. D.; Swope, W. C.; Pitera, J. W.; Dill, K. A. Long-time protein folding dynamics from short-time molecular dynamics simulations Multiscale Model. Simul. 2006, 5, 1214-1226 10.1137/06065146X
36. Pérez-Hernández, G.; Paul, F.; Giorgino, T.; De Fabritiis, G.; Noé, F. Identification of slow molecular order parameters for Markov model construction J. Chem. Phys. 2013, 139, 015102 10.1063/1.4811489
37. Stanley, N.; Esteban-Martin, S.; De Fabritiis, G. Kinetic modulation of a disordered protein domain by phosphorylation Nat. Commun. 2014, 5, 5272 10.1038/ncomms6272
38. Schor, M.; Mey, A. S. J. S.; Noé, F.; MacPhee, C. Shedding Light on the Dock-Lock Mechanism in Amyloid Fibril Growth Using Markov State Models J. Phys. Chem. Lett. 2015, 6, 1076-1081 10.1021/acs.jpclett.5b00330
39. Morcos, F.; Chatterjee, S.; McClendon, C. L.; Brenner, P. R.; López-Rendón, R.; Zintsmaster, J.; Ercsey-Ravasz, M.; Sweet, C. R.; Jacobson, M. P.; Peng, W. J.; Izaguirre, J. A. Modeling Conformational Ensembles of Slow Functional Motions in Pin1-WW PLoS Comput. Biol. 2010, 6, e1001015 10.1371/journal.pcbi.1001015
40. Faelber, K.; Posor, Y.; Gao, S.; Held, M.; Roske, Y.; Schulze, D.; Haucke, V.; Noé, F.; Daumke, O. Crystal structure of nucleotide-free dynamin Nature 2011, 477, 556-560 10.1038/nature10369
41. Sadiq, S. K.; Noé, F.; De Fabritiis, G. Kinetic characterization of the critical step in HIV-1 protease maturation Proc. Natl. Acad. Sci. U. S. A. 2012, 109, 20449-20454 10.1073/pnas.1210983109
42. Xia, J.; Deng, N.; Levy, R. M. NMR Relaxation in Proteins with Fast Internal Motions and Slow Conformational Exchange: Model-Free Framework and Markov State Simulations J. Phys. Chem. B 2013, 117, 6625-6634 10.1021/jp400797y
43. Kohlhoff, K. J.; Shukla, D.; Lawrenz, M.; Bowman, G. R.; Konerding, D. E.; Belov, D.; Altman, R. B.; Pande, V. S. Cloud-based simulations on Google Exacycle reveal ligand modulation of GPCR activation pathways Nat. Chem. 2014, 6, 15-21 10.1038/nchem.1821
44. Shukla, D.; Meng, Y.; Roux, B.; Pande, V. S. Activation pathway of Src kinase reveals intermediate states as targets for drug design Nat. Commun. 2014, 5, 3397 10.1038/ncomms4397
45. Pontiggia, F.; Pachov, D. V.; Clarkson, M. W.; Villali, J.; Hagan, M. F.; Pande, V. S.; Kern, D. Free energy landscape of activation in a signalling protein at atomic resolution Nat. Commun. 2015, 6, 7284 10.1038/ncomms8284
46. Banerjee, R.; Yan, H.; Cukier, R. I. Conformational Transition in Signal Transduction: Metastable States and Transition Pathways in the Activation of a Signaling Protein J. Phys. Chem. B 2015, 119, 6591-6602 10.1021/acs.jpcb.5b02582
47. Malmstrom, R. D.; Kornev, A. P.; Taylor, S. S.; Amaro, R. E. Allostery through the computational microscope: cAMP activation of a canonical signalling domain Nat. Commun. 2015, 6, 7588 10.1038/ncomms8588
48. Reubold, T. F.; Faelber, K.; Plattner, N.; Posor, Y.; Ketel, K.; Curth, U.; Schlegel, J.; Anand, R.; Manstein, D. J.; Noé, F.; Haucke, V.; Daumke, O.; Eschenburg, S. Crystal structure of the dynamin tetramer Nature 2015, 525, 404-408 10.1038/nature14880
49. Perkett, M. R.; Hagan, M. F. Using Markov state models to study self-assembly J. Chem. Phys. 2014, 140, 214101 10.1063/1.4878494
50. Peters, B.; Trout, B. L. Obtaining reaction coordinates by likelihood maximization J. Chem. Phys. 2006, 125, 054108 10.1063/1.2234477
51. Rohrdanz, M. A.; Zheng, W.; Maggioni, M.; Clementi, C. Determination of reaction coordinates via locally scaled diffusion map J. Chem. Phys. 2011, 134, 124116 10.1063/1.3569857
52. Schwantes, C. R.; Pande, V. S. Improvements in Markov State Model Construction Reveal Many Non-Native Interactions in the Folding of NTL9 J. Chem. Theory Comput. 2013, 9, 2000-2009 10.1021/ct300878a
53. Schwantes, C. R.; Pande, V. S. Modeling Molecular Kinetics with tICA and the Kernel Trick J. Chem. Theory Comput. 2015, 11, 600-608 10.1021/ct5007357
54. Noé, F.; Clementi, C. Kinetic distance and kinetic maps from molecular dynamics simulation J. Chem. Theory Comput. 2015, 10.1021/acs.jctc.5b00553
55. Altis, A.; Nguyen, P. H.; Hegger, R.; Stock, G. Dihedral angle principal component analysis of molecular dynamics simulations J. Chem. Phys. 2007, 126, 244111 10.1063/1.2746330
56. Nadler, B.; Lafon, S.; Coifman, R. R.; Kevrekidis, I. G. Diffusion Maps, Spectral Clustering and Eigenfunctions of Fokker-Planck Operators Adv. Neural Inf. Process Syst. 2005, 18, 955-962
57. Yao, Y.; Sun, J.; Huang, X.; Bowman, G. R.; Singh, G.; Lesnick, M.; Guibas, L. J.; Pande, V. S.; Carlsson, G. Topological methods for exploring low-density states in biomolecular folding pathways J. Chem. Phys. 2009, 130, 144115 10.1063/1.3103496
58. Keller, B. G.; Daura, X.; van Gunsteren, W. F. Comparing geometric and kinetic cluster algorithms for molecular simulation data J. Chem. Phys. 2010, 132, 074110 10.1063/1.3301140
59. Dasgupta, S.; Long, P. Performance guarantees for hierarchical clustering J. Comput. Syst. Sci. 2005, 70, 555-569 10.1016/j.jcss.2004.10.006
60. Arthur, D.; Vassilvitskii, S. k-means++: The advantages of careful seeding. Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms; Society for Industrial and Applied Mathematics: Philadelphia, PA,USA, 2007; pp 1027-1035.
61. Beauchamp, K. A.; Bowman, G. R.; Lane, T. J.; Maibaum, L.; Haque, I. S.; Pande, V. S. MSMBuilder2: Modeling Conformational Dynamics at the Picosecond to Millisecond Scale J. Chem. Theory Comput. 2011, 7, 3412-3419 10.1021/ct200463m
62. Prinz, J.-H.; Wu, H.; Sarich, M.; Keller, B. G.; Senne, M.; Held, M.; Chodera, J. D.; Schütte, C.; Noé, F. Markov models of molecular kinetics: Generation and Validation J. Chem. Phys. 2011, 134, 174105 10.1063/1.3565032
63. Hinrichs, N. S.; Pande, V. S. Calculation of the distribution of eigenvalues and eigenvectors in Markovian state models for molecular dynamics J. Chem. Phys. 2007, 126, 244101 10.1063/1.2740261
64. Chodera, J. D.; 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 10.1063/1.3463406
65. Bacallado, S.; Chodera, J. D.; Pande, V. S. Bayesian comparison of Markov models of molecular dynamics with detailed balance constraint J. Chem. Phys. 2009, 131, 045106 10.1063/1.3192309
66. Metzner, P.; Noé, F.; Schütte, C. Estimating the sampling error: Distribution of transition matrices and functions of transition matrices for given trajectory data Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 2009, 80, 021106 10.1103/PhysRevE.80.021106
67. Schütte, C.; Noé, F.; Lu, J.; Sarich, M.; Vanden-Eijnden, E. Markov State Models Based on Milestoning J. Chem. Phys. 2011, 134, 204105 10.1063/1.3590108
68. Trendelkamp-Schroer, B.; Noé, F. Efficient Bayesian estimation of Markov model transition matrices with given stationary distribution J. Chem. Phys. 2013, 138, 164113 10.1063/1.4801325
69. Trendelkamp-Schroer, B.; Wu, H.; Paul, F.; Noé, F. Estimation and uncertainty of reversible Markov models. J. Chem. Phys. 2015, (Preprint at http://arxiv.org/abs/1507.05990)manuscript in revision
70. Sarich, M.; Noé, F.; Schütte, C. On the approximation quality of Markov state models Multiscale Model. Simul. 2010, 8, 1154-1177 10.1137/090764049
71. Djurdjevac, N.; Sarich, M.; Schütte, C. Estimating the eigenvalue error of Markov State Models Multiscale Model. Simul. 2012, 10, 61-81 10.1137/100798910
72. McGibbon, R. T.; Pande, V. S. Variational cross-validation of slow dynamical modes in molecular kinetics J. Chem. Phys. 2015, 142, 124105 10.1063/1.4916292
73. Deuflhard, P.; Weber, M.; Dellnitz, M.; Kirkland, S.; Neumann, M.; Schütte, C. Robust Perron Cluster Analysis in Conformation Dynamics Linear Algebra Its Appl. 2005, 398, 161-184 (Dellnitz, M., Kirkland, S., Newmann, M., Schutte, C., Eds.) 10.1016/j.laa.2004.10.026
74. Kube, S.; Weber, M. A coarse graining method for the identification of transition rates between molecular conformations J. Chem. Phys. 2007, 126, 024103 10.1063/1.2404953
75. Bowman, G. R. Improved coarse-graining of Markov state models via explicit consideration of statistical uncertainty J. Chem. Phys. 2012, 137, 134111 10.1063/1.4755751
76. Yao, Y.; Cui, R. Z.; Bowman, G. R.; Silva, D.-A.; Sun, J.; Huang, X. Hierarchical Nyström methods for constructing Markov state models for conformational dynamics J. Chem. Phys. 2013, 138, 174106 10.1063/1.4802007
77. 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 10.1007/s11634-013-0134-6
78. Noé, F.; Wu, H.; Prinz, J.-H.; Plattner, N. Projected and Hidden Markov Models for calculating kinetics and metastable states of complex molecules J. Chem. Phys. 2013, 139, 184114 10.1063/1.4828816
79. Sheong, F. K.; Silva, D.-A.; 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 10.1021/ct5007168
80. Hummer, G.; Szabo, A. Optimal Dimensionality Reduction of Multistate Kinetic and Markov-State Models J. Phys. Chem. B 2015, 119, 9029-9037 10.1021/jp508375q
81. E, W.; Vanden-Eijnden, E. Towards a Theory of Transition Paths J. Stat. Phys. 2006, 123, 503-523 10.1007/s10955-005-9003-9
82. Jain, A.; Stock, G. Hierarchical Folding Free Energy Landscape of HP35 Revealed by Most Probable Path Clustering J. Phys. Chem. B 2014, 7750-7760
83. Metzner, P.; Schütte, C.; Vanden-Eijnden, E. Transition Path Theory for Markov Jump Processes Multiscale Model. Simul. 2009, 7, 1192-1219 10.1137/070699500
84. Berezhkovskii, A.; Hummer, G.; Szabo, A. Reactive flux and folding pathways in network models of coarse-grained protein dynamics J. Chem. Phys. 2009, 130, 205102 10.1063/1.3139063
85. Noé, F.; Doose, S.; Daidone, I.; Löllmann, M.; Chodera, J. D.; Sauer, M.; Smith, J. C. Dynamical fingerprints for probing individual relaxation processes in biomolecular dynamics with simulations and kinetic experiments Proc. Natl. Acad. Sci. U. S. A. 2011, 108, 4822-4827 10.1073/pnas.1004646108
86. Zhuang, W.; Cui, R. Z.; Silva, D.-A.; Huang, X. Simulating the T-Jump-Triggered Unfolding Dynamics of trpzip2 Peptide and Its Time-Resolved IR and Two-Dimensional IR Signals Using the Markov State Model Approach J. Phys. Chem. B 2011, 115, 5415-5424 10.1021/jp109592b
87. Bowman, G. R.; Voelz, V. A.; Pande, V. S. Atomistic Folding Simulations of the Five-Helix Bundle Protein Lambda 6-85 J. Am. Chem. Soc. 2011, 133, 664-667 10.1021/ja106936n
88. Keller, B. G.; Prinz, J.-H.; Noé, F. Markov models and dynamical fingerprints: Unraveling the complexity of molecular kinetics Chem. Phys. 2012, 396, 92-107 10.1016/j.chemphys.2011.08.021
89. Lindner, B.; Yi, Z.; Prinz, J.-H.; Smith, J. C.; Noé, F. Dynamic Neutron Scattering from Conformational Dynamics I: Theory and Markov models J. Chem. Phys. 2013, 139, 175101 10.1063/1.4824070
90. Yi, Z.; Lindner, B.; Prinz, J.-H.; Noé, F.; Smith, J. C. Dynamic Neutron Scattering from Conformational Dynamics II: Application Using Molecular Dynamics Simulation and Markov Modeling J. Chem. Phys. 2013, 139, 175102 10.1063/1.4824071
91. Bowman, G. R.; Bolin, E. R.; Hart, K. M.; Maguire, B.; Marqusee, S. Discovery of multiple hidden allosteric sites by combining Markov state models and experiments Proc. Natl. Acad. Sci. U. S. A. 2015, 112, 2734-2739 10.1073/pnas.1417811112
92. Bowman, G. R.; Pande, V. S.; Noé, F., Eds. An Introduction to Markov State Models and Their Application to Long Timescale Molecular Simulation Advances in Experimental Medicine and Biology; Springer: Heidelberg, Germany, 2014; Vol. 797.
93. Wu, H.; Noé, F. Gaussian Markov transition models of molecular kinetics J. Chem. Phys. 2015, 142, 084104 10.1063/1.4913214
94. McGibbon, R. T.; Ramsundar, B.; Sultan, M. M.; Kiss, G.; Pande, V. S. Understanding Protein Dynamics with L1-Regularized Reversible Hidden Markov Models Proc. Int. Conf. Mach. Learn. 2014, 1197-1205
95. Noé, F.; Nüske, F. A variational approach to modeling slow processes in stochastic dynamical systems Multiscale Model. Simul. 2013, 11, 635-655 10.1137/110858616
96. Nüske, F.; Keller, B. G.; Pérez-Hernández, G.; Mey, A. S. J. S.; Noé, F. Variational Approach to Molecular Kinetics J. Chem. Theory Comput. 2014, 10, 1739-1752 10.1021/ct4009156
97. Wu, H.; Noé, F. Optimal estimation of free energies and stationary densities from multiple biased simulations Multiscale Model. Simul. 2014, 12, 25-54 10.1137/120895883
98. Mey, A. S. J. S.; Wu, H.; Noé, F. xTRAM: Estimating equilibrium expectations from time-correlated simulation data at multiple thermodynamic states Phys. Rev. X 2014, 4, 041018 10.1103/PhysRevX.4.041018
99. Rosta, E.; Hummer, G. Free energies from dynamic weighted histogram analysis using unbiased Markov state model J. Chem. Theory Comput. 2015, 11, 276-285 10.1021/ct500719p
100. Wu, H.; Mey, A. S. J. S.; Rosta, E.; Noé, F. Statistically optimal analysis of state-discretized trajectory data from multiple thermodynamic states J. Chem. Phys. 2014, 141, 214106 10.1063/1.4902240
101. Senne, M.; Trendelkamp-Schroer, B.; Mey, A. S. J. S.; Schütte, C.; Noé, F. EMMA-A software package for Markov model building and analysis J. Chem. Theory Comput. 2012, 8, 2223-2238 10.1021/ct300274u
102. GNU Operating System, http://www.gnu.org/licenses/gpl-3.0.html.
103. Molgedey, L.; Schuster, H. G. Separation of a mixture of independent signals using time delayed correlations Phys. Rev. Lett. 1994, 72, 3634-3637 10.1103/PhysRevLett.72.3634
104. Ferrenberg, A. M.; Swendsen, R. H. Optimized Monte Carlo data analysis Phys. Rev. Lett. 1989, 63, 1195-1198 10.1103/PhysRevLett.63.1195
105. Vitalini, F.; Noé, F.; Keller, B. G. A basis set for peptides for the variational approach to conformational kinetics J. Chem. Theory Comput. 2015, 11, 3992-4004 10.1021/acs.jctc.5b00498
106. Nüske, F.; Schneider, R.; Vitalini, F.; Noé, F. Variational Tensor Approach for Approximating the Rare-Event Kinetics of Macromolecular Systems. J. Chem. Phys. 2015, manuscript in revision
107. McGibbon, R. T.; Beauchamp, K. A.; Schwantes, C. R.; Wang, L.-P.; Hernández, C. X.; Harrigan, M. P.; Lane, T. J.; Swails, J. M.; Pande, V. S. MDTraj: a modern, open library for the analysis of molecular dynamics trajectories. BioRxiv, 2014, DOI: 10.1101/008896.
108. ten Wolde, P. R.; Chandler, D. Drying-induced hydrophobic polymer collapse Proc. Natl. Acad. Sci. U. S. A. 2002, 99, 6539-6543 10.1073/pnas.052153299
109. Prada-Gracia, D.; Shevchuk, R.; Hamm, P.; Rao, F. Towards a microscopic description of the free-energy landscape of water J. Chem. Phys. 2012, 137, 144504 10.1063/1.4755746
110. Harrigan, M. P.; Shukla, D.; Pande, V. S. Conserve Water: A Method for the Analysis of Solvent in Molecular Dynamics J. Chem. Theory Comput. 2015, 11, 1094-1101 10.1021/ct5010017
111. Pearson, K. On Lines and Planes of Closest Fit to Systems of Points in Space Philos. Mag. 1901, 2, 559-572 10.1080/14786440109462720
112. Hotelling, H. Analysis of a complex of statistical variables into principal components J. Edu. Psych 1933, 24, 417-441 10.1037/h0071325
113. Voronoi, M. G. Nouvelles applications des parametres continus a la theorie des formes quadratiques J. Reine Angew. Math. 1908, 134, 198-287
114. Lloyd, S. P. Least squares quantization in PCM IEEE Trans. Inf. Theory 1982, 28, 129-137 10.1109/TIT.1982.1056489
115. Sculley, D. Web-scale k-means clustering. Proceedings of the 19th international conference on World wide web, Raleigh, NC, USA, Apr. 26-30, 2010; ACM: New York, NY, USA, 2010; pp 1177-1178, DOI: 10.1145/1772690.1772862.
116. Noé, F. Statistical inefficiency of Markov model count matrices. Preprint, http://publications.mi.fu-berlin.de/1699/ 2015.
117. Trendelkamp-Schroer, B.; Noé, F. Efficient estimation of rare-event kinetics. Phys. Rev. X 2015, (preprint at arXiv:1409.6439)manuscript in revision
118. Prinz, J.-H.; Chodera, J. D.; Noé, F. Spectral rate theory for two-state kinetics Phys. Rev. X 2014, 4, 011020 10.1103/PhysRevX.4.011020
119. Baum, L. E.; Petrie, T.; Soules, G.; Weiss, N. A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains Ann. Math. Stat. 1970, 41, 164-171 10.1214/aoms/1177697196
120. Welch, L. R. Hidden Markov Models and the Baum-Welch Algorithm IEEE Inf. Theory Soc. Newsl. 2003, 53, 1-13
121. Rabiner, L. R. A tutorial on hidden Markov models and selected applications in speech recognition Proc. IEEE 1989, 77, 257-286 10.1109/5.18626
122. Chodera, J. D.; Elms, P.; Noé, F.; Keller, B. G.; Kaiser, C. M.; Ewall-Wice, A.; Marqusee, S.; Bustamante, C.; Hinrichs, N. S. Bayesian hidden Markov model analysis of single-molecule force spectroscopy: Characterizing kinetics under measurement uncertainty, 2011; http://arxiv.org/abs/1108.1430.
123. Pohlhausen, E. Berechnung der Eigenschwingungen statisch-bestimmter Fachwerke Z. Angew. Math. Mech. 1921, 1, 28-42 10.1002/zamm.19210010104
124. Du, R.; Pande, V. S.; Grosberg, A. Y.; Tanaka, T.; Shakhnovich, E. S. On the transition coordinate for protein folding J. Chem. Phys. 1998, 108, 334-350 10.1063/1.475393
125. Bolhuis, P. G.; Dellago, C.; Chandler, D. Reaction coordinates of biomolecular isomerization Proc. Natl. Acad. Sci. U. S. A. 2000, 97, 5877-5882 10.1073/pnas.100127697
126. Pirchi, M.; Ziv, G.; Riven, I.; Cohen, S. S.; Zohar, N.; Barak, Y.; Haran, G. Single-molecule fluorescence spectroscopy maps the folding landscape of a large protein Nat. Commun. 2011, 2, 493 10.1038/ncomms1504
127. Blau, C.; Grubmüller, H. g-contacts: Fast contact search in bio-molecular ensemble data Comput. Phys. Commun. 2013, 184, 2856-2859 10.1016/j.cpc.2013.07.018
128. Doerr, S.; De Fabritiis, G. D. On-the-Fly Learning and Sampling of Ligand Binding by High-Throughput Molecular Simulations J. Chem. Theory Comput. 2014, 10, 2064-2069 10.1021/ct400919u
129. Limongelli, V.; Bonomi, M.; Parrinello, M. Funnel metadynamics as accurate binding free-energy method Proc. Natl. Acad. Sci. U. S. A. 2013, 110, 6358-6363 10.1073/pnas.1303186110
130. Tiwary, P.; Limongelli, V.; Salvalaglio, M.; Parrinello, M. Kinetics of protein-ligand unbinding: Predicting pathways, rates, and rate-limiting steps Proc. Natl. Acad. Sci. U. S. A. 2015, 112, E386-E391 10.1073/pnas.1424461112
131. Katz, B. A. et al. A novel serine protease inhibition motif involving a multi-centered short hydrogen bonding network at the active site J. Mol. Biol. 2001, 307, 1451-1486 10.1006/jmbi.2001.4516
132. Talhout, R.; Engberts, J. B. F. N. Thermodynamic analysis of binding of p-substituted benzamidines to trypsin Eur. J. Biochem. 2001, 268, 1554-1560 10.1046/j.1432-1327.2001.01991.x
133. Guillain, F.; Thusius, D. Use of proflavine as an indicator in temperature-jump studies of the binding of a competitive inhibitor to trypsin J. Am. Chem. Soc. 1970, 92, 5534-5536 10.1021/ja00721a051
134. Humphrey, W.; Dalke, A.; Schulten, K. VMD: visual molecular dynamics J. Mol. Graphics 1996, 14, 33-38 10.1016/0263-7855(96)00018-5