Improving Estimation of the Koopman Operator with Kolmogorov–Smirnov Indicator Functions

Ngo, Van A.; Lin, Yen Ting; Perez, Danny

doi:10.1021/acs.jctc.3c00632

Improving Estimation of the Koopman Operator with Kolmogorov–Smirnov Indicator Functions

Journal Article · Fri Oct 06 00:00:00 EDT 2023 · Journal of Chemical Theory and Computation

DOI:https://doi.org/10.1021/acs.jctc.3c00632· OSTI ID:2076196

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Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)
Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

It has become common to perform kinetic analysis using approximate Koopman operators that transform high-dimensional timeseries of observables into ranked dynamical modes. The key to the practical success of the approach is the identification of a set of observables that form a good basis on which to expand the slow relaxation modes. Good observables are, however, difficult to identify a priori and suboptimal choices can lead to significant underestimations of characteristic time scales. Leveraging the representation of slow dynamics in terms of Hidden Markov Models (HMM), we propose a simple and computationally efficient clustering procedure to infer surrogate observables that form a good basis for slow modes. Here, we apply the approach to an analytically solvable model system as well as on three protein systems of different complexities. We consistently demonstrate that the inferred indicator functions can significantly improve the estimation of the leading eigenvalues of Koopman operators and correctly identify key states and transition time scales of stochastic systems, even when good observables are not known a priori.

View Accepted Manuscript (DOE)

Research Organization:: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States); Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States). Oak Ridge Leadership Computing Facility (OLCF)

Sponsoring Organization:: USDOE Laboratory Directed Research and Development (LDRD) Program; USDOE National Nuclear Security Administration (NNSA); USDOE Office of Science (SC)

Grant/Contract Number:: 89233218CNA000001; AC05-00OR22725

OSTI ID:: 2076196

Alternate ID(s):: OSTI ID: 2203409

Report Number(s):: LA-UR--23-26045

Journal Information:: Journal of Chemical Theory and Computation, Journal Name: Journal of Chemical Theory and Computation Journal Issue: 20 Vol. 19; ISSN 1549-9618

Publisher:: American Chemical SocietyCopyright Statement

Country of Publication:: United States

Language:: English

References (54)

The ββα fold: explorations in sequence space11Edited by M. F. Summers Sarisky, Catherine A.; Mayo, Stephen L. Journal of Molecular Biology, Vol. 307, Issue 5 https://doi.org/10.1006/jmbi.2000.4345	journal	April 2001
A Data–Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition Williams, Matthew O.; Kevrekidis, Ioannis G.; Rowley, Clarence W. Journal of Nonlinear Science, Vol. 25, Issue 6 https://doi.org/10.1007/s00332-015-9258-5	journal	June 2015
Data-Driven Model Reduction and Transfer Operator Approximation Klus, Stefan; Nüske, Feliks; Koltai, Péter Journal of Nonlinear Science, Vol. 28, Issue 3 https://doi.org/10.1007/s00332-017-9437-7	journal	January 2018
Variational Approach for Learning Markov Processes from Time Series Data Wu, Hao; Noé, Frank Journal of Nonlinear Science, Vol. 30, Issue 1 https://doi.org/10.1007/s00332-019-09567-y	journal	August 2019
Spectral Properties of Dynamical Systems, Model Reduction and Decompositions Mezić, Igor Nonlinear Dynamics, Vol. 41, Issue 1-3 https://doi.org/10.1007/s11071-005-2824-x	journal	August 2005
Fuzzy spectral clustering by PCCA+: application to Markov state models and data classification Röblitz, Susanna; Weber, Marcus Advances in Data Analysis and Classification, Vol. 7, Issue 2 https://doi.org/10.1007/s11634-013-0134-6	journal	May 2013
Efficient agglomerative hierarchical clustering Bouguettaya, Athman; Yu, Qi; Liu, Xumin Expert Systems with Applications, Vol. 42, Issue 5 https://doi.org/10.1016/j.eswa.2014.09.054	journal	April 2015
Transition networks for modeling the kinetics of conformational change in macromolecules Noé, Frank; Fischer, Stefan Current Opinion in Structural Biology, Vol. 18, Issue 2 https://doi.org/10.1016/j.sbi.2008.01.008	journal	April 2008
Markov state models of biomolecular conformational dynamics Chodera, John D.; Noé, Frank Current Opinion in Structural Biology, Vol. 25 https://doi.org/10.1016/j.sbi.2014.04.002	journal	April 2014
Everything you wanted to know about Markov State Models but were afraid to ask Pande, Vijay S.; Beauchamp, Kyle; Bowman, Gregory R. Methods, Vol. 52, Issue 1 https://doi.org/10.1016/j.ymeth.2010.06.002	journal	September 2010
PyEMMA 2: A Software Package for Estimation, Validation, and Analysis of Markov Models Scherer, Martin K.; Trendelkamp-Schroer, Benjamin; Paul, Fabian Journal of Chemical Theory and Computation, Vol. 11, Issue 11 https://doi.org/10.1021/acs.jctc.5b00743	journal	October 2015
Recovery of Protein Folding Funnels from Single-Molecule Time Series by Delay Embeddings and Manifold Learning Wang, Jiang; Ferguson, Andrew L. The Journal of Physical Chemistry B, Vol. 122, Issue 50 https://doi.org/10.1021/acs.jpcb.8b08800	journal	November 2018
Transition Rates and Efficiency of Collective Variables from Time-Dependent Biased Simulations Palacio-Rodriguez, Karen; Vroylandt, Hadrien; Stelzl, Lukas S. The Journal of Physical Chemistry Letters, Vol. 13, Issue 32 https://doi.org/10.1021/acs.jpclett.2c01807	journal	August 2022
Machine Learning of Biomolecular Reaction Coordinates Brandt, Simon; Sittel, Florian; Ernst, Matthias The Journal of Physical Chemistry Letters, Vol. 9, Issue 9 https://doi.org/10.1021/acs.jpclett.8b00759	journal	April 2018
MSMBuilder2: Modeling Conformational Dynamics on the Picosecond to Millisecond Scale Beauchamp, Kyle A.; Bowman, Gregory R.; Lane, Thomas J. Journal of Chemical Theory and Computation, Vol. 7, Issue 10 https://doi.org/10.1021/ct200463m	journal	August 2011
Improvements in Markov State Model Construction Reveal Many Non-Native Interactions in the Folding of NTL9 Schwantes, Christian R.; Pande, Vijay S. Journal of Chemical Theory and Computation, Vol. 9, Issue 4 https://doi.org/10.1021/ct300878a	journal	March 2013
Assessing the Reliability of the Dynamics Reconstructed from Metadynamics Salvalaglio, Matteo; Tiwary, Pratyush; Parrinello, Michele Journal of Chemical Theory and Computation, Vol. 10, Issue 4 https://doi.org/10.1021/ct500040r	journal	March 2014
Modeling Molecular Kinetics with tICA and the Kernel Trick Schwantes, Christian R.; Pande, Vijay S. Journal of Chemical Theory and Computation, Vol. 11, Issue 2 https://doi.org/10.1021/ct5007357	journal	January 2015
Folding Simulations for Proteins with Diverse Topologies Are Accessible in Days with a Physics-Based Force Field and Implicit Solvent Nguyen, Hai; Maier, James; Huang, He Journal of the American Chemical Society, Vol. 136, Issue 40 https://doi.org/10.1021/ja5032776	journal	August 2014
VAMPnets for deep learning of molecular kinetics Mardt, Andreas; Pasquali, Luca; Wu, Hao Nature Communications, Vol. 9, Issue 1 https://doi.org/10.1038/s41467-017-02388-1	journal	January 2018
Learning dynamical information from static protein and sequencing data Pearce, Philip; Woodhouse, Francis G.; Forrow, Aden Nature Communications, Vol. 10, Issue 1 https://doi.org/10.1038/s41467-019-13307-x	journal	November 2019
Machine-guided path sampling to discover mechanisms of molecular self-organization Jung, Hendrik; Covino, Roberto; Arjun, A. Nature Computational Science, Vol. 3, Issue 4 https://doi.org/10.1038/s43588-023-00428-z	journal	April 2023
Slow dynamics in protein fluctuations revealed by time-structure based independent component analysis: The case of domain motions Naritomi, Yusuke; Fuchigami, Sotaro The Journal of Chemical Physics, Vol. 134, Issue 6 https://doi.org/10.1063/1.3554380	journal	February 2011
Markov models of molecular kinetics: Generation and validation Prinz, Jan-Hendrik; Wu, Hao; Sarich, Marco The Journal of Chemical Physics, Vol. 134, Issue 17 https://doi.org/10.1063/1.3565032	journal	May 2011
Identification of slow molecular order parameters for Markov model construction Pérez-Hernández, Guillermo; Paul, Fabian; Giorgino, Toni The Journal of Chemical Physics, Vol. 139, Issue 1 https://doi.org/10.1063/1.4811489	journal	July 2013
Projected and hidden Markov models for calculating kinetics and metastable states of complex molecules Noé, Frank; Wu, Hao; Prinz, Jan-Hendrik The Journal of Chemical Physics, Vol. 139, Issue 18 https://doi.org/10.1063/1.4828816	journal	November 2013
Variational Koopman models: Slow collective variables and molecular kinetics from short off-equilibrium simulations Wu, Hao; Nüske, Feliks; Paul, Fabian The Journal of Chemical Physics, Vol. 146, Issue 15 https://doi.org/10.1063/1.4979344	journal	April 2017
Perspective: Identification of collective variables and metastable states of protein dynamics Sittel, Florian; Stock, Gerhard The Journal of Chemical Physics, Vol. 149, Issue 15 https://doi.org/10.1063/1.5049637	journal	October 2018
Variational selection of features for molecular kinetics Scherer, Martin K.; Husic, Brooke E.; Hoffmann, Moritz The Journal of Chemical Physics, Vol. 150, Issue 19 https://doi.org/10.1063/1.5083040	journal	May 2019
Kernel methods for detecting coherent structures in dynamical data Klus, Stefan; Husic, Brooke E.; Mollenhauer, Mattes Chaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 29, Issue 12 https://doi.org/10.1063/1.5100267	journal	December 2019
A variational approach to assess reaction coordinates for two-step crystallization Finney, A. R.; Salvalaglio, M. The Journal of Chemical Physics, Vol. 158, Issue 9 https://doi.org/10.1063/5.0139842	journal	March 2023
Simple few-state models reveal hidden complexity in protein folding Beauchamp, K. A.; McGibbon, R.; Lin, Y. -S. Proceedings of the National Academy of Sciences, Vol. 109, Issue 44 https://doi.org/10.1073/pnas.1201810109	journal	July 2012
Native contacts determine protein folding mechanisms in atomistic simulations Best, R. B.; Hummer, G.; Eaton, W. A. Proceedings of the National Academy of Sciences, Vol. 110, Issue 44 https://doi.org/10.1073/pnas.1311599110	journal	October 2013
Energetically significant networks of coupled interactions within an unfolded protein Cho, J. -H.; Meng, W.; Sato, S. Proceedings of the National Academy of Sciences, Vol. 111, Issue 33 https://doi.org/10.1073/pnas.1402054111	journal	August 2014
Multiensemble Markov models of molecular thermodynamics and kinetics Wu, Hao; Paul, Fabian; Wehmeyer, Christoph Proceedings of the National Academy of Sciences, Vol. 113, Issue 23 https://doi.org/10.1073/pnas.1525092113	journal	May 2016
Hamiltonian Systems and Transformation in Hilbert Space Koopman, B. O. Proceedings of the National Academy of Sciences, Vol. 17, Issue 5 https://doi.org/10.1073/pnas.17.5.315	journal	May 1931
Dynamical Systems of Continuous Spectra Koopman, B. O.; Neumann, J. v. Proceedings of the National Academy of Sciences, Vol. 18, Issue 3 https://doi.org/10.1073/pnas.18.3.255	journal	March 1932
Fast protein folding is governed by memory-dependent friction Dalton, Benjamin A.; Ayaz, Cihan; Kiefer, Henrik Proceedings of the National Academy of Sciences, Vol. 120, Issue 31 https://doi.org/10.1073/pnas.2220068120	journal	July 2023
A multidimensional version of the Kolmogorov–Smirnov test Fasano, G.; Franceschini, A. Monthly Notices of the Royal Astronomical Society, Vol. 225, Issue 1 https://doi.org/10.1093/mnras/225.1.155	journal	March 1987
The Trp-cage: optimizing the stability of a globular miniprotein Barua, B.; Lin, J. C.; Williams, V. D. Protein Engineering Design and Selection, Vol. 21, Issue 3 https://doi.org/10.1093/protein/gzm082	journal	January 2008
Separation of a mixture of independent signals using time delayed correlations Molgedey, L.; Schuster, H. G. Physical Review Letters, Vol. 72, Issue 23 https://doi.org/10.1103/PhysRevLett.72.3634	journal	June 1994
A tutorial on hidden Markov models and selected applications in speech recognition Rabiner, L. R. Proceedings of the IEEE, Vol. 77, Issue 2 https://doi.org/10.1109/5.18626	journal	January 1989
How Fast-Folding Proteins Fold Lindorff-Larsen, K.; Piana, S.; Dror, R. O. Science, Vol. 334, Issue 6055 https://doi.org/10.1126/science.1208351	journal	October 2011
A Variational Approach to Modeling Slow Processes in Stochastic Dynamical Systems Noé, Frank; Nüske, Feliks Multiscale Modeling & Simulation, Vol. 11, Issue 2 https://doi.org/10.1137/110858616	journal	January 2013
Data-Driven Learning for the Mori--Zwanzig Formalism: A Generalization of the Koopman Learning Framework Lin, Yen Ting; Tian, Yifeng; Livescu, Daniel SIAM Journal on Applied Dynamical Systems, Vol. 20, Issue 4 https://doi.org/10.1137/21M1401759	journal	January 2021
Anton, a special-purpose machine for molecular dynamics simulation Shaw, David E.; Chao, Jack C.; Eastwood, Michael P. Communications of the ACM, Vol. 51, Issue 7 https://doi.org/10.1145/1364782.1364802	journal	July 2008
A Maximization Technique Occurring in the Statistical Analysis of Probabilistic Functions of Markov Chains Baum, Leonard E.; Petrie, Ted; Soules, George The Annals of Mathematical Statistics, Vol. 41, Issue 1 https://doi.org/10.1214/aoms/1177697196	journal	February 1970
Table for Estimating the Goodness of Fit of Empirical Distributions Smirnov, N. The Annals of Mathematical Statistics, Vol. 19, Issue 2 https://doi.org/10.1214/aoms/1177730256	journal	June 1948
Koopman Invariant Subspaces and Finite Linear Representations of Nonlinear Dynamical Systems for Control Brunton, Steven L.; Brunton, Bingni W.; Proctor, Joshua L. PLOS ONE, Vol. 11, Issue 2 https://doi.org/10.1371/journal.pone.0150171	journal	February 2016
A mathematical formalization of the parallel replica dynamics Le Bris, Claude; Lelièvre, Tony; Luskin, Mitchell Monte Carlo Methods and Applications, Vol. 18, Issue 2 https://doi.org/10.1515/mcma-2012-0003	journal	January 2012
Computing the Kolmogorov-Smirnov Distribution When the Underlying CDF is Purely Discrete, Mixed, or Continuous Dimitrova, Dimitrina S.; Kaishev, Vladimir K.; Tan, Senren Journal of Statistical Software, Vol. 95, Issue 10 https://doi.org/10.18637/jss.v095.i10	journal	January 2020
The Comparison of Dendrograms by Objective Methods Sokal, Robert R.; Rohlf, F. James TAXON, Vol. 11, Issue 2 https://doi.org/10.2307/1217208	journal	February 1962
Zur Operatorenmethode In Der Klassischen Mechanik v. Neumann, J. The Annals of Mathematics, Vol. 33, Issue 3 https://doi.org/10.2307/1968537	journal	July 1932
Learning Deep Neural Network Representations for Koopman Operators of Nonlinear Dynamical Systems Yeung, Enoch; Kundu, Soumya; Hodas, Nathan 2019 American Control Conference (ACC) https://doi.org/10.23919/ACC.2019.8815339	conference	July 2019

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