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Identification of slow molecular order parameters for Markov model construction
Open accessAbstract
A goal in the kinetic characterization of a macromolecular system is the description of its slow relaxation processes via (i) identification of the structural changes involved in these processes and (ii) estimation of the rates or timescales at which these slow processes occur. Most of the approaches to this task, including Markov models, master-equation models, and kinetic network models, start by discretizing the high-dimensional state space and then characterize relaxation processes in terms of the eigenvectors and eigenvalues of a discrete transition matrix. The practical success of such an approach depends very much on the ability to finely discretize the slow order parameters. How can this task be achieved in a high-dimensional configuration space without relying on subjective guesses of the slow order parameters? In this paper, we use the variational principle of conformation dynamics to derive an optimal way of identifying the “slow subspace” of a large set of prior order parameters – either generic internal coordinates or a user-defined set of parameters. Using a variational formulation of conformational dynamics, it is shown that an existing method—the time-lagged independent component analysis—provides the optional solution to this problem. In addition, optimal indicators—order parameters indicating the progress of the slow transitions and thus may serve as reaction coordinates—are readily identified. We demonstrate that the slow subspace is well suited to construct accurate kinetic models of two sets of molecular dynamics simulations, the 6-residue fluorescent peptide MR121-GSGSW and the 30-residue intrinsically disordered peptide kinase inducible domain (KID). The identified optimal indicators reveal the structural changes associated with the slow processes of the molecular system under analysis.
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“which was found to adopt a conformation that may precede its KIX-bound geometry even in the absence of its binding partner [21].”
From Markov state models of biomolecular conformational dynamics (2014) · cites this paper as [21]
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“This approach can also be shown to solve the variational problem for the specific choice of basis functions identical to the input coordinates [21].”
From Markov state models of biomolecular conformational dynamics (2014) · cites this paper as [21]
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“This approach, independently rediscovered by two groups [20, 21], linearly transforms the input coordinates (e.g. torsions, distances, contacts) into collective coordinates sorted by “slowness”, thus providing an excellent dimension reduction method for MSM construction.”
From Markov state models of biomolecular conformational dynamics (2014) · cites this paper as [21]
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“This can be understood as a generalization of principal component analysis, as it solves a generalized eigenvalue problem with the instantaneous and time-lagged correlation matrices of the input coordinates [20, 21].”
From Markov state models of biomolecular conformational dynamics (2014) · cites this paper as [21]
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“This proved critical in enabling the construction of MSMs for the intrinsically disordered KID peptide”
From Markov state models of biomolecular conformational dynamics (2014) · cites this paper as [21]
In the citing paper
“the main drawback being the large amount of simulation data that is necessary to reliably estimate the autocorrelation function.”
From Unsupervised machine learning in atomistic simulations, between predictions and understanding (2019) · cites this paper as [81]
In the citing paper
“The time-lagged independent component analysis method (TICA) 80,81 identifies linear combinations of features from a time-dependent trajectory that not only have the largest variance, but that also have the longest autocorrelation time”
From Unsupervised machine learning in atomistic simulations, between predictions and understanding (2019) · cites this paper as [81]
In the citing paper
“Dimensionality reduction algorithms as TICA 80,81 or SGOOP 82 , that incorporate information on the system's dynamics, implicitly ensure that the order parameter(s) perform well as descriptors of the slowest molecular motion.”
From Unsupervised machine learning in atomistic simulations, between predictions and understanding (2019) · cites this paper as [81]
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Cites
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“…the variational approach of conformation dynamics (VAC) (Noé and Nüske 2013; Nüske et al. 2014), time-lagged independent component analysis (TICA) (Perez-Hernandez et al. 2013; Schwantes and Pande 2013), dynamic mode decomposition (DMD) (Rowley et al. 2009; Schmid 2010; Tu et al. 2014), extended…”
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“[9], which consists of simulations for 24 systems and uses the amber ff03 force field [25].”
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“The time-lagged independent components are required (a) to be uncorrelated and (b) to maximize the autocovariances at lag time τ , see (Hyvärinen et al. 2001; Pérez-Hernández et al. 2013) for more details.”
BackgroundMethodInfluentialSelf-citationScalable emulation of protein equilibrium ensembles with generative deep learning
Lewis, S., Hempel, T., Jiménez-Luna, J., Gastegger, M., Xie, Y., Foong, A. Y. K., Satorras, V. G., Abdin, O., Veeling, B. S., Zaporozhets, I., et al. (2025) · Science · doi:10.1126/science.adv9817 · Cited by 390
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Nüske, F., Keller, B. G., Pérez-Hernández, G., Mey, A. S. J. S., & Noé, F. (2014) · Journal of Chemical Theory and Computation · doi:10.1021/ct4009156 · Cited by 308
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“Methods such as TICA [9] implement VAC directly by learning slow eigenvectors as linear combinations of pre-selected basis functions, and are at the heart of building MSMs.”
MethodBottom-up Coarse-Graining: Principles and Perspectives
Jin, J., Pak, A. J., Durumeric, A. E. P., Loose, T. D., & Voth, G. A. (2022) · Journal of Chemical Theory and Computation · doi:10.1021/acs.jctc.2c00643 · Cited by 321
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“In practice, this dimensionality reduction has proven to be very useful for capturing slow, biophysical conformational changes (20,21).”
BackgroundMarkov State Models Provide Insights into Dynamic Modulation of Protein Function
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“In addition, we have used the property that the matrix elements of Tτ can be written in terms of timecorrelation function [25].”
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“The identification of such order parameters poses another complicated problem, but as routinely done in other methods aimed at optimizing CVs (15, 18, 24), we assume such order parameters are a priori known.”
BackgroundTorchMD: A Deep Learning Framework for Molecular Simulations
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“The energy landscapes are obtained from binning the configurations over the first two TICA dimensions and computing the average of the equilibrium probability on each bin, obtained by Markov state model analysis of the mi-crostate of each configuration.”
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“motions [68] can reduce the required simulation time, taken together, by an order of”
BackgroundVariational cross-validation of slow dynamical modes in molecular kinetics
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Deeptime: a Python library for machine learning dynamical models from time series data
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“TICA [4] is a linear transformation method which was introduced for molecular dynamics in [5], was independently derived as a method for extracting the slow molecular order parameters by invoking the variational approach for conformation dynamics (see below) [80], and introduced as a method for constructing high-accur…”
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Flood, E., Boiteux, C., Lev, B. B., Vorobyov, I. V., & Allen, T. W. (2019) · Chemical Reviews · doi:10.1021/acs.chemrev.8b00630 · Cited by 146
McCarty, J., & Parrinello, M. (2017) · The Journal of Chemical Physics · doi:10.1063/1.4998598 · Cited by 130
Kitao, A. (2022) · ET Journal · doi:10.3390/j5020021 · Cited by 118
Tsai, S.-T., Kuo, E.-J., & Tiwary, P. (2020) · Nature Communications · doi:10.1038/s41467-020-18959-8 · Cited by 115
Timewarp: Transferable Acceleration of Molecular Dynamics by Learning Time-Coarsened DynamicsPreprint
Klein, L., Foong, A. Y. K., Fjelde, T. E., Mlodozeniec, B., Brockschmidt, M., Nowozin, S., No'e, F., & Tomioka, R. (2023) · Neural Information Processing Systems · doi:10.48550/arxiv.2302.01170 · Cited by 114
Structural basis of omega-3 fatty acid transport across the blood–brain barrier
Cater, R. J., Chua, G. L., Erramilli, S. K., Keener, J. E., Choy, B. C., Tokarz, P., Chin, C. F., Quek, D. Q. Y., Kloss, B., Pepe, J. G., et al. (2021) · Nature · doi:10.1038/s41586-021-03650-9 · Cited by 137
Role of conformational dynamics in the evolution of novel enzyme function
Maria‐Solano, M. A., Serrano‐Hervás, E., Romero‐Rivera, A., Iglesias‐Fernández, J., & Osuna, S. (2018) · Chemical Communications · doi:10.1039/c8cc02426j · Cited by 178
Transferable Neural Networks for Enhanced Sampling of Protein Dynamics
Sultan, M. M., Wayment-Steele, H. K., & Pande, V. S. (2018) · Journal of Chemical Theory and Computation · doi:10.1021/acs.jctc.8b00025 · Cited by 114
“…showed 3,11 that time-structure based independent component analysis (tICA), a relatively recent advance in the Markov state model (MSM) field 12–14 , yields an excellent set of linear and non-linear collective variables (CVs) for enhanced sampling via Metadynamics or other methods. tICA and…”
MethodMarkov State Models to Study the Functional Dynamics of Proteins in the Wake of Machine Learning
Konovalov, K. A., Unarta, I. C., Cao, S., Goonetilleke, E. C., & Huang, X. (2021) · JACS Au · doi:10.1021/jacsau.1c00254 · Cited by 124
Membrane proteins structures: A review on computational modeling tools
de Almeida, J. G., Preto, A. J., Koukos, P. I., Bonvin, A. M. J. J., & Moreira, I. S. (2017) · Biochimica et Biophysica Acta (BBA) - Biomembranes · doi:10.1016/j.bbamem.2017.07.008 · Cited by 115
“This requires an initial dimensionality reduction (which eliminates redundant data on atom coordinates) and grouping similar kinetic modes [133].”
Palma, J., & Pierdominici‐Sottile, G. (2022) · ChemPhysChem · doi:10.1002/cphc.202200491 · Cited by 122
Flow-Matching: Efficient Coarse-Graining of Molecular Dynamics without Forces
Köhler, J., Chen, Y., Krämer, A., Clementi, C., & Noé, F. (2023) · Journal of Chemical Theory and Computation · doi:10.1021/acs.jctc.3c00016 · Cited by 84
Self-citationEncoderMap: Dimensionality Reduction and Generation of Molecule Conformations.
Lemke, T., & Peter, C. (2019) · Journal of Chemical Theory and Computation · doi:10.1021/acs.jctc.8b00975 · Cited by 102
Combining experimental and simulation data of molecular processes via augmented Markov models
Olsson, S., Wu, H., Paul, F., Clementi, C., & Noé, F. (2017) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.1704803114 · Cited by 105
“The simulations are embedded in a joint space of time-lagged independent components (TICs) (11, 45).”
BackgroundSelf-citationContact- and distance-based principal component analysis of protein dynamics
Ernst, M., Sittel, F., & Stock, G. (2015) · The Journal of Chemical Physics · doi:10.1063/1.4938249 · Cited by 108
Decherchi, S., Berteotti, A., Bottegoni, G., Rocchia, W., & Cavalli, A. (2015) · Nature Communications · doi:10.1038/ncomms7155 · Cited by 112
“Notable examples include the TSA inhibitors of human purine nucleoside phosphorylase (PNP), a homotrimeric enzyme that catalyzes the reversible phosphorolysis of 6-oxopurine nucleosides and deoxynucleosides to the corresponding purine base and a -d-(deoxy)ribose 1-phosphate 10 .”
BackgroundGenerative Modeling of Molecular Dynamics Trajectories
Jing, B., Stärk, H., Jaakkola, T., & Berger, B. (2024) · Neural Information Processing Systems · doi:10.48550/arxiv.2409.17808 · Cited by 100
“We then reduce dimensionality with Time-lagged Independent Components Analysis (TICA) (Pérez-Hernández et al., 2013) in PyEMMA (Scherer et al., 2015).”
InfluentialMultiensemble Markov models of molecular thermodynamics and kinetics
Wu, H., Paul, F., Wehmeyer, C., & Noé, F. (2016) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.1525092113 · Cited by 202
“μðxÞ= 1 P kN k iðxÞexp h f k iðxÞ − bkðxÞ i, for x∈ Si, [29]”
Self-citationCapturing the continuous complexity of behaviour in Caenorhabditis elegans
Ahamed, T., Costa, A. C., & Stephens, G. J. (2020) · Nature Physics · doi:10.1038/s41567-020-01036-8 · Cited by 80
Kutz, J., Proctor, J. L., & Brunton, S. (2018) · Complex · doi:10.1155/2018/6010634 · Cited by 98
“The authors further show that time-lagged independent component analysis (TICA) [20, 21], which was originally developed in 1994, is closely related to the DMD algorithm [17].”
ResultBonati, L., Trizio, E., Rizzi, A., & Parrinello, M. (2023) · The Journal of Chemical Physics · doi:10.1063/5.0156343 · Cited by 90
Understanding and Modeling Polymers: The Challenge of Multiple Scales
Schmid, F. (2022) · ACS Polymers Au · doi:10.1021/acspolymersau.2c0004 · Cited by 95
“Techniques are available how to optimize the partitioning into macrostates [414], how to determine transition rates and the associated uncertainty [415], and even, how to use MSMs to connect theoretical models to experimental trajectory data [416].”
BackgroundCoarse-grained modelling out of equilibrium
Schilling, T. (2022) · Physics Reports · doi:10.1016/j.physrep.2022.04.006 · Cited by 90
Time-Lagged Independent Component Analysis of Random Walks and Protein Dynamics
Schultze, S., & Grubmüller, H. (2021) · bioRxiv · doi:10.1101/2021.03.18.435940 · Cited by 92
“This property also renders these dimension reduction techniques highly useful as a preprocessing step to describing the conformational dynamics of macromolecules in terms of a discrete Markov process [17, 18, 19].”
Protein Conformation Generation via Force-Guided SE(3) Diffusion ModelsPreprint
Wang, Y., Wang, L., Shen, Y., Wang, Y., Yuan, H., Wu, Y., & Gu, Q. (2024) · arXiv (Cornell University) · doi:10.48550/arxiv.2403.14088 · Cited by 43
Navigating protein landscapes with a machine-learned transferable coarse-grained model
Charron, N. E., Bonneau, K., Pasos-Trejo, A. S., Guljas, A., Chen, Y., Musil, F., Venturin, J., Gusew, D., Zaporozhets, I., Krämer, A., et al. (2025) · Nature Chemistry · doi:10.1038/s41557-025-01874-0 · Cited by 93
“In the case of Homeodomain, the CG model stabilizes a state very close to the reference native structure, with higher relative flexibility at the N and C termini comparable with the all-atom simulations.”
Self-citationDistant residues modulate conformational opening in SARS-CoV-2 spike protein
Ray, D., Le, L., & Andricioaei, I. (2020) · Proceedings of the National Academy of Sciences of the United States of America · doi:10.1101/2020.12.07.415596 · Cited by 89
Identification of simple reaction coordinates from complex dynamics
McGibbon, R. T., Husic, B. E., & Pande, V. S. (2017) · The Journal of Chemical Physics · doi:10.1063/1.4974306 · Cited by 92
Gating mechanism of the extracellular entry to the lipid pathway in a TMEM16 scramblase
Lee, B., Khelashvili, G., Falzone, M. E., Menon, A. K., Weinstein, H., & Accardi, A. (2018) · Nature Communications · doi:10.1038/s41467-018-05724-1 · Cited by 93
Perspective: Markov models for long-timescale biomolecular dynamics
Schwantes, C. R., McGibbon, R. T., & Pande, V. S. (2014) · The Journal of Chemical Physics · doi:10.1063/1.4895044 · Cited by 87
Galerkin approximation of dynamical quantities using trajectory data
Thiede, E. H., Giannakis, D., Dinner, A. R., & Weare, J. (2019) · The Journal of Chemical Physics · doi:10.1063/1.5063730 · Cited by 91
Pantsar, T., Rissanen, S., Dauch, D., Laitinen, T., Vattulainen, I., & Poso, A. (2018) · PLOS Computational Biology · doi:10.1371/journal.pcbi.1006458 · Cited by 85
“In our work, we unlocked this issue through the analysis of the simulation data using PCA, interaction network analysis and MSMs that indeed revealed the differences, not only between the wild-type and the mutants, but also between the mutants.”
MethodInfluentialWhat Makes a Kinase Promiscuous for Inhibitors?
Hanson, S. M., Georghiou, G. P., Thakur, M. K., Miller, W. T., Rest, J. S., Chodera, J. D., & Seeliger, M. A. (2019) · Cell chemical biology · doi:10.1016/j.chembiol.2018.11.005 · Cited by 110
“The time-lagged independent component analysis (TICA) projection was then calculated from these coordinates to find the most kinetically relevant linear combination (Perez-Hernandez et al., 2013), and cumulative sum of the eigenvalues began to flatten despite the small number coordinates used (Figure S4A).”
Da, L., Pardo‐Ávila, F., Xu, L., Silva, D., Zhang, L., Gao, X., Wang, D., & Huang, X. (2016) · Nature Communications · doi:10.1038/ncomms11244 · Cited by 91
Machine learning implicit solvation for molecular dynamics
Chen, Y., Krämer, A., Charron, N. E., Husic, B. E., Clementi, C., & Noé, F. (2021) · The Journal of Chemical Physics · doi:10.1063/5.0059915 · Cited by 82
Appadurai, R., Nagesh, J., & Srivastava, A. K. (2020) · Nature Communications · doi:10.1101/2020.06.02.124628 · Cited by 80
Str2Str: A Score-based Framework for Zero-shot Protein Conformation SamplingPreprint
Lu, J., Zhong, B., Zhang, Z., & Tang, J. (2023) · arXiv (Cornell University) · doi:10.48550/arxiv.2306.03117 · Cited by 35
“To better demonstrate the performance of S TR 2S TR , we present the TICA plots (P´erez-Hern´andez et al., 2013) for the sampled conformations from each method.”
BackgroundMethodMethods for data-driven multiscale model discovery for materials
Brunton, S. L., & Kutz, J. N. (2019) · Journal of Physics Materials · doi:10.1088/2515-7639/ab291e · Cited by 79
Accurate Estimation of Protein Folding and Unfolding Times: Beyond Markov State Models
Suárez, E., Adelman, J. L., & Zuckerman, D. M. (2016) · Journal of Chemical Theory and Computation · doi:10.1021/acs.jctc.6b00339 · Cited by 74
Markov Models of Molecular Kinetics
Noé, F., & Rosta, E. (2019) · The Journal of Chemical Physics · doi:10.1063/1.5134029 · Cited by 82
Metadata
- Authors
- Pérez-Hernández, G., Paul, F., Giorgino, T., De Fabritiis, G., & Noé, F.
- Affiliations
- Freie Universität Berlin (DE)
- Max Planck Institute of Colloids and Interfaces (DE)
- National Research Council (IT)
- GRIB 4 , Barcelona Biomedical Research Park (PRBB), C/Dr. Aiguader 88, 08003 Barcelona, Spain
- Venue
- The Journal of Chemical Physics
- Publisher
- AIP Publishing
- Published
- 2013-07-03
- Volume
- 139
- Issue
- 1
- Pages
- 015102-015102
- ISSN
- 0021-9606 (ISSN-L)
- 1089-7690
- OpenAlex
- W2143592395
- Cited by
- 990
- References
- 98
- Citations by year
- 2026: 92 · 2025: 89 · 2024: 95 · 2023: 106 · 2022: 103 · 2021: 102 · 2020: 80 · 2019: 87 · 2018: 89 · 2017: 61 · 2016: 42 · 2015: 27 · 2014: 14 · 2013: 2
- FWCI
- 12.69
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- The Journal of Chemical Physics (publishedVersion)
- PubMed (publishedVersion)
- Repositori digital de la UPF (Universitat Pompeu Fabra) (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- Refubium (Universitätsbibliothek der Freien Universität Berlin) (submittedVersion)
- arXiv (Cornell University)
- Funders
- Deutsche Forschungsgemeinschaft
- Generalitat de Catalunya
- Max-Planck-Gesellschaft
- Ministerio de Ciencia e Innovación (BIO2011/27450)
- Agència de Gestió d'Ajuts Universitaris i de Recerca
- FP7 Ideas: European Research Council (307494)
- Topics
- Life Sciences › Biochemistry, Genetics and Molecular Biology › Molecular Biology › Protein Structure and Dynamics
- Physical Sciences › Chemistry › Physical and Theoretical Chemistry › Advanced Physical and Chemical Molecular Interactions
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Model Reduction and Neural Networks
- Keywords
- Markov model, conformational dynamics, variational principle, time-lagged independent component analysis, reaction coordinate, master equation, peptide dynamics, intrinsically disordered proteins
- MeSH terms
- Algorithms, Kinetics, Markov Chains, Molecular Conformation, Phosphotransferases, Temperature, Protein Structure, Tertiary, Molecular Dynamics Simulation
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