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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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Rational design of a Kappa opioid receptor peptide agonist with attenuated β-arrestin signaling
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“Examples include time-lagged independent component analysis (TICA) [1], variational approaches to Markov processes (VAMP, VAMPnets) [2], approaches based on normalizing flows [3] and dominant subspace learning methods such as ISOKANN [4], which minimize the deviation between effective and full…”
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“…number of paths collected, x are the starting and end points of the k th path sample, 1 S i ( · ), 1 S j ( · ) are indicator functions of whether the phase space coordinates x 0 or x τ belong to the i th or j th state and Ω d ω , Γ d x 0 are integrals of all ω or x 0 over the whole path space…”
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“…, JS-Rg assess the distance between the model distribution and reference distribution, which includes the pairwise distance (PwD), time-lagged independent components (TIC) (Naritomi & Fuchigami, 2011; P´erez-Hern´andez et al., 2013) and radius of gyration (Rg), which are the lower the better ( ↓ ).”
BackgroundCooperative Membrane Binding of HIV-1 Matrix Proteins
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“Subsequently, we performed a time-lagged independent component analysis (TICA) on the feature selected to reduce the dimensionality of the feature space, using a lag-time of 5.0 ns and preserving 95% of the kinetic variance.”
InfluentialEnhanced sampling in explicit solvent by deep learning module in FSATOOL
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“The output of the HDXer method is a large number of structures (frames) that can be difficult to interpret and clustering methods such as tICA (61) can be used to identify similarities between structures facilitating structural interpretations of the HDX-MS data.”
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“36 tICA was developed to obtain the slowest collective motions by linear combinations of input degrees of freedom.”
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“We again used the fluctuation-amplification of specific traits (FAST) approach[56] to adaptive sampling, with clustering performed through time-lagged independent component analysis (TICA)[57, 58] using a lag time of 5 ns and Cα distances fed to the k-means clustering algorithm[59] to yield 128 clusters.”
MethodMarkov state models from hierarchical density-based assignment
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“Dimensionality reduction was achieved using tIC analysis, previously shown to form a good basis for discretization of conformational space into Markov states, as the tIC eigenvetors linearly approximate the MSM eigenvectors [54, 55].”
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“Time-lagged Independent Component Analysis (tICA) is a dimensionality reduction method that identifies the slowest dynamical processes in time-dependent data. tICA finds the combination of input variables which maximize self time-correlation after a chosen lag-time τ [39, 45–47].”
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“Second, many widely used statistical learning meth-10 ods, such as principal component analysis (PCA) and time-lagged independent component analysis (TICA) [58, 59], construct CVs through full-rank linear transformations, which are inherently bijective.”
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“42 The advantage of selecting physics-based reaction coordinates is that, when comparing the same system under slightly different conditions, the changes may always be followed intuitively and comparatively.”
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“…can be challenging; however, since the eigenfunctions are maximally autocorrelated, its eigenfunctions can be obtained with Time-lagged Independent Component Analysis (TICA), 54,56 a statistical method that searches for the linear projection of a set of descriptors with maximum autocorrelation.”
Automatic forward model parameterization with Bayesian inference of conformational populations
Raddi, R. M., Marshall, T., & Voelz, V. A. (2026) · APL Machine Learning · doi:10.1063/5.0287423 · Cited by 1
Tracking the protein conformational motions driving HIV-1 membrane fusion
Unarta, I., Crotzer, S. M., & Gnanakaran, S. (2026) · Scientific Reports · doi:10.1038/s41598-025-33350-7 · Cited by 1
“To visualize our simulations, we perform dimensionality reduction of all the structures from the simulations using time-lagged Independent Component Analysis (tICA) 71,72 .”
Bansal, P. D., & Shukla, D. (2026) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.64898/2026.01.05.697796 · Cited by 2
Sampling Function-Related Metastable States of Proteins With DASH
Zha, J., Zheng, Z., Zhong, J., Wang, W., Li, Q.-H., Shen, Q., Li, M.-Y., Wu, C., Xiao, Q.-J., Ren, Q.-H., et al. (2025) · bioRxiv · doi:10.64898/2025.12.24.696448 · Cited by 1
“2a, CTMAE demonstrate well performance in all metrics, whereas other methods excel in fewer areas: PCA 48 and IsoMap 49 achieve better marginal quality; autoencoder-based 35 methods do well in motion preservation and TICA 50 excel at separating different secondary-structure states.”
Prypoten, V., Norton, R. S., & Chalmers, D. K. (2025) · Journal of Chemical Information and Modeling · doi:10.1021/acs.jcim.5c02118 · Cited by 1
Pedroni, L., Perugino, F., Dall'Asta, C., Alvarez-Erviti, L., Mally, A., Vettorazzi, A., & Dellafiora, L. (2025) · Food Research International · doi:10.1016/j.foodres.2025.117532 · Cited by 1
Oh, M., Oh, C., & Tabassum, E. (2025) · Journal of Chemical Theory and Computation · doi:10.1021/acs.jctc.5c01402 · Cited by 1
Entangled Schrödinger Bridge Matching
Tang, S., Zhang, Y., & Chatterjee, P. (2025) · arXiv.org · doi:10.48550/arxiv.2511.07406 · Cited by 1
Coarse-Graining Self-Assembly by the Stochastic Landscape Method
Faran, M., & Bisker, G. (2025) · Journal of Chemical Theory and Computation · doi:10.1021/acs.jctc.5c01241 · Cited by 1
Effect of Glycosylation on the Reorganization at the Active Site of Human Carbonic Anhydrase IX.
Dey, R., Shukla, K., Das, A., & Taraphder, S. (2025) · ChemPhysChem · doi:10.1002/cphc.202500573 · Cited by 1
Random Functions as Data Compressors for Machine Learning of Molecular Processes
Debnath, J., & Hummer, G. (2025) · Journal of Chemical Theory and Computation · doi:10.1021/acs.jctc.5c01638 · Cited by 1
“…in the context of time dependent data is Time-structure Independent Component Analysis (TICA) where the data are projected onto the generalized eigenvectors of time-lagged covariance matrix, accounting for static correlations, to separate slow from fast relaxation processes 16 that generated it.”
Shi, D., Lv, E., Li, W., Xia, Q., Du, H., & Tian, J. (2025) · International Journal of Biological Macromolecules · doi:10.1016/j.ijbiomac.2025.147705 · Cited by 2
ESMDynamic: Fast and Accurate Prediction of Protein Dynamic Contact Maps from Single Sequences
Kleiman, D., Feng, J.-Y., Xue, Z., & Shukla, D. (2025) · bioRxiv · doi:10.1101/2025.08.20.671365 · Cited by 1
“These distances are then used to featurize the MD followed by time-lagged independent component analysis (tICA) 52,53 for dimensionality reduction. tICA identifies the slowest decorrelating collective motions in the trajectory by maximizing autocorrelation over a specified lag time.”
Locating Transition States for Biomolecular Dynamics via Invertible Dimensionality Reduction.
Yang, J.-Y., Guo, H., Liu, S., Xi, K., Wu, Q., Li, Y.-X., Fang, K., Zhou, K., Su, C., Jing, B.-Y., et al. (2025) · Journal of Chemical Theory and Computation · doi:10.1021/acs.jctc.4c01624 · Cited by 1
Song, X., Wang, D.-D., Ji, J., Weng, J.-W., & Wang, W.-N. (2025) · Journal of Chemical Theory and Computation · doi:10.1021/acs.jctc.4c01308 · Cited by 1
Calculating linear and nonlinear multi-ensemble slow collective variables for protein folding.
Wu, M.-C., Liao, J., Meng, F., & Chen, C.-J. (2024) · Journal of Chemical Physics · doi:10.1063/5.0232102 · Cited by 1
A Mode Evolution Metric to Extract Reaction Coordinates for Biomolecular Conformational Transitions
Das, M., & Venkatramani, R. (2024) · Journal of Chemical Theory and Computation · doi:10.1021/acs.jctc.4c00744 · Cited by 2
Karasawa, N., Mitsutake, A., & Takano, H. (2024) · Journal of Chemical Physics · doi:10.1063/5.0211504 · Cited by 1
Weigle, A. T., & Shukla, D. (2024) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.1101/2024.06.12.598708 · Cited by 2
“The number of processes plotted for implied timescale analysis was determined via spectral decomposition analysis (62, 63) ( Figure S2 ).”
MethodIsomerization Pathways of a Mismatched Base Pair of A:8OG in Free Duplex DNA
Kim, H., & Pak, Y. (2024) · Journal of Chemical Information and Modeling · doi:10.1021/acs.jcim.4c00563 · Cited by 1
Understanding the Fast-Triggering Unfolding Dynamics of FK-11 upon Photoexcitation of Azobenzene
Xu, T., Li, Y., Gao, X., & Zhang, L. (2024) · The Journal of Physical Chemistry Letters · doi:10.1021/acs.jpclett.4c00091 · Cited by 1
Binding Position Dependent Modulation of Smoothened Activity by CyclopaminePreprint
Kim, K., Bansal, P. D., & Shukla, D. (2024) · bioRxiv · doi:10.1101/2024.02.08.579369 · Cited by 1
“Prior to MSM construction, time-lagged Independent Component Analysis (tICA) was performed to reduce the high dimensionality of data 70 (Fig.”
MethodSarkar, D. K., Surpeta, B., & Brezovský, J. (2023) · bioRxiv · doi:10.1101/2023.09.21.558608 · Cited by 1
Transferable Coarse Graining via Contrastive Learning of Graph Neural Networks
Airas, J., Ding, X.-Q., & Zhang, B. (2023) · bioRxiv · doi:10.1101/2023.09.08.556923 · Cited by 1
Salamh, S., & Sayyed–Ahmad, A. (2023) · Journal of Biomolecular Structure and Dynamics · doi:10.1080/07391102.2023.2238080 · Cited by 1
“Initially time-lagged independent component analysis (TICA) (P � erez-Hern � andez et al., 2013) with a 1 : 0 ns lag-time was used to map the distances among C a atoms of the relatively dynamic amino acids D12, H27, Y32, P34, I36, E37, D38, S39, I46, D47, L56, E63, Y64, S65, A66, T74, I94, H94, H95, R102, V103, L104,…”
Improving Estimation of the Koopman Operator with Kolmogorov–Smirnov Indicator Functions
Ngo, V. A., Lin, Y. T., & Pérez, D. (2023) · Journal of Chemical Theory and Computation · doi:10.1021/acs.jctc.3c00632
Solvent’s Role in Cavity–Ligand Recognition Would Depend on the Mode of Ligand Diffusion
Bandyopadhyay, S., Majumdar, B. B., & Mondal, J. (2022) · The Journal of Physical Chemistry B · doi:10.1021/acs.jpcb.1c09645 · Cited by 3
Ahalawat, N., & Mondal, J. (2022) · bioRxiv · doi:10.1101/2022.01.07.475334 · Cited by 1
He, X.-H., Yang, Z.-Y., Chai, Z.-T., Wang, J.-Y., Rehman, A. U., Ni, D., Pu, J., Sun, J.-P., Zhang, J., & Lu, S.-Y. (2020) · doi:10.21203/rs.3.rs-104911/v1 · Cited by 1
Benchmarking ensemble docking methods as a scientific outreach projectPreprint
Gan, J. L., Kumar, D., Chen, C., Taylor, B. C., Jagger, B. R., Amaro, R., & Lee, C. T. (2020) · bioRxiv · doi:10.1101/2020.10.02.324343 · Cited by 1
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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