Identifying a reduced set of collective variables is critical for understanding atomistic simulations and accelerating them through enhanced sampling techniques. Recently, several methods have been proposed to learn these variables directly from atomistic data. Depending on the type of data available, the learning process can be framed as dimensionality reduction, classification of metastable states, or identification of slow modes. Here, we present mlcolvar, a Python library that simplifies the construction of these variables and their use in the context of enhanced sampling through a contributed interface to the PLUMED software. The library is organized modularly to facilitate the extension and cross-contamination of these methodologies. In this spirit, we developed a general multi-task learning framework in which multiple objective functions and data from different simulations can be combined to improve the collective variables. The library’s versatility is demonstrated through simple examples that are prototypical of realistic scenarios.
REFERENCES
1.
D.
Frenkel
and B.
Smit
, Understanding Molecular Simulation: From Algorithms to Applications
(Elsevier
, 2001
), Vol. 1
.2.
J.
Behler
and M.
Parrinello
, “Generalized neural-network representation of high-dimensional potential-energy surfaces
,” Phys. Rev. Lett.
98
, 146401
(2007
).3.
J.
Behler
and G.
Csányi
, “Machine learning potentials for extended systems: A perspective
,” Eur. Phys. J. B
94
, 142
(2021
).4.
O. T.
Unke
, S.
Chmiela
, H. E.
Sauceda
, M.
Gastegger
, I.
Poltavsky
, K. T.
Schütt
, A.
Tkatchenko
, and K. R.
Müller
, “Machine learning force fields
,” Chem. Rev.
121
, 10142
(2021
).5.
F.
Noé
, A.
Tkatchenko
, K.-R.
Müller
, and C.
Clementi
, “Machine learning for molecular simulation
,” Annu. Rev. Phys. Chem.
71
, 361
(2020
).6.
L.
Bonati
, “Machine learning and enhanced sampling simulations
,” Ph.D. thesis, Swiss Federal Institute of Technology (ETH) Zürich
, 2021
.7.
M.
Chen
, “Collective variable-based enhanced sampling and machine learning
,” Eur. Phys. J. B
94
, 211
(2021
).8.
H.
Sidky
, W.
Chen
, and A. L.
Ferguson
, “Machine learning for collective variable discovery and enhanced sampling in biomolecular simulation
,” Mol. Phys.
118
, 1737742
(2020
).9.
Y.
Wang
, J. M.
Lamim Ribeiro
, and P.
Tiwary
, “Machine learning approaches for analyzing and enhancing molecular dynamics simulations
,” Curr. Opin. Struct. Biol.
61
, 139
–145
(2020
).10.
O.
Valsson
, P.
Tiwary
, and M.
Parrinello
, “Enhancing important fluctuations: Rare events and metadynamics from a conceptual viewpoint
,” Annu. Rev. Phys. Chem.
67
, 159
–184
(2016
).11.
J.
Hénin
, T.
Lelièvre
, M.
Shirts
, O.
Valsson
, and L.
Delemotte
, “Enhanced sampling methods for molecular dynamics simulations [article v1. 0]
,” Living J. Comput. Mol. Sci.
4
, 1583
(2022
).12.
Y. I.
Yang
, Q.
Shao
, J.
Zhang
, L.
Yang
, and Y. Q.
Gao
, “Enhanced sampling in molecular dynamics
,” J. Chem. Phys.
151
, 070902
(2019
).13.
G.
Bussi
and D.
Branduardi
, Free-Energy Calculations with Metadynamics: Theory and Practice
(Wiley
, 2015
), pp. 1
–49
.14.
J.
Rogal
, E.
Schneider
, and M. E.
Tuckerman
, “Neural-network-based path collective variables for enhanced sampling of phase transformations
,” Phys. Rev. Lett.
123
, 245701
(2019
).15.
T.
Karmakar
, M.
Invernizzi
, V.
Rizzi
, and M.
Parrinello
, “Collective variables for the study of crystallisation
,” Mol. Phys.
119
, e1893848
(2021
).16.
O.
Elishav
, R.
Podgaetsky
, O.
Meikler
, and B.
Hirshberg
, “Collective variables for conformational polymorphism in molecular crystals
,” J. Phys. Chem. Lett.
14
, 971
–976
(2023
).17.
G.
Piccini
, D.
Mendels
, and M.
Parrinello
, “Metadynamics with discriminants: A tool for understanding chemistry
,” J. Chem. Theory Comput.
14
, 5040
–5044
(2018
).18.
D.
Mendels
, G.
Piccini
, Z. F.
Brotzakis
, Y. I.
Yang
, M.
Parrinello
, Z. F.
Brotzakis
, Y. I.
Yang
, and M.
Parrinello
, “Folding a small protein using harmonic linear discriminant analysis
,” J. Chem. Phys.
149
, 194113
(2018
).19.
U.
Raucci
, V.
Rizzi
, and M.
Parrinello
, “Discover, sample, and refine: Exploring chemistry with enhanced sampling techniques
,” J. Phys. Chem. Lett.
13
, 1424
(2022
).20.
S.
Das
, U.
Raucci
, R. P. P.
Neves
, M. J.
Ramos
, and M.
Parrinello
, “How and when does an enzyme react? Unraveling α-amylase catalytic activity with enhanced sampling techniques
,” ACS Catal.
13
, 8092
–8098
(2023
).21.
M.
Bertazzo
, D.
Gobbo
, S.
Decherchi
, and A.
Cavalli
, “Machine learning and enhanced sampling simulations for computing the potential of mean force and standard binding free energy
,” J. Chem. Theory Comput.
17
, 5287
–5300
(2021
).22.
N.
Ansari
, V.
Rizzi
, and M.
Parrinello
, “Water regulates the residence time of benzamidine in trypsin
,” Nat. Commun.
13
, 5438
(2022
).23.
J. M.
Lamim Ribeiro
, D.
Provasi
, and M.
Filizola
, “A combination of machine learning and infrequent metadynamics to efficiently predict kinetic rates, transition states, and molecular determinants of drug dissociation from g protein-coupled receptors
,” J. Chem. Phys.
153
, 124105
(2020
).24.
V.
Rizzi
, L.
Bonati
, N.
Ansari
, and M.
Parrinello
, “The role of water in host-guest interaction
,” Nat. Commun.
12
, 93
(2021
).25.
M.
Badaoui
, P. J.
Buigues
, D.
Berta
, G. M.
Mandana
, H.
Gu
, T.
Földes
, C. J.
Dickson
, V.
Hornak
, M.
Kato
, C.
Molteni
et al, “Combined free-energy calculation and machine learning methods for understanding ligand unbinding kinetics
,” J. Chem. Theory Comput.
18
, 2543
–2555
(2022
).26.
M. M.
Sultan
, H. K.
Wayment-Steele
, and V. S.
Pande
, “Transferable neural networks for enhanced sampling of protein dynamics
,” J. Chem. Theory Comput.
14
, 1887
–1894
(2018
).27.
D.
Ray
, E.
Trizio
, and M.
Parrinello
, “Deep learning collective variables from transition path ensemble
,” J. Chem. Phys.
158
, 204102
(2023
).28.
H.
Chen
, H.
Liu
, H.
Feng
, H.
Fu
, W.
Cai
, X.
Shao
, and C.
Chipot
, “MLCV: Bridging machine-learning-based dimensionality reduction and free-energy calculation
,” J. Chem. Inf. Model.
62
, 1
(2021
).29.
R.
Ketkaew
and S.
Luber
, “DeepCV: A deep learning framework for blind search of collective variables in expanded configurational space
,” J. Chem. Inf. Model.
64
, 6352
(2022
).30.
D.
Trapl
, I.
Horvacanin
, V.
Mareska
, F.
Ozcelik
, G.
Unal
, and V.
Spiwok
, “Anncolvar: Approximation of complex collective variables by artificial neural networks for analysis and biasing of molecular simulations
,” Front. Mol. Biosci.
6
, 25
(2019
).31.
G. A.
Tribello
, M.
Bonomi
, D.
Branduardi
, C.
Camilloni
, and G.
Bussi
, “PLUMED 2: New feathers for an old bird
,” Comput. Phys. Commun.
185
, 604
–613
(2014
).32.
B.
Hashemian
, D.
Millán
, and M.
Arroyo
, “Modeling and enhanced sampling of molecular systems with smooth and nonlinear data-driven collective variables
,” J. Chem. Phys.
139
, 214101
(2013
).33.
W.
Chen
, A. R.
Tan
, and A. L.
Ferguson
, “Collective variable discovery and enhanced sampling using autoencoders: Innovations in network architecture and error function design
,” J. Chem. Phys.
149
, 072312
(2018
).34.
Neha
, V.
Tiwari
, S.
Mondal
, N.
Kumari
, and T.
Karmakar
, “Collective variables for crystallization simulations—From early developments to recent advances
,” ACS Omega
8
, 127
(2022
).35.
O.
Fleetwood
, M. A.
Kasimova
, A. M.
Westerlund
, and L.
Delemotte
, “Molecular insights from conformational ensembles via machine learning
,” Biophys. J.
118
, 765
–780
(2020
).36.
H.
Jung
, R.
Covino
, A.
Arjun
, C.
Leitold
, C.
Dellago
, P. G.
Bolhuis
, and G.
Hummer
, “Machine-guided path sampling to discover mechanisms of molecular self-organization
,” Nat. Comput. Sci.
3
, 334
(2023
).37.
P.
Novelli
, L.
Bonati
, M.
Pontil
, and M.
Parrinello
, “Characterizing metastable states with the help of machine learning
,” J. Chem. Theory Comput.
18
, 5195
–5202
(2022
).38.
I.
Goodfellow
, Y.
Bengio
, and A.
Courville
, Deep Learning
(MIT Press
, 2017
), http://www.deeplearningbook.org.39.
J. M. L.
Ribeiro
, P.
Bravo
, Y.
Wang
, and P.
Tiwary
, “Reweighted autoencoded variational Bayes for enhanced sampling (RAVE)
,” J. Chem. Phys.
149
, 072301
(2018
).40.
T.
Lemke
and C.
Peter
, “EncoderMap: Dimensionality reduction and generation of molecule conformations
,” J. Chem. Theory Comput.
15
, 1209
–1215
(2019
).41.
D.
Mendels
, G.
Piccini
, and M.
Parrinello
, “Collective variables from local fluctuations
,” J. Phys. Chem. Lett.
9
, 2776
–2781
(2018
).42.
M. M.
Sultan
and V. S.
Pande
, “Automated design of collective variables using supervised machine learning
,” J. Chem. Phys.
149
, 094106
(2018
).43.
L.
Bonati
, V.
Rizzi
, and M.
Parrinello
, “Data-driven collective variables for enhanced sampling
,” J. Phys. Chem. Lett.
11
, 2998
–3004
(2020
).44.
R. T.
McGibbon
, B. E.
Husic
, and V. S.
Pande
, “Identification of simple reaction coordinates from complex dynamics
,” J. Chem. Phys.
146
, 044109
(2017
).45.
C.
Wehmeyer
and F.
Noé
, “Time-lagged autoencoders: Deep learning of slow collective variables for molecular kinetics
,” J. Chem. Phys.
148
, 241703
(2018
).46.
M.
Schöberl
, N.
Zabaras
, and P.-S.
Koutsourelakis
, “Predictive collective variable discovery with deep Bayesian models
,” J. Chem. Phys.
150
, 024109
(2019
).47.
Y.
Wang
, J. M. L.
Ribeiro
, and P.
Tiwary
, “Past-future information bottleneck for sampling molecular reaction coordinate simultaneously with thermodynamics and kinetics
,” Nat. Commun.
10
, 3573
(2019
).48.
A.
Mardt
, L.
Pasquali
, H.
Wu
, and F.
Noé
, “VAMPnets for deep learning of molecular kinetics
,” Nat. Commun.
9
, 5
(2018
).49.
P.
Tiwary
and B. J.
Berne
, “Spectral gap optimization of order parameters for sampling complex molecular systems
,” Proc. Natl. Acad. Sci. U. S. A.
113
, 2839
–2844
(2016
).50.
F.
Noé
and C.
Clementi
, “Collective variables for the study of long-time kinetics from molecular trajectories: Theory and methods
,” Curr. Opin. Struct. Biol.
43
, 141
–147
(2017
).51.
W.
Chen
and A. L.
Ferguson
, “Molecular enhanced sampling with autoencoders: On-the-fly collective variable discovery and accelerated free energy landscape exploration
,” J. Comput. Chem.
39
, 2079
–2102
(2018
).52.
L.
Bonati
, G.
Piccini
, and M.
Parrinello
, “Deep learning the slow modes for rare events sampling
,” Proc. Natl. Acad. Sci. U. S. A.
118
, e2113533118
(2021
).53.
Z.
Belkacemi
, P.
Gkeka
, T.
Lelièvre
, and G.
Stoltz
, “Chasing collective variables using autoencoders and biased trajectories
,” J. Chem. Theory Comput.
18
, 59
–78
(2022
).54.
H.
Chen
and C.
Chipot
, “Chasing collective variables using temporal data-driven strategies
,” QRB Discovery
4
, e2
(2023
).55.
G. A.
Tribello
and P.
Gasparotto
, “Using dimensionality reduction to analyze protein trajectories
,” Front. Mol. Biosci.
6
, 46
(2019
).56.
M.
Ceriotti
, “Unsupervised machine learning in atomistic simulations, between predictions and understanding
,” J. Chem. Phys.
150
, 150901
(2019
).57.
E.
Trizio
and M.
Parrinello
, “From enhanced sampling to reaction profiles
,” J. Phys. Chem. Lett.
12
, 8621
–8626
(2021
).58.
K.
Lindorff-Larsen
, S.
Piana
, R. O.
Dror
, and D. E.
Shaw
, “How fast-folding proteins fold
,” Science
334
, 517
–520
(2011
).59.
L.
Molgedey
and H. G.
Schuster
, “Separation of a mixture of independent signals using time delayed correlations
,” Phys. Rev. Lett.
72
, 3634
(1994
).60.
Y.
Naritomi
and S.
Fuchigami
, “Slow dynamics in protein fluctuations revealed by time-structure based independent component analysis: The case of domain motions
,” J. Chem. Phys.
134
, 065101
(2011
).61.
G.
Pérez-Hernández
, F.
Paul
, T.
Giorgino
, G. D.
Fabritiis
, and F.
Noè
, “Identification of slow molecular order parameters for Markov model construction
,” J. Chem. Phys.
139
, 015102
(2013
).62.
W.
Chen
, H.
Sidky
, and A. L.
Ferguson
, “Nonlinear discovery of slow molecular modes using state-free reversible VAMPnets
,” J. Chem. Phys.
150
, 214114
(2019
).63.
C. X.
Hernández
, H. K.
Wayment-Steele
, M. M.
Sultan
, B. E.
Husic
, and V. S.
Pande
, “Variational encoding of complex dynamics
,” Phys. Rev. E
97
, 062412
(2018
).64.
65.
L.
Sun
, J.
Vandermause
, S.
Batzner
, Y.
Xie
, D.
Clark
, W.
Chen
, and B.
Kozinsky
, “Multitask machine learning of collective variables for enhanced sampling of rare events
,” J. Chem. Theory Comput.
18
, 2341
–2353
(2022
).66.
M.
Ceriotti
, G. A.
Tribello
, and M.
Parrinello
, “Simplifying the representation of complex free-energy landscapes using sketch-map
,” Proc. Natl. Acad. Sci. U. S. A.
108
, 13023
–13028
(2011
).67.
A.
Paszke
, S.
Gross
, F.
Massa
, A.
Lerer
, J.
Bradbury
, G.
Chanan
, T.
Killeen
, Z.
Lin
, N.
Gimelshein
, L.
Antiga
et al, “PyTorch: An imperative style, high-performance deep learning library
,” in Advances in Neural Information Processing Systems 32
(Curran Associates
, 2019
); arXiv.1912.01703.68.
W.
Falcon
and PyTorch Lightning Team
(2023
). “PyTorch Lightning (2.0.4),” Zenodo
. https://doi.org/10.5281/zenodo.807171069.
G. M.
Torrie
and J. P.
Valleau
, “Nonphysical sampling distributions in Monte Carlo free-energy estimation: Umbrella sampling
,” J. Comput. Phys.
23
, 187
–199
(1977
).70.
A.
Laio
and M.
Parrinello
, “Escaping free-energy minima
,” Proc. Natl. Acad. Sci. U. S. A.
99
, 12562
–12566
(2002
).71.
G.
Bussi
and A.
Laio
, “Using metadynamics to explore complex free-energy landscapes
,” Nat. Rev. Phys.
2
, 200
–212
(2020
).72.
O.
Valsson
and M.
Parrinello
, “Variational approach to enhanced sampling and free energy calculations
,” Phys. Rev. Lett.
113
, 090601
(2014
).73.
L.
Bonati
, Y.-Y.
Zhang
, and M.
Parrinello
, “Neural networks-based variationally enhanced sampling
,” Proc. Natl. Acad. Sci. U. S. A.
116
, 17641
–17647
(2019
).74.
M.
Invernizzi
and M.
Parrinello
, “Rethinking metadynamics: From bias potentials to probability distributions
,” J. Phys. Chem. Lett.
11
, 2731
–2736
(2020
).75.
T.
Giorgino
, “PYCV: A PLUMED 2 module enabling the rapid prototyping of collective variables in Python
,” J. Open Source Software
4
, 1773
(2019
).76.
C. R.
Harris
, K. J.
Millman
, S. J.
van der Walt
, R.
Gommers
, P.
Virtanen
, D.
Cournapeau
, E.
Wieser
, J.
Taylor
, S.
Berg
, N. J.
Smith
, R.
Kern
, M.
Picus
, S.
Hoyer
, M. H.
van Kerkwijk
, M.
Brett
, A.
Haldane
, J. F.
del Río
, M.
Wiebe
, P.
Peterson
, P.
Gérard-Marchant
, K.
Sheppard
, T.
Reddy
, W.
Weckesser
, H.
Abbasi
, C.
Gohlke
, and T. E.
Oliphant
, “Array programming with NumPy
,” Nature
585
, 357
–362
(2020
).77.
Pandas Development Team
, pandas-dev/pandas: Pandas, 2020
.78.
I. T.
Jolliffe
, Principal Component Analysis for Special Types of Data
(Springer
, 2002
).79.
D. P.
Kingma
and M.
Welling
, “Auto-encoding variational Bayes
,” presented at the 2nd International Conference on Learning Representations (ICLR 2014), Banff, AB, Canada, 14-16 April 2014; arXiv:1312.6114 (2013
).80.
G. A.
Tribello
, M.
Ceriotti
, and M.
Parrinello
, “Using sketch-map coordinates to analyze and bias molecular dynamics simulations
,” Proc. Natl. Acad. Sci. U. S. A.
109
, 5196
–5201
(2012
).81.
J.
Rydzewski
and O.
Valsson
, “Multiscale reweighted stochastic embedding: Deep learning of collective variables for enhanced sampling
,” J. Phys. Chem. A
125
, 6286
–6302
(2021
).82.
M.
Welling
, Fisher Linear Discriminant Analysis
(Department of Computer Science, University of Toronto
, 2005
).83.
M.
Dorfer
, R.
Kelz
, and G.
Widmer
, “Deep linear discriminant analysis
,” in Proceedings of the 4th International Conference on Learning Representations
(ICLR 2016); arXiv:1511.04707 (2015
).84.
J.-H.
Prinz
, H.
Wu
, M.
Sarich
, B.
Keller
, M.
Senne
, M.
Held
, J. D.
Chodera
, C.
Schütte
, and F.
Noé
, “Markov models of molecular kinetics: Generation and validation
,” J. Chem. Phys.
134
, 174105
(2011
).85.
M. M.
Sultan
and V. S.
Pande
, “tICA-metadynamics: Accelerating metadynamics by using kinetically selected collective variables
,” J. Chem. Theory Comput.
13
, 2440
–2447
(2017
).86.
J.
McCarty
and M.
Parrinello
, “A variational conformational dynamics approach to the selection of collective variables in metadynamics
,” J. Chem. Phys.
147
, 204109
(2017
).87.
Y. I.
Yang
and M.
Parrinello
, “Refining collective coordinates and improving free energy representation in variational enhanced sampling
,” J. Chem. Theory Comput.
14
, 2889
–2894
(2018
).88.
V.
Kostic
, P.
Novelli
, A.
Maurer
, C.
Ciliberto
, L.
Rosasco
, and M.
Pontil
, “Learning dynamical systems via Koopman operator regression in reproducing kernel Hilbert spaces
,” in Advances in Neural Information Processing Systems (NeurIPS) 2022
(Neural Information Processing Systems Foundation, 2022); arXiv:2205.14027 (2022
).89.
W.
Chen
, H.
Sidky
, and A. L.
Ferguson
, “Capabilities and limitations of time-lagged autoencoders for slow mode discovery in dynamical systems
,” J. Chem. Phys.
151
, 064123
(2019
).90.
K. T.
Schütt
, H. E.
Sauceda
, P.-J.
Kindermans
, A.
Tkatchenko
, and K.-R.
Müller
, “SchNet—A deep learning architecture for molecules and materials
,” J. Chem. Phys.
148
, 241722
(2018
).91.
PLUMED Consortium
, “Promoting transparency and reproducibility in enhanced molecular simulations
,” Nat. Methods
16
, 670
–673
(2019
).© 2023 Author(s). Published under an exclusive license by AIP Publishing.
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