In this work, we introduce a flow based machine learning approach called reaction coordinate (RC) flow for the discovery of low-dimensional kinetic models of molecular systems. The RC flow utilizes a normalizing flow to design the coordinate transformation and a Brownian dynamics model to approximate the kinetics of RC, where all model parameters can be estimated in a data-driven manner. In contrast to existing model reduction methods for molecular kinetics, RC flow offers a trainable and tractable model of reduced kinetics in continuous time and space due to the invertibility of the normalizing flow. Furthermore, the Brownian dynamics-based reduced kinetic model investigated in this work yields a readily discernible representation of metastable states within the phase space of the molecular system. Numerical experiments demonstrate how effectively the proposed method discovers interpretable and accurate low-dimensional representations of given full-state kinetics from simulations.
REFERENCES
1.
F.
Noé
and F.
Nuske
, Multiscale Model. Simul.
11
, 635
(2013
).2.
F.
Nuske
, B. G.
Keller
, G.
Pérez-Hernández
, A. S.
Mey
, and F.
Noé
, J. Chem. Theory Comput.
10
, 1739
(2014
).3.
4.
S.
Klus
, F.
Nüske
, P.
Koltai
, H.
Wu
, I.
Kevrekidis
, C.
Schütte
, and F.
Noé
, J. Nonlinear Sci.
28
, 985
(2018
).5.
K. A.
Konovalov
, I. C.
Unarta
, S.
Cao
, E. C.
Goonetilleke
, and X.
Huang
, JACS Au
1
, 1330
(2021
).6.
G.
Pérez-Hernández
, F.
Paul
, T.
Giorgino
, G.
De Fabritiis
, and F.
Noé
, J. Chem. Phys.
139
, 015102
(2013
).7.
C. R.
Schwantes
and V. S.
Pande
, J. Chem. Theory Comput.
9
, 2000
(2013
).8.
I.
Mezić
, Nonlinear Dyn.
41
, 309
(2005
).9.
H.
Wu
and F.
Noé
, J. Nonlinear Sci.
30
, 23
(2020
).10.
L.
Boninsegna
, G.
Gobbo
, F.
Noé
, and C.
Clementi
, J. Chem. Theory Comput.
11
, 5947
(2015
).11.
M. O.
Williams
, I. G.
Kevrekidis
, and C. W.
Rowley
, J. Nonlinear Sci.
25
, 1307
(2015
).12.
C. R.
Schwantes
and V. S.
Pande
, J. Chem. Theory Comput.
11
, 600
(2015
).13.
F.
Nüske
, R.
Schneider
, F.
Vitalini
, and F.
Noé
, J. Chem. Phys.
144
, 054105
(2016
).14.
H.
Wu
, F.
Nüske
, F.
Paul
, S.
Klus
, P.
Koltai
, and F.
Noé
, J. Chem. Phys.
146
, 154104
(2017
).15.
M. K.
Scherer
, B. E.
Husic
, M.
Hoffmann
, F.
Paul
, H.
Wu
, and F.
Noé
, J. Chem. Phys.
150
, 194108
(2019
).16.
17.
H.
Sidky
, W.
Chen
, and A. L.
Ferguson
, J. Phys. Chem. B
123
, 7999
(2019
).18.
L.
Bonati
, G.
Piccini
, and M.
Parrinello
, Proc. Natl. Acad. Sci. U. S. A.
118
, e2113533118
(2021
).19.
M.
Ghorbani
, S.
Prasad
, J. B.
Klauda
, and B. R.
Brooks
, J. Chem. Phys.
156
, 184103
(2022
).20.
C.
Wehmeyer
and F.
Noé
, J. Chem. Phys.
148
, 241703
(2018
).21.
A.
Bittracher
, P.
Koltai
, S.
Klus
, R.
Banisch
, M.
Dellnitz
, and C.
Schütte
, J. Nonlinear Sci.
28
, 471
(2018
).22.
A.
Bittracher
, S.
Klus
, B.
Hamzi
, P.
Koltai
, and C.
Schütte
, J. Nonlinear Sci.
31
, 1
(2021
).23.
24.
25.
Z. D.
Pozun
, K.
Hansen
, D.
Sheppard
, M.
Rupp
, K.-R.
Müller
, and G.
Henkelman
, J. Chem. Phys.
136
, 174101
(2012
).26.
J.
Lu
and E.
Vanden-Eijnden
, J. Chem. Phys.
141
, 044109
(2014
).27.
F.
Legoll
and T.
Lelievre
, Nonlinearity
23
, 2131
(2010
).28.
W.
Zhang
, C.
Hartmann
, and C.
Schütte
, Faraday Discuss.
195
, 365
(2016
).29.
B.
Lusch
, J. N.
Kutz
, and S. L.
Brunton
, Nat. Commun.
9
, 1
(2018
).30.
K.
Champion
, B.
Lusch
, J. N.
Kutz
, and S. L.
Brunton
, Proc. Natl. Acad. Sci. U. S. A.
116
, 22445
(2019
).31.
I.
Kobyzev
, S. J.
Prince
, and M. A.
Brubaker
, IEEE Trans. Pattern Anal. Mach. Intell.
43
, 3964
(2021
).32.
F.
Noé
, S.
Olsson
, J.
Köhler
, and H.
Wu
, Science
365
, eaaw1147
(2019
).33.
34.
J.
Köhler
, L.
Klein
, and F.
Noé
, International Conference on Machine Learning
(PMLR
, 2020
), pp. 5361
–5370
.35.
36.
M.
Invernizzi
, A.
Krämer
, C.
Clementi
, and F.
Noé
, J. Phys. Chem. Lett.
13
, 11643
(2022
).37.
M.
Dibak
, L.
Klein
, A.
Krämer
, and F.
Noé
, Phys. Rev. Res.
4
, L042005
(2022
).38.
J.
Köhler
, M.
Invernizzi
, P.
De Haan
, and F.
Noé
, Proceedings of the 40th International Conference on Machine Learning,
Proceedings of Machine Learning Research Vol. 202 (PMLR, 2023), pp. 17301
–17326
.39.
S.-H.
Li
and L.
Wang
, Phys. Rev. Lett.
121
, 260601
(2018
).40.
G.
Kanwar
, M. S.
Albergo
, D.
Boyda
, K.
Cranmer
, D. C.
Hackett
, S.
Racaniere
, D. J.
Rezende
, and P. E.
Shanahan
, Phys. Rev. Lett.
125
, 121601
(2020
).41.
L.
Klein
, A. Y. K.
Foong
, T. E.
Fjelde
, B.
Mlodozeniec
, M.
Brockschmidt
, S.
Nowozin
, F.
Noé
, and R.
Tomioka
, in Proceedings of the 37th Conference on Neural Information Processing Systems, arXiv:2302.01170 (2023
).42.
G.
Papamakarios
, E. T.
Nalisnick
, D. J.
Rezende
, S.
Mohamed
, and B.
Lakshminarayanan
, J. Mach. Learn. Res.
22
, 1
(2021
).43.
L.
Dinh
, D.
Krueger
, and Y.
Bengio
, in 3rd International Conference on Learning Representations, {ICLR} 2015, San Diego, CA, 7-9 May 2015; arXiv:1410.8516 (2014
).44.
D.
Rezende
and S.
Mohamed
, International Conference on Machine Learning
(PMLR
, 2015
), pp. 1530
–1538
.45.
L.
Dinh
, J.
Sohl-Dickstein
, and S.
Bengio
, 5th International Conference on Learning Representations, {ICLR} 2017, Toulon, France, 24-26 April 2017
(OpenReview.net, 2017).46.
W.
Grathwohl
, R. T.
Chen
, J.
Bettencourt
, I.
Sutskever
, and D.
Duvenaud
, International Conference on Learning Representations
(2018
).47.
K.
Tang
, X.
Wan
, and Q.
Liao
, Theor. Appl. Mech. Lett.
10
, 143
(2020
).48.
49.
N.
Singhal
, C. D.
Snow
, and V. S.
Pande
, J. Chem. Phys.
121
, 415
(2004
).50.
D. H.
Anderson
, Compartmental Modeling and Tracer Kinetics
(Springer Science & Business Media
, 2013
), Vol. 50
.51.
52.
B.
Jensen
and R.
Poulsen
, J. Deriv.
9
, 18
(2002
).53.
G. B.
Durham
and A. R.
Gallant
, J. Bus. Econ. Stat.
20
, 297
(2002
).54.
D. P.
Kingma
and J.
Ba
, in 3rd International Conference on Learning Representations, {ICLR} 2015, San Diego, CA, 7-9 May 2015; arXiv:1412.6980 (2014
).55.
P.
Kidger
, J.
Foster
, X. C.
Li
, and T.
Lyons
, Adv. Neural Inf. Process. Syst.
34
, 18747
(2021
).56.
B.
Liu
, M.
Xue
, Y.
Qiu
, K. A.
Konovalov
, M. S.
O’Connor
, and X.
Huang
, J. Chem. Phys.
159
, 094901
(2023
).57.
58.
D.
Wang
and P.
Tiwary
, J. Chem. Phys.
154
, 134111
(2021
).59.
60.
J.-H.
Prinz
, H.
Wu
, M.
Sarich
, B.
Keller
, M.
Senne
, M.
Held
, J. D.
Chodera
, C.
Schütte
, and F.
Noé
, J. Chem. Phys.
134
, 174105
(2011
).61.
B. E.
Husic
and V. S.
Pande
, J. Am. Chem. Soc.
140
, 2386
(2018
).62.
G.
Puccetti
, Inf. Sci.
607
, 1328
(2022
).63.
S.
Cao
, Y.
Qiu
, M. L.
Kalin
, and X.
Huang
, J. Chem. Phys.
159
, 134106
(2023
).64.
A. J.
Dominic
III, T.
Sayer
, S.
Cao
, T. E.
Markland
, X.
Huang
, and A.
Montoya-Castillo
, Proc. Natl. Acad. Sci. U. S. A.
120
, e2221048120
(2023
).65.
I. C.
Unarta
, S.
Cao
, and X.
Huang
, Biophys. J.
122
, 445a
(2023
).66.
M. K.
Scherer
, B.
Trendelkamp-Schroer
, F.
Paul
, G.
Pérez-Hernández
, M.
Hoffmann
, N.
Plattner
, C.
Wehmeyer
, J.-H.
Prinz
, and F.
Noé
, J. Chem. Theory Comput.
11
, 5525
(2015
).67.
See https://github.com/noegroup/bgflow for the RealNVP code.
68.
M.
Hoffmann
, M.
Scherer
, T.
Hempel
, A.
Mardt
, B.
de Silva
, B. E.
Husic
, S.
Klus
, H.
Wu
, N.
Kutz
, S. L.
Brunton
, and F.
Noé
, Mach. Learn.: Sci. Technol.
3
, 015009
(2021
).69.
See https://markovmodel.github.io/mdshare/ for MD simulation data for the pentapeptide.
70.
J.
Kim
, D.
Bloore
, K.
Kapoor
, J.
Feng
, M.-H.
Hao
, and M.
Wang
, “Scalable normalizing flows enable Boltzmann generators for macromolecules
,” Poster presented at Workshop: New Frontiers of AI for Drug Discovery and Development
(NeurIPS
, 2023
).© 2024 Author(s). Published under an exclusive license by AIP Publishing.
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