Artificial intelligence (AI)-based approaches have had indubitable impact across the sciences through the ability to extract relevant information from raw data. Recently, AI has also found use in enhancing the efficiency of molecular simulations, wherein AI derived slow modes are used to accelerate the simulation in targeted ways. However, while typical fields where AI is used are characterized by a plethora of data, molecular simulations, per construction, suffer from limited sampling and thus limited data. As such, the use of AI in molecular simulations can suffer from a dangerous situation where the AI-optimization could get stuck in spurious regimes, leading to incorrect characterization of the reaction coordinate (RC) for the problem at hand. When such an incorrect RC is then used to perform additional simulations, one could start to deviate progressively from the ground truth. To deal with this problem of spurious AI-solutions, here, we report a novel and automated algorithm using ideas from statistical mechanics. It is based on the notion that a more reliable AI-solution will be one that maximizes the timescale separation between slow and fast processes. To learn this timescale separation even from limited data, we use a maximum caliber-based framework. We show the applicability of this automatic protocol for three classic benchmark problems, namely, the conformational dynamics of a model peptide, ligand-unbinding from a protein, and folding/unfolding energy landscape of the C-terminal domain of protein G. We believe that our work will lead to increased and robust use of trustworthy AI in molecular simulations of complex systems.

1.
R. C.
Bernardi
,
M. C. R.
Melo
, and
K.
Schulten
, “
Enhanced sampling techniques in molecular dynamics simulations of biological systems
,”
Biochim. Biophys. Acta
1850
,
872
877
(
2015
).
2.
M.
Karplus
and
G. A.
Petsko
, “
Molecular dynamics simulations in biology
,”
Nature
347
,
631
639
(
1990
).
3.
M. A.
Rohrdanz
,
W.
Zheng
, and
C.
Clementi
, “
Discovering mountain passes via torchlight: Methods for the definition of reaction coordinates and pathways in complex macromolecular reactions
,”
Annu. Rev. Phys. Chem.
64
,
295
316
(
2013
).
4.
C.
Abrams
and
G.
Bussi
, “
Enhanced sampling in molecular dynamics using metadynamics, replica-exchange, and temperature-acceleration
,”
Entropy
16
,
163
199
(
2014
).
5.
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
).
6.
J. N.
Onuchic
and
P. G.
Wolynes
, “
Theory of protein folding
,”
Curr. Opin. Struct. Biol.
14
,
70
75
(
2004
).
7.
K. A.
Dill
,
S. B.
Ozkan
,
M. S.
Shell
, and
T. R.
Weikl
, “
The protein folding problem
,”
Annu. Rev. Biophys.
37
,
289
316
(
2008
).
8.
P.
Tiwary
,
V.
Limongelli
,
M.
Salvalaglio
, and
M.
Parrinello
, “
Kinetics of protein–ligand unbinding: Predicting pathways, rates, and rate-limiting steps
,”
Proc. Natl. Acad. Sci. U. S. A.
112
,
E386
E391
(
2015
).
9.
P.
Tiwary
,
J.
Mondal
, and
B. J.
Berne
, “
How and when does an anticancer drug leave its binding site?
,”
Sci. Adv.
3
,
e1700014
(
2017
).
10.
M.
Moradi
and
E.
Tajkhorshid
, “
Mechanistic picture for conformational transition of a membrane transporter at atomic resolution
,”
Proc. Natl. Acad. Sci. U. S. A.
110
,
18916
18921
(
2013
).
11.
M.
Moradi
and
E.
Tajkhorshid
, “
Computational recipe for efficient description of large-scale conformational changes in biomolecular systems
,”
J. Chem. Theory Comput.
10
,
2866
2880
(
2014
).
12.
M.
Moradi
,
G.
Enkavi
, and
E.
Tajkhorshid
, “
Atomic-level characterization of transport cycle thermodynamics in the glycerol-3-phosphate:phosphate transporter
,”
Nat. Commun.
6
,
8393
(
2015
).
13.
S.
Pant
and
E.
Tajkhorshid
, “
Microscopic characterization of GRP1 PH domain interaction with anionic membranes
,”
J. Comput. Chem.
41
,
489
499
(
2019
).
14.
S. P.
Brooks
and
B. J.
Morgan
, “
Optimization using simulated annealing
,”
J. R. Stat. Soc.: D
44
,
241
257
(
1995
).
15.
U. H.
Hansmann
, “
Parallel tempering algorithm for conformational studies of biological molecules
,”
Chem. Phys. Lett.
281
,
140
150
(
1997
).
16.
Y.
Sugita
and
Y.
Okamoto
, “
Replica-exchange molecular dynamics method for protein folding
,”
Chem. Phys. Lett.
314
,
141
151
(
1999
).
17.
Y.
Sugita
,
A.
Kitao
, and
Y.
Okamoto
, “
Multidimensional replica-exchange method for free-energy calculations
,”
J. Chem. Phys.
113
,
6042
6051
(
2000
).
18.
A.
Mitsutake
,
Y.
Sugita
, and
Y.
Okamoto
, “
Generalized-ensemble algorithms for molecular simulations of biopolymers
,”
Biopolymers
60
,
96
123
(
2001
).
19.
H.
Fukunishi
,
O.
Watanabe
, and
S.
Takada
, “
On the Hamiltonian replica exchange method for efficient sampling of biomolecular systems: Application to protein structure prediction
,”
J. Chem. Phys.
116
,
9058
9067
(
2002
).
20.
L.
Maragliano
and
E.
Vanden-Eijnden
, “
A temperature accelerated method for sampling free energy and determining reaction pathways in rare events simulations
,”
Chem. Phys. Lett.
426
,
168
175
(
2006
).
21.
Y.
Miao
,
V. A.
Feher
, and
J. A.
McCammon
, “
Gaussian accelerated molecular dynamics: Unconstrained enhanced sampling and free energy calculation
,”
J. Chem. Theory Comput.
11
,
3584
3595
(
2015
).
22.
A.
Laio
and
M.
Parrinello
, “
Escaping free-energy minima
,”
Proc. Natl. Acad. Sci. U. S. A.
99
,
12562
12566
(
2002
).
23.
A.
Laio
and
F. L.
Gervasio
, “
Metadynamics: A method to simulate rare events and reconstruct the free energy in biophysics, chemistry and material science
,”
Rep. Progr. Phys.
71
,
126601
(
2008
).
24.
A.
Barducci
,
G.
Bussi
, and
M.
Parrinello
, “
Well-tempered metadynamics: A smoothly converging and tunable free-energy method
,”
Phys. Rev. Lett.
100
,
020603
(
2008
).
25.
G. M.
Torrie
and
J. P.
Valleau
, “
Nonphysical sampling distributions in Monte Carlo free-energy estimation: Umbrella sampling
,”
J. Chem. Phys.
23
,
187
199
(
1977
).
26.
T.
Lelièvre
,
M.
Rousset
, and
G.
Stoltz
, “
Computation of free energy profiles with parallel adaptive dynamics
,”
J. Chem. Phys.
126
,
134111
(
2007
).
27.
E.
Darve
,
D.
Rodríguez-Gómez
, and
A.
Pohorille
, “
Adaptive biasing force method for scalar and vector free energy calculations
,”
J. Chem. Phys.
128
,
144120
(
2008
).
28.
J.
Comer
,
J. C.
Gumbart
,
J.
Hénin
,
T.
Lelièvre
,
A.
Pohorille
, and
C.
Chipot
, “
The adaptive biasing force method: Everything you always wanted to know but were afraid to ask
,”
J. Phys. Chem. B
119
,
1129
1151
(
2015
).
29.
H.
Fu
,
X.
Shao
,
C.
Chipot
, and
W.
Cai
, “
Extended adaptive biasing force algorithm. An on-the-fly implementation for accurate free-energy calculations
,”
J. Chem. Theory Comput.
12
,
3506
3513
(
2016
).
30.
A.
Lesage
,
T.
Lelièvre
,
G.
Stoltz
, and
J.
Hénin
, “
Smoothed biasing forces yield unbiased free energies with the extended-system adaptive biasing force method
,”
J. Phys. Chem. B
121
,
3676
3685
(
2017
).
31.
J. B.
Abrams
and
M. E.
Tuckerman
, “
Efficient and direct generation of multidimensional free energy surfaces via adiabatic dynamics without coordinate transformations
,”
J. Phys. Chem. B
112
,
15742
15757
(
2008
).
32.
J. G.
Kirkwood
, “
Statistical mechanics of fluid mixtures
,”
J. Chem. Phys.
3
,
300
313
(
1935
).
33.
W. K.
den Otter
and
W. J.
Briels
, “
The calculation of free-energy differences by constrained molecular-dynamics simulations
,”
J. Chem. Phys.
109
,
4139
4146
(
1998
).
34.
A.
Hazel
,
C.
Chipot
, and
J. C.
Gumbart
, “
Thermodynamics of deca-alanine folding in water
,”
J. Chem. Theory Comput.
10
,
2836
2844
(
2014
).
35.
P.
Liu
,
B.
Kim
,
R. A.
Friesner
, and
B.
Berne
, “
Replica exchange with solute tempering: A method for sampling biological systems in explicit water
,”
Proc. Natl. Acad. Sci. U. S. A.
102
,
13749
13754
(
2005
).
36.
X.
Huang
,
M.
Hagen
,
B.
Kim
,
R. A.
Friesner
,
R.
Zhou
, and
B. J.
Berne
, “
Replica exchange with solute tempering: Efficiency in large scale systems
,”
J. Phys. Chem. B
111
,
5405
5410
(
2007
).
37.
Y.
Wang
,
J. M. L.
Ribeiro
, and
P.
Tiwary
, “
Machine learning approaches for analyzing and enhancing molecular dynamics simulations
,”
Curr. Opin. Struct. Biol.
61
,
139
145
(
2020
).
38.
F.
Noé
,
G.
De Fabritiis
, and
C.
Clementi
, “
Machine learning for protein folding and dynamics
,”
Curr. Opin. Struct. Biol.
60
,
77
84
(
2020
).
39.
F.
Noé
,
S.
Olsson
,
J.
Köhler
, and
H.
Wu
, “
Boltzmann generators: Sampling equilibrium states of many-body systems with deep learning
,”
Science
365
,
eaaw1147
(
2019
).
40.
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
).
41.
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. Comput. Chem.
149
,
072312
(
2018
).
42.
M. E.
Tuckerman
, “
Machine learning transforms how microstates are sampled
,”
Science
365
,
982
983
(
2019
).
43.
S.-L. J.
Lahey
and
C. N.
Rowley
, “
Simulating protein–ligand binding with neural network potentials
,”
Chem. Sci.
11
,
2362
2368
(
2020
).
44.
G.
Rotskoff
and
E.
Vanden-Eijnden
, “
Parameters as interacting particles: Long time convergence and asymptotic error scaling of neural networks
,” in
Advances in Neural Information Processing Systems
(
Curran Associates
,
2018
), pp.
7146
7155
.
45.
G.
Cybenko
, “
Approximation by superpositions of a sigmoidal function
,”
Math. Control, Signals, Syst.
5
,
455
(
1992
).
46.
A. R.
Barron
, “
Universal approximation bounds for superpositions of a sigmoidal function
,”
IEEE Trans. Inf. Theory
39
,
930
945
(
1993
).
47.
F.
Bach
, “
Breaking the curse of dimensionality with convex neural networks
,”
J. Mach. Learn. Res.
18
,
629
681
(
2017
).
48.
I.
Evtimov
,
K.
Eykholt
,
E.
Fernandes
,
T.
Kohno
,
B.
Li
,
A.
Prakash
,
A.
Rahmati
, and
D.
Song
, “
Robust physical-world attacks on deep learning models
,” arXiv:1707.08945 (
2017
).
49.
F.
Noé
,
I.
Horenko
,
C.
Schütte
, and
J. C.
Smith
, “
Hierarchical analysis of conformational dynamics in biomolecules: Transition networks of metastable states
,”
J. Chem. Phys.
126
,
155102
(
2007
).
50.
M. A.
Rohrdanz
,
W.
Zheng
,
M.
Maggioni
, and
C.
Clementi
, “
Determination of reaction coordinates via locally scaled diffusion map
,”
J. Chem. Phys.
134
,
124116
(
2011
).
51.
F.
Noé
and
F.
Nüske
, “
A variational approach to modeling slow processes in stochastic dynamical systems
,”
Multiscale Model. Simul.
11
,
635
655
(
2013
).
52.
G.
Pérez-Hernández
,
F.
Paul
,
T.
Giorgino
,
G.
De Fabritiis
, and
F.
Noé
, “
Identification of slow molecular order parameters for Markov model construction
,”
J. Chem. Phys.
139
,
015102
(
2013
).
53.
Q.
Li
,
F.
Dietrich
,
E. M.
Bollt
, and
I. G.
Kevrekidis
, “
Extended dynamic mode decomposition with dictionary learning: A data-driven adaptive spectral decomposition of the Koopman operator
,”
Chaos
27
,
103111
(
2017
).
54.
P.
Tiwary
and
B.
Berne
, “
Spectral gap optimization of order parameters for sampling complex molecular systems
,”
Proc. Natl. Acad. Sci. U. S. A.
113
,
2839
2844
(
2016
).
55.
K.
Ghosh
,
P. D.
Dixit
,
L.
Agozzino
, and
K. A.
Dill
, “
The maximum caliber variational principle for nonequilibria
,”
Annu. Rev. Phys. Chem.
71
,
213
238
(
2020
).
56.
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
).
57.
P.
Ravindra
,
Z.
Smith
, and
P.
Tiwary
, “
Automatic mutual information noise omission (AMINO): Generating order parameters for molecular systems
,”
Mol. Syst. Des. Eng.
5
,
339
(
2020
).
58.
N.
Tishby
,
F. C.
Pereira
, and
W.
Bialek
, “
The information bottleneck method
,” arXiv:physics/0004057 (
2000
).
59.
S. E.
Palmer
,
O.
Marre
,
M. J.
Berry
, and
W.
Bialek
, “
Predictive information in a sensory population
,”
Proc. Natl. Acad. Sci. U. S. A.
112
,
6908
6913
(
2015
).
60.
G. J.
Berman
,
W.
Bialek
, and
J. W.
Shaevitz
, “
Predictability and hierarchy in Drosophila behavior
,”
Proc. Natl. Acad. Sci. U. S. A.
113
,
11943
11948
(
2016
).
61.
A. A.
Alemi
,
I.
Fischer
,
J. V.
Dillon
, and
K.
Murphy
, “
Deep variational information bottleneck
,” arXiv:1612.00410 (
2016
).
62.
S.
Still
, “
Information bottleneck approach to predictive inference
,”
Entropy
16
,
968
989
(
2014
).
63.
S. V.
Krivov
, “
On reaction coordinate optimality
,”
J. Chem. Theory Comput.
9
,
135
146
(
2013
).
64.
Z.
Smith
,
P.
Ravindra
,
Y.
Wang
,
R.
Cooley
, and
P.
Tiwary
, “
Discovering loop conformational flexibility in T4 lysozyme mutants through Artificial Intelligence aided Molecular Dynamics
,”
J. Phys. Chem. B
124
,
8221
8229
(
2020
).
65.
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
).
66.
T. M.
Cover
and
J. A.
Thomas
,
Elements of Information Theory
(
John Wiley & Sons
,
2012
).
67.
I.
Goodfellow
,
Y.
Bengio
, and
A.
Courville
,
Deep Learning
(
MIT Press
,
2016
), http://www.deeplearningbook.org.
68.
Y.
Wang
and
P.
Tiwary
, “
Understanding the role of predictive time delay and biased propagator in RAVE
,”
J. Chem. Phys.
152
,
144102
(
2020
).
69.
I. N.
Levine
,
D. H.
Busch
, and
H.
Shull
,
Quantum Chemistry
(
Pearson Prentice Hall Upper
,
Saddle River, NJ
,
2009
), Vol. 6.
70.
D. L.
Nelson
,
A. L.
Lehninger
, and
M. M.
Cox
,
Lehninger Principles of Biochemistry
(
Macmillan
,
2008
).
71.
D. G.
Truhlar
and
B. C.
Garrett
, “
Variational transition state theory
,”
Annu. Rev. Phys. Chem.
35
,
159
189
(
1984
).
72.
P. D.
Dixit
and
K. A.
Dill
, “
Caliber corrected Markov modeling (C2M2): Correcting equilibrium Markov models
,”
J. Chem. Theory Comput.
14
,
1111
1119
(
2018
).
73.
Z.
Smith
,
D.
Pramanik
,
S.-T.
Tsai
, and
P.
Tiwary
, “
Multi-dimensional spectral gap optimization of order parameters (SGOOP) through conditional probability factorization
,”
J. Chem. Phys.
149
,
234105
(
2018
).
74.
D.
Meral
,
D.
Provasi
, and
M.
Filizola
, “
An efficient strategy to estimate thermodynamics and kinetics of G protein-coupled receptor activation using metadynamics and maximum caliber
,”
J. Chem. Phys.
149
,
224101
(
2018
).
75.
P.
Tiwary
and
A.
van de Walle
,
Multiscale Materials Modeling for Nanomechanics
(
Springer
,
2016
), pp.
195
221
.
76.
J.
Debnath
and
M.
Parrinello
, “
Gaussian mixture based enhanced sampling for statics and dynamics
,”
J. Phys. Chem. Lett.
11
,
5076
(
2020
).
77.
M. J.
Abraham
,
T.
Murtola
,
R.
Schulz
,
S.
Páll
,
J. C.
Smith
,
B.
Hess
, and
E.
Lindahl
, “
GROMACS: High performance molecular simulations through multi-level parallelism from laptops to supercomputers
,”
SoftwareX
1-2
,
19
25
(
2015
).
78.
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
).
79.
M.
Bonomi
,
G.
Bussi
,
C.
Camilloni
,
G. A.
Tribello
,
P.
Banáš
,
A.
Barducci
,
M.
Bernetti
,
P. G.
Bolhuis
,
S.
Bottaro
,
D.
Branduardi
 et al, “
Promoting transparency and reproducibility in enhanced molecular simulations
,”
Nat. Methods
16
,
670
673
(
2019
).
80.
Y.
Mu
,
P. H.
Nguyen
, and
G.
Stock
, “
Energy landscape of a small peptide revealed by dihedral angle principal component analysis
,”
Proteins
58
,
45
52
(
2005
).
81.
A.
Altis
,
P. H.
Nguyen
,
R.
Hegger
, and
G.
Stock
, “
Dihedral angle principal component analysis of molecular dynamics simulations
,”
J. Chem. Phys.
126
,
244111
(
2007
).
82.
M.
Salvalaglio
,
P.
Tiwary
, and
M.
Parrinello
, “
Assessing the reliability of the dynamics reconstructed from metadynamics
,”
J. Chem. Theory Comput.
10
,
1420
1425
(
2014
).
83.
V.
Hornak
,
R.
Abel
,
A.
Okur
,
B.
Strockbine
,
A.
Roitberg
, and
C.
Simmerling
, “
Comparison of multiple Amber force fields and development of improved protein backbone parameters
,”
Proteins
65
,
712
725
(
2006
).
84.
R. B.
Best
and
G.
Hummer
, “
Optimized molecular dynamics force fields applied to the helix- coil transition of polypeptides
,”
J. Phys. Chem. B
113
,
9004
9015
(
2009
).
85.
K.
Lindorff-Larsen
,
S.
Piana
,
K.
Palmo
,
P.
Maragakis
,
J. L.
Klepeis
,
R. O.
Dror
, and
D. E.
Shaw
, “
Improved side-chain torsion potentials for the Amber ff99SB protein force field
,”
Proteins
78
,
1950
1958
(
2010
).
86.
J. C.
Gumbart
,
B.
Roux
, and
C.
Chipot
, “
Standard binding free energies from computer simulations: What is the best strategy?
,”
J. Chem. Theory Comput.
9
,
794
802
(
2013
).
87.
P.
Burkhard
,
P.
Taylor
, and
M. D.
Walkinshaw
, “
X-ray structures of small ligand-FKBP complexes provide an estimate for hydrophobic interaction energies
,”
J. Mol. Biol.
295
,
953
962
(
2000
).
88.
D.
Pramanik
,
Z.
Smith
,
A.
Kells
, and
P.
Tiwary
, “
Can one trust kinetic and thermodynamic observables from biased metadynamics simulations?: Detailed quantitative benchmarks on millimolar drug fragment dissociation
,”
J. Phys. Chem. B
123
,
3672
3678
(
2019
).
89.
A. C.
Pan
,
H.
Xu
,
T.
Palpant
, and
D. E.
Shaw
, “
Quantitative characterization of the binding and unbinding of millimolar drug fragments with molecular dynamics simulations
,”
J. Chem. Theory Comput.
13
,
3372
3377
(
2017
).
90.
N.
Ahalawat
and
J.
Mondal
, “
Assessment and optimization of collective variables for protein conformational landscape: GB1 β-hairpin as a case study
,”
J. Chem. Phys.
149
,
094101
(
2018
).
91.
V.
Muñoz
,
P. A.
Thompson
,
J.
Hofrichter
, and
W. A.
Eaton
, “
Folding dynamics and mechanism of β-hairpin formation
,”
Nature
390
,
196
199
(
1997
).
92.
R. M.
Fesinmeyer
,
F. M.
Hudson
, and
N. H.
Andersen
, “
Enhanced hairpin stability through loop design: The case of the protein G B1 domain hairpin
,”
J. Am. Chem. Soc.
126
,
7238
7243
(
2004
).
93.
A. J.
Hazel
,
E. T.
Walters
,
C. N.
Rowley
, and
J. C.
Gumbart
, “
Folding free energy landscapes of β-sheets with non-polarizable and polarizable CHARMM force fields
,”
J. Chem. Phys.
149
,
072317
(
2018
).
94.
R. B.
Best
and
J.
Mittal
, “
Free-energy landscape of the GB1 hairpin in all-atom explicit solvent simulations with different force fields: Similarities and differences
,”
Proteins
79
,
1318
1328
(
2011
).
95.
A.
Ardevol
,
G. A.
Tribello
,
M.
Ceriotti
, and
M.
Parrinello
, “
Probing the unfolded configurations of a β-hairpin using sketch-map
,”
J. Chem. Theory Comput.
11
,
1086
1093
(
2015
).
96.
F.
Nüske
,
B. G.
Keller
,
G.
Pérez-Hernández
,
A. S.
Mey
, and
F.
Noé
, “
Variational approach to molecular kinetics
,”
J. Chem. Theory Comput.
10
,
1739
1752
(
2014
).
97.
R. T.
McGibbon
and
V. S.
Pande
, “
Variational cross-validation of slow dynamical modes in molecular kinetics
,”
J. Chem. Phys.
142
,
124105
(
2015
).
98.
J. M.
Lamim Ribeiro
and
P.
Tiwary
, “
Toward achieving efficient and accurate Ligand–Protein unbinding with deep learning and molecular dynamics through RAVE
,”
J. Chem. Theory Comput.
15
,
708
719
(
2018
).
99.
M.
Bonomi
,
D.
Branduardi
,
F. L.
Gervasio
, and
M.
Parrinello
, “
The unfolded ensemble and folding mechanism of the C-terminal GB1 β-hairpin
,”
J. Am. Chem. Soc.
130
,
13938
13944
(
2008
).
100.
G.
Bussi
,
F. L.
Gervasio
,
A.
Laio
, and
M.
Parrinello
, “
Free-energy landscape for β hairpin folding from combined parallel tempering and metadynamics
,”
J. Am. Chem. Soc.
128
,
13435
13441
(
2006
).
101.
G.
Saladino
,
S.
Pieraccini
,
S.
Rendine
,
T.
Recca
,
P.
Francescato
,
G.
Speranza
, and
M.
Sironi
, “
Metadynamics study of a β-hairpin stability in mixed solvents
,”
J. Am. Chem. Soc.
133
,
2897
2903
(
2011
).
102.
S.
Pressé
,
K.
Ghosh
,
J.
Lee
, and
K. A.
Dill
, “
Principles of maximum entropy and maximum caliber in statistical physics
,”
Rev. Mod. Phys.
85
,
1115
(
2013
).
103.
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
).
104.
G.
Bussi
,
D.
Donadio
, and
M.
Parrinello
, “
Canonical sampling through velocity rescaling
,”
J. Chem. Phys.
126
,
014101
(
2007
).
105.
J.
Wang
,
R. M.
Wolf
,
J. W.
Caldwell
,
P. A.
Kollman
, and
D. A.
Case
, “
Development and testing of a general amber force field
,”
J. Comput. Chem.
25
,
1157
1174
(
2004
).
106.
G. J.
Martyna
,
D. J.
Tobias
, and
M. L.
Klein
, “
Constant pressure molecular dynamics algorithms
,”
J. Chem. Phys.
101
,
4177
4189
(
1994
).
107.
P.
Tiwary
and
M.
Parrinello
, “
A time-independent free energy estimator for metadynamics
,”
J. Phys. Chem. B
119
,
736
742
(
2015
).

Supplementary Material

You do not currently have access to this content.