The present work shows that the free energy landscape associated with alanine dipeptide isomerization can be effectively represented by specific interatomic distances without explicit reference to dihedral angles. Conventionally, two stable states of alanine dipeptide in vacuum, i.e., C7eq (β-sheet structure) and C7ax (left handed α-helix structure), have been primarily characterized using the main chain dihedral angles, φ (C–N–Cα–C) and ψ (N–Cα–C–N). However, our recent deep learning combined with the “Explainable AI” (XAI) framework has shown that the transition state can be adequately captured by a free energy landscape using φ and θ (O–C–N–Cα) [Kikutsuji et al., J. Chem. Phys. 156, 154108 (2022)]. In the perspective of extending these insights to other collective variables, a more detailed characterization of the transition state is required. In this work, we employ interatomic distances and bond angles as input variables for deep learning rather than the conventional and more elaborate dihedral angles. Our approach utilizes deep learning to investigate whether changes in the main chain dihedral angle can be expressed in terms of interatomic distances and bond angles. Furthermore, by incorporating XAI into our predictive analysis, we quantified the importance of each input variable and succeeded in clarifying the specific interatomic distance that affects the transition state. The results indicate that constructing a free energy landscape based on the identified interatomic distance can clearly distinguish between the two stable states and provide a comprehensive explanation for the energy barrier crossing.
REFERENCES
1.
D. M.
Zuckerman
, Statistical Physics of Biomolecules: An Introduction
(CRC Press
, Boca Raton, FL
, 2010
).2.
3.
G. N.
Ramachandran
and V.
Sasisekharan
, “Conformation of polypeptides and proteins
,” Adv. Protein Chem.
23
, 283
–437
(1968
).4.
P. G.
Bolhuis
, C.
Dellago
, and D.
Chandler
, “Reaction coordinates of biomolecular isomerization
,” Proc. Natl. Acad. Sci. U. S. A.
97
, 5877
–5882
(2000
).5.
R.
Du
, V. S.
Pande
, A. Y.
Grosberg
, T.
Tanaka
, and E. S.
Shakhnovich
, “On the transition coordinate for protein folding
,” J. Chem. Phys.
108
, 334
–350
(1998
).6.
P. L.
Geissler
, C.
Dellago
, and D.
Chandler
, “Kinetic pathways of ion pair dissociation in water
,” J. Phys. Chem. B
103
, 3706
–3710
(1999
).7.
P. G.
Bolhuis
, D.
Chandler
, C.
Dellago
, and P. L.
Geissler
, “TRANSITION PATH SAMPLING: Throwing ropes over rough mountain passes, in the dark
,” Annu. Rev. Phys. Chem.
53
, 291
–318
(2002
).8.
G.
Hummer
, “From transition paths to transition states and rate coefficients
,” J. Chem. Phys.
120
, 516
–523
(2004
).9.
R. B.
Best
and G.
Hummer
, “Reaction coordinates and rates from transition paths
,” Proc. Natl. Acad. Sci. U. S. A.
102
, 6732
–6737
(2005
).10.
W.
E
, W.
Ren
, and E.
Vanden-Eijnden
, “Transition pathways in complex systems: Reaction coordinates, isocommittor surfaces, and transition tubes
,” Chem. Phys. Lett.
413
, 242
–247
(2005
).11.
B.
Peters
, “Using the histogram test to quantify reaction coordinate error
,” J. Chem. Phys.
125
, 241101
(2006
).12.
B.
Peters
and B. L.
Trout
, “Obtaining reaction coordinates by likelihood maximization
,” J. Chem. Phys.
125
, 054108
(2006
).13.
B.
Peters
, G. T.
Beckham
, and B. L.
Trout
, “Extensions to the likelihood maximization approach for finding reaction coordinates
,” J. Chem. Phys.
127
, 034109
(2007
).14.
B.
Peters
, “p(TP|q) peak maximization: Necessary but not sufficient for reaction coordinate accuracy
,” Chem. Phys. Lett.
494
, 100
–103
(2010
).15.
B.
Peters
, “Reaction coordinates and mechanistic hypothesis tests
,” Annu. Rev. Phys. Chem.
67
, 669
–690
(2016
).16.
H.
Jung
, K.-i.
Okazaki
, and G.
Hummer
, “Transition path sampling of rare events by shooting from the top
,” J. Chem. Phys.
147
, 152716
(2017
).17.
M. F.
Hagan
, A. R.
Dinner
, D.
Chandler
, and A. K.
Chakraborty
, “Atomistic understanding of kinetic pathways for single base-pair binding and unbinding in DNA
,” Proc. Natl. Acad. Sci. U. S. A.
100
, 13922
–13927
(2003
).18.
A. C.
Pan
and D.
Chandler
, “Dynamics of nucleation in the Ising model
,” J. Phys. Chem. B
108
, 19681
–19686
(2004
).19.
Y. M.
Rhee
and V. S.
Pande
, “One-dimensional reaction coordinate and the corresponding potential of mean force from commitment probability distribution
,” J. Phys. Chem. B
109
, 6780
–6786
(2005
).20.
A.
Berezhkovskii
and A.
Szabo
, “One-dimensional reaction coordinates for diffusive activated rate processes in many dimensions
,” J. Chem. Phys.
122
, 014503
(2005
).21.
D.
Moroni
, P. R.
ten Wolde
, and P. G.
Bolhuis
, “Interplay between structure and size in a critical crystal nucleus
,” Phys. Rev. Lett.
94
, 235703
(2005
).22.
D.
Branduardi
, F. L.
Gervasio
, and M.
Parrinello
, “From A to B in free energy space
,” J. Chem. Phys.
126
, 054103
(2007
).23.
S. L.
Quaytman
and S. D.
Schwartz
, “Reaction coordinate of an enzymatic reaction revealed by transition path sampling
,” Proc. Natl. Acad. Sci. U. S. A.
104
, 12253
–12258
(2007
).24.
G. T.
Beckham
, B.
Peters
, C.
Starbuck
, N.
Variankaval
, and B. L.
Trout
, “Surface-mediated nucleation in the solid-state polymorph transformation of terephthalic acid
,” J. Am. Chem. Soc.
129
, 4714
–4723
(2007
).25.
B.
Peters
, N. E. R.
Zimmermann
, G. T.
Beckham
, J. W.
Tester
, and B. L.
Trout
, “Path sampling calculation of methane diffusivity in natural gas hydrates from a water-vacancy assisted mechanism
,” J. Am. Chem. Soc.
130
, 17342
–17350
(2008
).26.
G. T.
Beckham
, B.
Peters
, and B. L.
Trout
, “Evidence for a size dependent nucleation mechanism in solid state polymorph transformations
,” J. Phys. Chem. B
112
, 7460
–7466
(2008
).27.
D.
Antoniou
and S. D.
Schwartz
, “The stochastic separatrix and the reaction coordinate for complex systems
,” J. Chem. Phys.
130
, 151103
(2009
).28.
L.
Xi
, M.
Shah
, and B. L.
Trout
, “Hopping of water in a glassy polymer studied via transition path sampling and likelihood maximization
,” J. Phys. Chem. B
117
, 3634
–3647
(2013
).29.
S.
Jungblut
, A.
Singraber
, and C.
Dellago
, “Optimising reaction coordinates for crystallisation by tuning the crystallinity definition
,” Mol. Phys.
111
, 3527
–3533
(2013
).30.
B. C.
Barnes
, B. C.
Knott
, G. T.
Beckham
, D. T.
Wu
, and A. K.
Sum
, “Reaction coordinate of incipient methane clathrate hydrate nucleation
,” J. Phys. Chem. B
118
, 13236
–13243
(2014
).31.
R. G.
Mullen
, J.-E.
Shea
, and B.
Peters
, “Transmission coefficients, committors, and solvent coordinates in ion-pair dissociation
,” J. Chem. Theory Comput.
10
, 659
–667
(2014
).32.
R. G.
Mullen
, J.-E.
Shea
, and B.
Peters
, “Easy transition path sampling methods: Flexible-length aimless shooting and permutation shooting
,” J. Chem. Theory Comput.
11
, 2421
–2428
(2015
).33.
L.
Lupi
, B.
Peters
, and V.
Molinero
, “Pre-ordering of interfacial water in the pathway of heterogeneous ice nucleation does not lead to a two-step crystallization mechanism
,” J. Chem. Phys.
145
, 211910
(2016
).34.
M.
Ernst
, S.
Wolf
, and G.
Stock
, “Identification and validation of reaction coordinates describing protein functional motion: Hierarchical dynamics of T4 lysozyme
,” J. Chem. Theory Comput.
13
, 5076
–5088
(2017
).35.
G.
Díaz Leines
and J.
Rogal
, “Maximum likelihood analysis of reaction coordinates during solidification in Ni
,” J. Phys. Chem. B
122
, 10934
–10942
(2018
).36.
M. N.
Joswiak
, M. F.
Doherty
, and B.
Peters
, “Ion dissolution mechanism and kinetics at kink sites on NaCl surfaces
,” Proc. Natl. Acad. Sci. U. S. A.
115
, 656
–661
(2018
).37.
K.-i.
Okazaki
, D.
Wöhlert
, J.
Warnau
, H.
Jung
, Ö.
Yildiz
, W.
Kühlbrandt
, and G.
Hummer
, “Mechanism of the electroneutral sodium/proton antiporter PaNhaP from transition-path shooting
,” Nat. Commun.
10
, 1742
(2019
).38.
T.
Mori
and S.
Saito
, “Dissecting the dynamics during enzyme catalysis: A case study of Pin1 peptidyl-prolyl isomerase
,” J. Chem. Theory Comput.
16
, 3396
–3407
(2020
).39.
N.
Schwierz
, “Kinetic pathways of water exchange in the first hydration shell of magnesium
,” J. Chem. Phys.
152
, 224106
(2020
).40.
R. L.
Silveira
, B. C.
Knott
, C. S.
Pereira
, M. F.
Crowley
, M. S.
Skaf
, and G. T.
Beckham
, “Transition path sampling study of the feruloyl esterase mechanism
,” J. Phys. Chem. B
125
, 2018
–2030
(2021
).41.
F.
Sittel
and G.
Stock
, “Robust density-based clustering to identify metastable conformational states of proteins
,” J. Chem. Theory Comput.
12
, 2426
–2435
(2016
).42.
M. M.
Sultan
and V. S.
Pande
, “Automated design of collective variables using supervised machine learning
,” J. Chem. Phys.
149
, 094106
(2018
).43.
C.
Wehmeyer
and F.
Noé
, “Time-lagged autoencoders: Deep learning of slow collective variables for molecular kinetics
,” J. Chem. Phys.
148
, 241703
(2018
).44.
A.
Mardt
, L.
Pasquali
, H.
Wu
, and F.
Noé
, “VAMPnets for deep learning of molecular kinetics
,” Nat. Commun.
9
, 5
(2018
).45.
A.
Bittracher
, R.
Banisch
, and C.
Schütte
, “Data-driven computation of molecular reaction coordinates
,” J. Chem. Phys.
149
, 154103
(2018
).46.
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
).47.
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
).48.
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
).49.
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
).50.
L.
Bonati
, V.
Rizzi
, and M.
Parrinello
, “Data-driven collective variables for enhanced sampling
,” J. Phys. Chem. Lett.
11
, 2998
–3004
(2020
).51.
H.
Sidky
, W.
Chen
, and A. L.
Ferguson
, “Machine learning for collective variable discovery and enhanced sampling in biomolecular simulation
,” Mol. Phys.
118
, e1737742
(2020
).52.
D.
Wang
and P.
Tiwary
, “State predictive information bottleneck
,” J. Chem. Phys.
154
, 134111
(2021
).53.
J.
Zhang
, Y.-K.
Lei
, Z.
Zhang
, X.
Han
, M.
Li
, L.
Yang
, Y. I.
Yang
, and Y. Q.
Gao
, “Deep reinforcement learning of transition states
,” Phys. Chem. Chem. Phys.
23
, 6888
–6895
(2021
).54.
M.
Frassek
, A.
Arjun
, and P. G.
Bolhuis
, “An extended autoencoder model for reaction coordinate discovery in rare event molecular dynamics datasets
,” J. Chem. Phys.
155
, 064103
(2021
).55.
F.
Hooft
, A.
Pérez de Alba Ortíz
, and B.
Ensing
, “Discovering collective variables of molecular transitions via genetic algorithms and neural networks
,” J. Chem. Theory Comput.
17
, 2294
–2306
(2021
).56.
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
).57.
M.
Chen
, “Collective variable-based enhanced sampling and machine learning
,” Eur. Phys. J. B
94
, 211
(2021
).58.
Z.
Belkacemi
, P.
Gkeka
, T.
Lelièvre
, and G.
Stoltz
, “Chasing collective variables using autoencoders and biased trajectories
,” J. Chem. Theory Comput.
18
, 59
–78
(2021
).59.
J.
Neumann
and N.
Schwierz
, “Artificial intelligence resolves kinetic pathways of magnesium binding to RNA
,” J. Chem. Theory Comput.
18
, 1202
–1212
(2022
).60.
D.
Ray
, E.
Trizio
, and M.
Parrinello
, “Deep learning collective variables from transition path ensemble
,” J. Chem. Phys.
158
, 204102
(2023
).61.
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
–345
(2023
).62.
L.
Bonati
, E.
Trizio
, A.
Rizzi
, and M.
Parrinello
, “A unified framework for machine learning collective variables for enhanced sampling simulations: mlcolvar
,” J. Chem. Phys.
159
, 014801
(2023
).63.
A. N.
Singh
and D. T.
Limmer
, “Variational deep learning of equilibrium transition path ensembles
,” J. Chem. Phys.
159
, 024124
(2023
).64.
S.
Liang
, A. N.
Singh
, Y.
Zhu
, D. T.
Limmer
, and C.
Yang
, “Probing reaction channels via reinforcement learning
,” Mach. Learn.: Sci. Technol.
4
, 045003
(2023
).65.
G.
Lazzeri
, H.
Jung
, P. G.
Bolhuis
, and R.
Covino
, “Molecular free energies, rates, and mechanisms from data-efficient path sampling simulations
,” J. Chem. Theory Comput.
19
, 9060
–9076
(2023
).66.
A.
Ma
and A. R.
Dinner
, “Automatic method for identifying reaction coordinates in complex systems
,” J. Phys. Chem. B
109
, 6769
–6779
(2005
).67.
Y.
Mori
, K.-i.
Okazaki
, T.
Mori
, K.
Kim
, and N.
Matubayasi
, “Learning reaction coordinates via cross-entropy minimization: Application to alanine dipeptide
,” J. Chem. Phys.
153
, 054115
(2020
).68.
T.
Kikutsuji
, Y.
Mori
, K.-i.
Okazaki
, T.
Mori
, K.
Kim
, and N.
Matubayasi
, “Explaining reaction coordinates of alanine dipeptide isomerization obtained from deep neural networks using Explainable Artificial Intelligence (XAI)
,” J. Chem. Phys.
156
, 154108
(2022
).69.
F.
Manuchehrfar
, H.
Li
, W.
Tian
, A.
Ma
, and J.
Liang
, “Exact topology of the dynamic probability surface of an activated process by persistent homology
,” J. Phys. Chem. B
125
, 4667
–4680
(2021
).70.
N.
Naleem
, C. R. A.
Abreu
, K.
Warmuz
, M.
Tong
, S.
Kirmizialtin
, and M. E.
Tuckerman
, “An exploration of machine learning models for the determination of reaction coordinates associated with conformational transitions
,” J. Chem. Phys.
159
, 034102
(2023
).71.
M. T.
Ribeiro
, S.
Singh
, and C.
Guestrin
, “‘Why should I trust you?’: Explaining the predictions of any classifier
,” in Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining
(Association for Computing Machinery
, San Francisco, CA
, 2016
), pp. 1135
–1144
.72.
S. M.
Lundberg
and S.-I.
Lee
, “A unified approach to interpreting model predictions
,” in Proceedings of the 31st International Conference on Neural Information Processing Systems
(Curran Associates Inc., Red Hook, NY
, 2017
), pp. 4768
–4777
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