We derive a matrix formalism for the simulation of long range proton dynamics for extended systems and timescales. On the basis of an ab initio molecular dynamics simulation, we construct a Markov chain, which allows us to store the entire proton dynamics in an M × M transition matrix (where M is the number of oxygen atoms). In this article, we start from common topology features of the hydrogen bond network of good proton conductors and utilize them as constituent constraints of our dynamic model. We present a thorough mathematical derivation of our approach and verify its uniqueness and correct asymptotic behavior. We propagate the proton distribution by means of transition matrices, which contain kinetic data from both ultra-short (sub-ps) and intermediate (ps) timescales. This concept allows us to keep the most relevant features from the microscopic level while effectively reaching larger time and length scales. We demonstrate the applicability of the transition matrices for the description of proton conduction trends in proton exchange membrane materials.

1.
A.
Katok
and
B.
Hasselblatt
,
Introduction to the Modern Theory of Dynamical Systems
(
Cambridge University Press
,
1997
), Vol. 54.
2.
S. H.
Strogatz
,
Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering
(
CRC Press
,
2018
).
3.
See https://www.nature.com/subjects/dynamical-systems for a definition of a dynamical system.
4.
L.
Delle Site
, “
What is a multiscale problem in molecular dynamics?
,”
Entropy
16
,
23
40
(
2014
).
5.
M.
Praprotnik
,
L.
Delle Site
, and
K.
Kremer
, “
Multiscale simulation of soft matter: From scale bridging to adaptive resolution
,”
Annu. Rev. Phys. Chem.
59
,
545
571
(
2008
).
6.
K.
Wendler
,
F.
Dommert
,
Y. Y.
Zhao
,
R.
Berger
,
C.
Holm
, and
L.
Delle Site
, “
Ionic liquids studied across different scales: A computational perspective
,”
Faraday Discuss.
154
,
111
132
(
2012
).
7.
J.
Behler
, “
Constructing high-dimensional neural network potentials: A tutorial review
,”
Int. J. Quantum Chem.
115
,
1032
1050
(
2015
).
8.
J.
Behler
, “
First principles neural network potentials for reactive simulations of large molecular and condensed systems
,”
Angew. Chem., Int. Ed.
56
,
12828
12840
(
2017
).
9.
B.
Nebgen
,
N.
Lubbers
,
J. S.
Smith
,
A. E.
Sifain
,
A.
Lokhov
,
O.
Isayev
,
A. E.
Roitberg
,
K.
Barros
, and
S.
Tretiak
, “
Transferable dynamic molecular charge assignment using deep neural networks
,”
J. Chem. Theory Comput.
14
,
4687
4698
(
2018
).
10.
K. T.
Butler
,
D. W.
Davies
,
H.
Cartwright
,
O.
Isayev
, and
A.
Walsh
, “
Machine learning for molecular and materials science
,”
Nature
559
,
547
(
2018
).
11.
M.
Hellström
and
J.
Behler
, “
Concentration-dependent proton transfer mechanisms in aqueous NaOH solutions: From acceptor-driven to donor-driven and back
,”
J. Phys. Chem. Lett.
7
,
3302
3306
(
2016
).
12.
M.
Gastegger
,
C.
Kauffmann
,
J.
Behler
, and
P.
Marquetand
, “
Comparing the accuracy of high-dimensional neural network potentials and the systematic molecular fragmentation method: A benchmark study for all-trans alkanes
,”
J. Chem. Phys.
144
,
194110
(
2016
).
13.
V.
Quaranta
,
M.
Hellström
, and
J.
Behler
, “
Proton-transfer mechanisms at the water–ZnO interface: The role of presolvation
,”
J. Phys. Chem. Lett.
8
,
1476
1483
(
2017
).
14.
F.
Noé
,
J.
Chodera
,
G.
Bowman
,
V.
Pande
, and
F.
Noé
,
An Introduction to Markov State Models and Their Application to Long Timescale Molecular Simulation
, Advances in Experimental Medicine and Biology Vol. 797 (
Springer
,
2014
).
15.
G. R.
Bowman
,
E. R.
Bolin
,
K. M.
Hart
,
B. C.
Maguire
, and
S.
Marqusee
, “
Discovery of multiple hidden allosteric sites by combining Markov state models and experiments
,”
Proc. Natl. Acad. Sci. U. S. A.
112
(
9
),
2734
2739
(
2015
).
16.
B. E.
Husic
and
V. S.
Pande
, “
Markov state models: From an art to a science
,”
J. Am. Chem. Soc.
140
,
2386
2396
(
2018
).
17.
V. S.
Pande
,
K.
Beauchamp
, and
G. R.
Bowman
, “
Everything you wanted to know about Markov State Models but were afraid to ask
,”
Methods
52
,
99
105
(
2010
), Protein Folding.
18.
C.
Schütte
and
M.
Sarich
, “
A critical appraisal of Markov state models
,”
Eur. Phys. J. Spec. Top.
224
,
2445
2462
(
2015
).
19.
C. R.
Schwantes
,
R. T.
McGibbon
, and
V. S.
Pande
, “
Perspective: Markov models for long-timescale biomolecular dynamics
,”
J. Chem. Phys.
141
,
090901
(
2014
).
20.
L. J.
LaBerge
and
J. C.
Tully
, “
A rigorous procedure for combining molecular dynamics and Monte Carlo simulation algorithms
,”
Chem. Phys.
260
,
183
191
(
2000
).
21.
E. C.
Neyts
and
A.
Bogaerts
, “
Combining molecular dynamics with Monte Carlo simulations: Implementations and applications
,”
Theor. Chem. Acc.
132
,
1320
(
2012
).
22.
B. M.
Forrest
and
U. W.
Suter
, “
Hybrid Monte Carlo simulations of dense polymer systems
,”
J. Chem. Phys.
101
,
2616
2629
(
1994
).
23.
D. G.
Gromov
and
J. J.
de Pablo
, “
Structure of binary polymer blends: Multiple time step hybrid Monte Carlo simulations and self-consistent integral-equation theory
,”
J. Chem. Phys.
103
,
8247
8256
(
1995
).
24.
A.
Irbäck
, “
Hybrid Monte Carlo simulation of polymer chains
,”
J. Chem. Phys.
101
,
1661
1667
(
1994
).
25.
D. W.
Heermann
and
L.
Yixue
, “
A global-update simulation method for polymer systems
,”
Macromol. Chem. Phys.
2
,
299
308
(
1993
).
26.
I.
Martin-Bragado
,
R.
Borges
,
J. P.
Balbuena
, and
M.
Jaraiz
, “
Kinetic Monte Carlo simulation for semiconductor processing: A review
,”
Prog. Mater. Sci.
92
,
1
32
(
2018
).
27.
G.
Betz
and
W.
Husinsky
, “
A combined molecular dynamics and kinetic Monte Carlo calculation to study sputter erosion and beam assisted deposition
,”
Nucl. Instrum. Methods Phys. Res., Sect. B
193
,
352
358
(
2002
).
28.
A.
Ghoufi
and
G.
Maurin
, “
Hybrid Monte Carlo simulations combined with a phase mixture model to predict the structural transitions of a porous metal-organic framework material upon adsorption of guest molecules
,”
J. Phys. Chem. C
114
,
6496
6502
(
2010
).
29.
A. A.
Knizhnik
,
A. A.
Bagaturyants
,
I. V.
Belov
,
B. V.
Potapkin
, and
A. A.
Korkin
, “
An integrated kinetic Monte Carlo molecular dynamics approach for film growth modeling and simulation: ZrO2 deposition on Si(100) surface
,”
Comput. Mater. Sci.
24
,
128
132
(
2002
).
30.
U. H. E.
Hansmann
and
Y.
Okamoto
, “
New Monte Carlo algorithms for protein folding
,”
Curr. Opin. Struct. Biol.
9
,
177
183
(
1999
).
31.
E. K.
Peter
and
J.-E.
Shea
, “
A hybrid MD-kMC algorithm for folding proteins in explicit solvent
,”
Phys. Chem. Chem. Phys.
16
,
6430
6440
(
2014
).
32.
H.
Zhang
, “
A new hybrid Monte Carlo algorithm for protein potential function test and structure refinement
,”
Proteins: Struct., Funct., Genet.
34
,
464
471
(
1999
).
33.
F.
Noé
and
E.
Rosta
, “
Markov models of molecular kinetics
,”
J. Chem. Phys.
151
,
190401
(
2019
).
34.
M.
Dibak
,
M. J.
del Razo
,
D.
De Sancho
,
C.
Schütte
, and
F.
Noé
, “
MSM/RD: Coupling Markov state models of molecular kinetics with reaction-diffusion simulations
,”
J. Chem. Phys.
148
,
214107
(
2018
).
35.
B. G.
Keller
,
J.-H.
Prinz
, and
F.
Noé
, “
Markov models and dynamical fingerprints: Unraveling the complexity of molecular kinetics
,”
Chem. Phys.
396
,
92
107
(
2012
), part of special issue: Experimental and theoretical studies of protein dynamics and function: From femtoseconds to milliseconds.
36.
B. G.
Keller
,
A.
Kobitski
,
A.
Jäschke
,
G. U.
Nienhaus
, and
F.
Noé
, “
Complex RNA folding kinetics revealed by single-molecule FRET and hidden Markov models
,”
J. Am. Chem. Soc.
136
,
4534
4543
(
2014
).
37.
F.
Nüske
,
B. G.
Keller
,
G.
Pérez-Hernández
,
A. S. J. S.
Mey
, and
F.
Noé
, “
Variational approach to molecular kinetics
,”
J. Chem. Theory Comput.
10
,
1739
1752
(
2014
).
38.
F.
Nüske
,
H.
Wu
,
J.-H.
Prinz
,
C.
Wehmeyer
,
C.
Clementi
, and
F.
Noé
, “
Markov state models from short non-equilibrium simulations—Analysis and correction of estimation bias
,”
J. Chem. Phys.
146
,
094104
(
2017
).
39.
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
).
40.
C.
Schütte
,
F.
Noé
,
J.
Lu
,
M.
Sarich
, and
E.
Vanden-Eijnden
, “
Markov state models based on milestoning
,”
J. Chem. Phys.
134
,
204105
(
2011
).
41.
B.
Trendelkamp-Schroer
,
H.
Wu
,
F.
Paul
, and
F.
Noé
, “
Estimation and uncertainty of reversible Markov models
,”
J. Chem. Phys.
143
,
174101
(
2015
).
42.
F.
Vitalini
,
A. S. J. S.
Mey
,
F.
Noé
, and
B. G.
Keller
, “
Dynamic properties of force fields
,”
J. Chem. Phys.
142
,
084101
(
2015
).
43.
H.
Wu
,
F.
Paul
,
C.
Wehmeyer
, and
F.
Noé
, “
Multiensemble Markov models of molecular thermodynamics and kinetics
,”
Proc. Natl. Acad. Sci. U. S. A.
113
(
23
),
E3221
E3230
(
2016
).
44.
E.
Hruska
,
J. R.
Abella
,
F.
Nüske
,
L. E.
Kavraki
, and
C.
Clementi
, “
Quantitative comparison of adaptive sampling methods for protein dynamics
,”
J. Chem. Phys.
149
,
244119
(
2018
).
45.
U.
Sengupta
,
M.
Carballo-Pacheco
, and
B.
Strodel
, “
Automated Markov state models for molecular dynamics simulations of aggregation and self-assembly
,”
J. Chem. Phys.
150
,
115101
(
2019
).
46.
F.
Noé
,
C.
Schütte
,
E.
Vanden-Eijnden
,
L.
Reich
, and
T. R.
Weikl
, “
Constructing the equilibrium ensemble of folding pathways from short off-equilibrium simulations
,”
Proc. Natl. Acad. Sci. U. S. A.
106
,
19011
19016
(
2009
).
47.
T. J.
Lane
,
G. R.
Bowman
,
K.
Beauchamp
,
V. A.
Voelz
, and
V. S.
Pande
, “
Markov state model reveals folding and functional dynamics in ultra-long MD trajectories
,”
J. Am. Chem. Soc.
133
,
18413
18419
(
2011
).
48.
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
).
49.
N.
Stanley
,
S.
Esteban-Martín
, and
G.
De Fabritiis
, “
Kinetic modulation of a disordered protein domain by phosphorylation
,”
Nat. Commun.
5
,
5272
(
2014
).
50.
I.
Buch
,
T.
Giorgino
, and
G.
De Fabritiis
, “
Complete reconstruction of an enzyme-inhibitor binding process by molecular dynamics simulations
,”
Proc. Natl. Acad. Sci. U. S. A.
108
,
10184
10189
(
2011
).
51.
M.
Held
,
P.
Metzner
,
J.-H.
Prinz
, and
F.
Noé
, “
Mechanisms of protein-ligand association and its modulation by protein mutations
,”
Biophys. J.
100
,
701
710
(
2011
).
52.
D.-A.
Silva
,
G. R.
Bowman
,
A.
Sosa-Peinado
, and
X.
Huang
, “
A role for both conformational selection and induced fit in ligand binding by the LAO protein
,”
PLoS Comput. Biol.
7
,
1
11
(
2011
).
53.
G. R.
Bowman
and
P. L.
Geissler
, “
Equilibrium fluctuations of a single folded protein reveal a multitude of potential cryptic allosteric sites
,”
Proc. Natl. Acad. Sci. U. S. A.
109
,
11681
11686
(
2012
).
54.
N.
Plattner
and
F.
Noé
, “
Protein conformational plasticity and complex ligand-binding kinetics explored by atomistic simulations and Markov models
,”
Nat. Commun.
6
,
7653
(
2015
).
55.
D.
Chakraborty
and
D. J.
Wales
, “
Dynamics of an adenine-adenine RNA conformational switch from discrete path sampling
,”
J. Chem. Phys.
150
,
125101
(
2019
).
56.
G.
Pinamonti
,
F.
Paul
,
F.
Noé
,
A.
Rodriguez
, and
G.
Bussi
, “
The mechanism of RNA base fraying: Molecular dynamics simulations analyzed with core-set Markov state models
,”
J. Chem. Phys.
150
,
154123
(
2019
).
57.
A. M.
Berezhkovskii
and
A.
Szabo
, “
Committors, first-passage times, fluxes, Markov states, milestones, and all that
,”
J. Chem. Phys.
150
,
054106
(
2019
).
58.
J. A.
Morrone
,
K. E.
Haslinger
, and
M. E.
Tuckerman
, “
Ab initio molecular dynamics simulation of the structure and proton transport dynamics of methanol–water solutions
,”
J. Phys. Chem. B
110
,
3712
3720
(
2006
).
59.
T. C.
Berkelbach
,
H.-S.
Lee
, and
M. E.
Tuckerman
, “
Concerted hydrogen-bond dynamics in the transport mechanism of the hydrated proton: A first-principles molecular dynamics study
,”
Phys. Rev. Lett.
103
,
238302
(
2009
).
60.
L.
Vilčiauskas
,
M. E.
Tuckerman
,
G.
Bester
,
S. J.
Paddison
, and
K.-D.
Kreuer
, “
The mechanism of proton conduction in phosphoric acid
,”
Nat. Chem.
4
,
461
(
2012
).
61.
C.
Dreßler
,
G.
Kabbe
, and
D.
Sebastiani
, “
Insight from atomistic simulations of protonation dynamics at the nanoscale
,”
Fuel Cells
16
,
682
694
(
2016
).
62.
R.
Vuilleumier
and
D.
Borgis
, “
Proton conduction: Hopping along hydrogen bonds
,”
Nat. Chem.
4
,
432
(
2012
).
63.
D.
Marx
,
A.
Chandra
, and
M. E.
Tuckerman
, “
Aqueous basic solutions: Hydroxide solvation, structural diffusion, and comparison to the hydrated proton
,”
Chem. Rev.
110
,
2174
2216
(
2010
).
64.
D.
Marx
, “
Proton transfer 200 years after von Grotthuss: Insights from ab initio simulations
,”
ChemPhysChem
7
,
1848
1870
(
2006
).
65.
M. E.
Tuckerman
,
A.
Chandra
, and
D.
Marx
, “
A statistical mechanical theory of proton transport kinetics in hydrogen-bonded networks based on population correlation functions with applications to acids and bases
,”
J. Chem. Phys.
133
,
124108
(
2010
).
66.
A.
Chandra
,
M. E.
Tuckerman
, and
D.
Marx
, “
Connecting solvation shell structure to proton transport kinetics in hydrogen–bonded networks via population correlation functions
,”
Phys. Rev. Lett.
99
,
145901
(
2007
).
67.
J.
Schmidt
,
J.
VandeVondele
,
I.-F. W.
Kuo
,
D.
Sebastiani
,
J. I.
Siepmann
,
J.
Hutter
, and
C. J.
Mundy
, “
Isobaric–isothermal molecular dynamics simulations utilizing density functional theory: An assessment of the structure and density of water at near-ambient conditions
,”
J. Phys. Chem. B
113
,
11959
11964
(
2009
).
68.
J. C.
Grossman
,
E.
Schwegler
,
E. W.
Draeger
,
F.
Gygi
, and
G.
Galli
, “
Towards an assessment of the accuracy of density functional theory for first principles simulations of water
,”
J. Chem. Phys.
120
,
300
311
(
2004
).
69.
E.
Schwegler
,
J. C.
Grossman
,
F.
Gygi
, and
G.
Galli
, “
Towards an assessment of the accuracy of density functional theory for first principles simulations of water. II
,”
J. Chem. Phys.
121
,
5400
5409
(
2004
).
70.
G.
Kabbe
,
C.
Wehmeyer
, and
D.
Sebastiani
, “
A coupled molecular dynamics/kinetic Monte Carlo approach for protonation dynamics in extended systems
,”
J. Chem. Theory Comput.
10
,
4221
(
2014
).
71.
G.
Kabbe
,
C.
Dreßler
, and
D.
Sebastiani
, “
Toward realistic transfer rates within the coupled molecular dynamics/lattice Monte Carlo approach
,”
J. Phys. Chem. C
120
,
19905
19912
(
2016
).
72.
C.
Dreßler
,
G.
Kabbe
, and
D.
Sebastiani
, “
Proton conductivity in hydrogen phosphate/sulfates from a coupled molecular dynamics/lattice Monte Carlo (cMD/LMC) approach
,”
J. Phys. Chem. C
120
,
19913
19922
(
2016
).
73.
G.
Kabbe
,
C.
Dreßler
, and
D.
Sebastiani
, “
Proton mobility in aqueous systems: Combining ab initio accuracy with millisecond timescales
,”
Phys. Chem. Chem. Phys.
19
,
28604
28609
(
2017
).
74.
H.-S.
Lee
and
M. E.
Tuckerman
, “
The structure and proton transport mechanisms in the superprotonic phase of CsH2PO4: An ab initio molecular dynamics study
,”
J. Phys. Chem. C
112
,
9917
9930
(
2008
).
75.
T.
Müller-Gronbach
,
E.
Novak
, and
K.
Ritter
,
Monte Carlo-Algorithmen
(
Springer-Verlag
,
2012
).
76.
C.
Wehmeyer
,
M.
Schrader
,
D.
Andrienko
, and
D.
Sebastiani
, “
Water-free proton conduction in hexakis(p-phosphonatophenyl)benzene nanochannels
,”
J. Phys. Chem. C
117
,
12366
12372
(
2013
).
77.
M. N.
Garaga
,
V.
Dracopoulos
,
U.
Werner-Zwanziger
,
J. W.
Zwanziger
,
M.
Maréchal
,
M.
Persson
,
L.
Nordstierna
, and
A.
Martinelli
, “
A long-chain protic ionic liquid inside silica nanopores: Enhanced proton mobility due to efficient self-assembly and decoupled proton transport
,”
Nanoscale
10
,
12337
12348
(
2018
).
78.
J.
Hutter
,
M.
Iannuzzi
,
F.
Schiffmann
, and
J.
VandeVondele
, “
cp2k: atomistic simulations of condensed matter systems
,”
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
4
,
15
25
(
2014
).
79.
J.
VandeVondele
,
M.
Krack
,
F.
Mohamed
,
M.
Parrinello
,
T.
Chassaing
, and
J.
Hutter
, “
Quickstep: Fast and accurate density functional calculations using a mixed Gaussian and plane waves approach
,”
Comput. Phys. Commun.
167
,
103
128
(
2005
).
80.
J.
VandeVondele
and
J.
Hutter
, “
An efficient orbital transformation method for electronic structure calculations
,”
J. Chem. Phys.
118
,
4365
4369
(
2003
).
81.
Y.
Zhang
and
W.
Yang
, “
Comment on “Generalized gradient approximation made simple”
,”
Phys. Rev. Lett.
80
,
890
(
1998
).
82.
J. P.
Perdew
,
A.
Ruzsinszky
,
G. I.
Csonka
,
O. A.
Vydrov
,
G. E.
Scuseria
,
L. A.
Constantin
,
X.
Zhou
, and
K.
Burke
, “
Restoring the density-gradient expansion for exchange in solids and surfaces
,”
Phys. Rev. Lett.
100
,
136406
(
2008
).
83.
J. P.
Perdew
,
K.
Burke
, and
M.
Ernzerhof
, “
Generalized gradient approximation made simple
,”
Phys. Rev. Lett.
77
,
3865
3868
(
1996
).
84.
C.
Lee
,
W.
Yang
, and
R. G.
Parr
, “
Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density
,”
Phys. Rev. B
37
,
785
789
(
1988
).
85.
A. D.
Becke
, “
Density-functional exchange-energy approximation with correct asymptotic behavior
,”
Phys. Rev. A
38
,
3098
3100
(
1988
).
86.
J.
VandeVondele
and
J.
Hutter
, “
Gaussian basis sets for accurate calculations on molecular systems in gas and condensed phases
,”
J. Chem. Phys.
127
,
114105
(
2007
).
87.
C.
Hartwigsen
,
S.
Goedecker
, and
J.
Hutter
, “
Relativistic separable dual-space Gaussian pseudopotentials from H to Rn
,”
Phys. Rev. B
58
,
3641
3662
(
1998
).
88.
M.
Krack
, “
Pseudopotentials for H to Kr optimized for gradient-corrected exchange-correlation functionals
,”
Theor. Chem. Acc.
114
,
145
152
(
2005
).
89.
S.
Grimme
,
J.
Antony
,
S.
Ehrlich
, and
H.
Krieg
, “
A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu
,”
J. Chem. Phys.
132
,
154104
(
2010
).
90.
S.
Nosé
, “
A unified formulation of the constant temperature molecular dynamics methods
,”
J. Chem. Phys.
81
,
511
519
(
1984
).
91.
G. J.
Martyna
,
M. L.
Klein
, and
M.
Tuckerman
, “
Nosé-Hoover chains: The canonical ensemble via continuous dynamics
,”
J. Chem. Phys.
97
,
2635
2643
(
1992
).
92.
S.
Nosé
, “
A molecular dynamics method for simulations in the canonical ensemble
,”
Mol. Phys.
52
,
255
268
(
1970
).

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