We analyze the probability distribution of rare first passage times corresponding to transitions between product and reactant states in a kinetic transition network. The mean first passage times and the corresponding rate constants are analyzed in detail for two model landscapes and the double funnel landscape corresponding to an atomic cluster. Evaluation schemes based on eigendecomposition and kinetic path sampling, which both allow access to the first passage time distribution, are benchmarked against mean first passage times calculated using graph transformation. Numerical precision issues severely limit the useful temperature range for eigendecomposition, but kinetic path sampling is capable of extending the first passage time analysis to lower temperatures, where the kinetics of interest constitute rare events. We then investigate the influence of free energy based state regrouping schemes for the underlying network. Alternative formulations of the effective transition rates for a given regrouping are compared in detail to determine their numerical stability and capability to reproduce the true kinetics, including recent coarse-graining approaches that preserve occupancy cross correlation functions. We find that appropriate regrouping of states under the simplest local equilibrium approximation can provide reduced transition networks with useful accuracy at somewhat lower temperatures. Finally, a method is provided to systematically interpolate between the local equilibrium approximation and exact intergroup dynamics. Spectral analysis is applied to each grouping of states, employing a moment-based mode selection criterion to produce a reduced state space, which does not require any spectral gap to exist, but reduces to gap-based coarse graining as a special case. Implementations of the developed methods are freely available online.
REFERENCES
1.
F.
Noé
and S.
Fischer
, Curr. Opin. Struct. Biol.
18
, 154
(2008
).2.
D.
Prada-Gracia
, J.
Gómez-Gardeñes
, P.
Echenique
, and F.
Falo
, PLoS Comput. Biol.
5
, e1000415
(2009
).3.
D. J.
Wales
, Curr. Opin. Struct. Biol.
20
, 3
(2010
).4.
D. J.
Wales
, Mol. Phys.
100
, 3285
(2002
).5.
D. J.
Wales
, Mol. Phys.
102
, 891
(2004
).6.
7.
8.
9.
N. G.
van Kampen
, Stochastic Processes in Physics and Chemistry
(North-Holland
, Amsterdam
, 1981
).10.
11.
C.
Schütte
, A.
Fischer
, W.
Huisinga
, and P.
Deuflhard
, J. Comput. Phys.
151
, 146
(1999
).12.
M.
Shirts
and V. S.
Pande
, Science
290
, 1903
(2000
).13.
N.
Singhal
, C. D.
Snow
, and V. S.
Pande
, J. Chem. Phys.
121
, 415
(2004
).14.
W. C.
Swope
, J. W.
Pitera
, and F.
Suits
, J. Phys. Chem. B
108
, 6571
(2004
).15.
J. D.
Chodera
, N.
Singhal
, V. S.
Pande
, K. A.
Dill
, W. C.
Swope
, and J. W.
Pitera
, J. Chem. Phys.
126
, 155101
(2007
).16.
V. S.
Pande
, K.
Beauchamp
, and G. R.
Bowman
, Methods
52
, 99
(2010
).17.
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
).18.
B. E.
Husic
and V. S.
Pande
, J. Am. Chem. Soc.
140
, 2386
(2018
).19.
T. D.
Swinburne
and D.
Perez
, Phys. Rev. Mater.
2
, 053802
(2018
).20.
J. A.
Joseph
, K.
Röder
, D.
Chakraborty
, R. G.
Mantell
, and D. J.
Wales
, Chem. Commun.
53
, 6974
(2017
).21.
D. J.
Wales
, Annu. Rev. Phys. Chem.
69
, 401
(2018
).22.
K.
Röder
, J. A.
Joseph
, B. E.
Husic
, and D. J.
Wales
, Adv. Theory Simul.
2
, 1800175
(2019
).23.
G. A.
Huber
and S.
Kim
, Biophys. J.
70
, 97
(1996
).24.
C.
Dellago
, P. G.
Bolhuis
, F. S.
Csajka
, and D.
Chandler
, J. Chem. Phys.
108
, 1964
(1998
).25.
P. G.
Bolhuis
, D.
Chandler
, C.
Dellago
, and P. L.
Geissler
, Annu. Rev. Phys. Chem.
53
, 291
(2002
).26.
T. S.
van Erp
, D.
Moroni
, and P. G.
Bolhuis
, J. Chem. Phys.
118
, 7762
(2003
).27.
A. K.
Faradjian
and R.
Elber
, J. Chem. Phys.
120
, 10880
(2004
).28.
A. M. A.
West
, R.
Elber
, and D.
Shalloway
, J. Chem. Phys.
126
, 145104
(2007
).29.
G.
Bussi
, F. L.
Gervasio
, A.
Laio
, and M.
Parrinello
, J. Am. Chem. Soc.
128
, 13435
(2006
).30.
R. J.
Allen
, D.
Frenkel
, and P. R.
ten Wolde
, J. Chem. Phys.
124
, 024102
(2006
).31.
E.
Vanden-Eijnden
, M.
Venturoli
, G.
Ciccotti
, and R.
Elber
, J. Chem. Phys.
129
, 174102
(2008
).32.
K.
Kuczera
, G. S.
Jas
, and R.
Elber
, J. Phys. Chem. A
113
, 7461
(2009
).33.
B. W.
Zhang
, D.
Jasnow
, and D. M.
Zuckerman
, J. Chem. Phys.
132
, 054107
(2010
).34.
E.
Vanden-Eijnden
and M.
Venturoli
, J. Chem. Phys.
130
, 194101
(2009
).35.
R.
Elber
, J.
Bello-Rivas
, P.
Ma
, A.
Cardenas
, and A.
Fathizadeh
, Entropy
19
, 219
(2017
).36.
37.
D. J.
Wales
, J. Chem. Phys.
130
, 204111
(2009
).38.
J. D.
Stevenson
and D. J.
Wales
, J. Chem. Phys.
141
, 041104
(2014
).39.
T. J.
Frankcombe
and S. C.
Smith
, Theor. Chem. Acc.
124
, 303
(2009
).40.
S. A.
Trygubenko
and D. J.
Wales
, J. Chem. Phys.
124
, 234110
(2006
).41.
R. S.
MacKay
and J. D.
Robinson
, Philos. Trans. R. Soc., A
376
, 20170232
(2018
).42.
T. D.
Swinburne
and D. J.
Wales
, J. Chem. Theory Comput.
16
, 2661
(2020
).43.
M.
Athènes
and V. V.
Bulatov
, Phys. Rev. Lett.
113
, 230601
(2014
).44.
M.
Athènes
, S.
Kaur
, G.
Adjanor
, T.
Vanacker
, and T.
Jourdan
, Phys. Rev. Mater.
3
, 103802
(2019
).45.
H. S.
Chung
, K.
McHale
, J. M.
Louis
, and W. A.
Eaton
, Science
335
, 981
(2012
).46.
F. H.
Stillinger
and T. A.
Weber
, Science
225
, 983
(1984
).47.
J. P. K.
Doye
and D. J.
Wales
, J. Chem. Phys.
116
, 3777
(2002
).48.
J. G.
Kemeny
, Lin. Algebra Appl.
38
, 193
(1981
).49.
P.
Deuflhard
, W.
Huisinga
, A.
Fischer
, and C.
Schütte
, Lin. Algebra Appl.
315
, 39
(2000
).50.
P.
Deuflhard
and M.
Weber
, Lin. Algebra Appl.
398
, 161
(2005
).51.
S.
Kube
and M.
Weber
, J. Chem. Phys.
126
, 024103
(2007
).52.
W.
E
, T.
Li
, and E.
Vanden-Eijnden
, Proc. Natl. Acad. Sci. U. S. A.
105
, 7907
(2008
).53.
N.
Lempesis
, D. G.
Tsalikis
, G. C.
Boulougouris
, and D. N.
Theodorou
, J. Chem. Phys.
135
, 204507
(2011
).54.
G. R.
Bowman
, L.
Meng
, and X.
Huang
, J. Chem. Phys.
139
, 121905
(2013
).55.
S.
Orioli
and P.
Faccioli
, J. Chem. Phys.
145
, 124120
(2016
).56.
W.
Wang
, T.
Liang
, F. K.
Sheong
, X.
Fan
, and X.
Huang
, J. Chem. Phys.
149
, 072337
(2018
).57.
J. A.
Ward
and M.
López-García
, Appl. Netw. Sci.
4
, 108
(2019
).58.
D. J.
Wales
and P.
Salamon
, Proc. Natl. Acad. Sci. U. S. A.
111
, 617
(2014
).59.
D. A.
Evans
and D. J.
Wales
, J. Chem. Phys.
118
, 3891
(2003
).60.
J. M.
Carr
and D. J.
Wales
, J. Phys. Chem. B
112
, 8760
(2008
).61.
T. D.
Swinburne
and D.
Kannan
, “PyGT: Graph transformation and dimensionality reduction in python
,” 2020
, https://github.com/tomswinburne/PyGT.62.
D. J.
Sharpe
, “DISCOTRESS: The discrete state continuous time rare event simulation suite
,” 2020
, https://github.com/danieljsharpe/DISCOTRESS.63.
M.
Manhart
, W.
Kion-Crosby
, and A. V.
Morozov
, J. Chem. Phys.
143
, 214106
(2015
).64.
M.
Manhart
and A. V.
Morozov
, Phys. Rev. Lett.
111
, 088102
(2013
).65.
M.
Manhart
and A. V.
Morozov
, Proc. Natl. Acad. Sci. U. S. A.
112
, 1797
(2015
).66.
D. J.
Sharpe
and D. J.
Wales
, J. Chem. Phys.
151
, 124101
(2019
).67.
D. J.
Sharpe
and D. J.
Wales
, J. Chem. Phys.
153
, 024121
(2020
).68.
N.-V.
Buchete
and G.
Hummer
, J. Phys. Chem. B
112
, 6057
(2008
).69.
E.
Anderson
, Z.
Bai
, C.
Bischof
, S.
Blackford
, J.
Demmel
, J.
Dongarra
, J.
Du Croz
, A.
Greenbaum
, S.
Hammarling
, A.
McKenney
, and D.
Sorensen
, LAPACK Users’ Guide
, 3rd ed. (Society for Industrial and Applied Mathematics
, Philadelphia, PA
, 1999
).70.
R.
Lehoucq
, K.
Maschhoff
, D.
Sorensen
, and C.
Yang
, ARPACK, a collection of Fortran 77 subroutines designed to solve large scale eigenvalue problems, https://www.caam.rice.edu/software/ARPACK/.71.
S. A.
Trygubenko
and D. J.
Wales
, Mol. Phys.
104
, 1497
(2006
).72.
P.
Metzner
, C.
Schütte
, and E.
Vanden-Eijnden
, Multiscale Model. Simul.
7
, 1192
(2009
).73.
A. J.
Ballard
, S.
Martiniani
, J. D.
Stevenson
, S.
Somani
, and D. J.
Wales
, Wiley Interdiscip. Rev.: Comput. Mol. Sci.
5
, 273
(2015
).74.
M. A.
Novotny
, Phys. Rev. Lett.
74
, 1
(1995
).75.
M. A.
Novotny
, in Annual Reviews of Computational Physics
, edited by D.
Stauffer
(World Scientific
, Singapore
, 2001
), Vol. 9, pp. 153
–210
.76.
M. A.
Novotny
, Comput. Phys. Commun.
147
, 659
(2002
).77.
O. M.
Becker
and M.
Karplus
, J. Chem. Phys.
106
, 1495
(1997
).78.
D. J.
Wales
, M. A.
Miller
, and T. R.
Walsh
, Nature
394
, 758
(1998
).79.
J. E.
Jones
and A. E.
Ingham
, Proc. R. Soc. A
107
, 636
(1925
).80.
M. A.
Miller
, J. P. K.
Doye
, and D. J.
Wales
, J. Chem. Phys.
110
, 328
(1999
).81.
J. P. K.
Doye
, M. A.
Miller
, and D. J.
Wales
, J. Chem. Phys.
110
, 6896
(1999
).82.
D. J.
Wales
and T. V.
Bogdan
, J. Phys. Chem. B
110
, 20765
(2006
).83.
D. J.
Wales
and J. P. K.
Doye
, J. Phys. Chem. A
101
, 5111
(1997
).84.
D. J.
Wales
and H. A.
Scheraga
, Science
285
, 1368
(1999
).85.
M. T.
Oakley
, R. L.
Johnston
, and D. J.
Wales
, Phys. Chem. Chem. Phys.
15
, 3965
(2013
).86.
M.
Dittner
and B.
Hartke
, Comput. Theor. Chem.
1107
, 7
(2017
).87.
J. P. K.
Doye
, D. J.
Wales
, and M. A.
Miller
, J. Chem. Phys.
109
, 8143
(1998
).88.
F.
Calvo
, J. P.
Neirotti
, D. L.
Freeman
, and J. D.
Doll
, J. Chem. Phys.
112
, 10350
(2000
).89.
J. P.
Neirotti
, F.
Calvo
, D. L.
Freeman
, and J. D.
Doll
, J. Chem. Phys.
112
, 10340
(2000
).90.
P. A.
Frantsuzov
and V. A.
Mandelshtam
, Phys. Rev. E
72
, 037102
(2005
).91.
C.
Predescu
, P. A.
Frantsuzov
, and V. A.
Mandelshtam
, J. Chem. Phys.
122
, 154305
(2005
).92.
H.
Liu
and K. D.
Jordan
, J. Phys. Chem. B
109
, 5203
(2005
).93.
G.
Adjanor
, M.
Athènes
, and F.
Calvo
, Eur. Phys. J. B
53
, 47
(2006
).94.
V. A.
Sharapov
and V. A.
Mandelshtam
, J. Phys. Chem. A
111
, 10284
(2007
).95.
V. A.
Sharapov
, D.
Meluzzi
, and V. A.
Mandelshtam
, Phys. Rev. Lett.
98
, 105701
(2007
).96.
F.
Calvo
, Phys. Rev. E
82
, 046703
(2010
).97.
M.
Picciani
, M.
Athènes
, J.
Kurchan
, and J.
Tailleur
, J. Chem. Phys.
135
, 034108
(2011
).98.
M.
Cameron
and E.
Vanden-Eijnden
, J. Stat. Phys.
156
, 427
(2014
).99.
D. J.
Wales
, J. Chem. Phys.
142
, 130901
(2015
).100.
D. J.
Wales
, The Cambridge Landscapes Database, http://www-wales.ch.cam.ac.uk/CCD.html.101.
G.
Hummer
and A.
Szabo
, J. Phys. Chem. B
119
, 9029
(2015
).102.
M. E. J.
Newman
, Eur. Phys. J. B
38
, 321
(2004
).103.
L.
Danon
, A.
Díaz-Guilera
, J.
Duch
, and A.
Arenas
, J. Stat. Mech.
2005
, P09008
.104.
A.
Lancichinetti
and S.
Fortunato
, Phys. Rev. E
80
, 056117
(2009
).105.
S.
Fortunato
, Phys. Rep.
486
, 75
(2010
).106.
F. D.
Malliaros
and M.
Vazirgiannis
, Phys. Rep.
533
, 95
(2013
).107.
S.
Fortunato
and D.
Hric
, Phys. Rep.
659
, 1
(2016
).108.
Z.
Yang
, R.
Algesheimer
, and C. J.
Tessone
, Sci. Rep.
6
, 30750
(2016
).109.
S.
Emmons
, S.
Kobourov
, M.
Gallant
, and K.
Börner
, PLoS One
11
, e0159161
(2016
).110.
X.
Zhang
, Z.
Ma
, Z.
Zhang
, Q.
Sun
, and J.
Yan
, J. Phys.: Conf. Ser.
1069
, 012123
(2018
).111.
A.
Kells
, V.
Koskin
, E.
Rosta
, and A.
Annibale
, J. Chem. Phys.
152
, 104108
(2020
).112.
S.
Pigolotti
and A.
Vulpiani
, J. Chem. Phys.
128
, 154114
(2008
).113.
114.
J. M.
Carr
, S. A.
Trygubenko
, and D. J.
Wales
, J. Chem. Phys.
122
, 234903
(2005
).115.
A.
Kells
, Z. É.
Mihálka
, A.
Annibale
, and E.
Rosta
, J. Chem. Phys.
150
, 134107
(2019
).116.
L.
Martini
, A.
Kells
, R.
Covino
, G.
Hummer
, N.-V.
Buchete
, and E.
Rosta
, Phys. Rev. X
7
, 031060
(2017
).© 2020 Author(s).
2020
Author(s)
You do not currently have access to this content.