We introduce an extension to the weighted ensemble (WE) path sampling method to restrict sampling to a one-dimensional path through a high dimensional phase space. Our method, which is based on the finite-temperature string method, permits efficient sampling of both equilibrium and non-equilibrium systems. Sampling obtained from the WE method guides the adaptive refinement of a Voronoi tessellation of order parameter space, whose generating points, upon convergence, coincide with the principle reaction pathway. We demonstrate the application of this method to several simple, two-dimensional models of driven Brownian motion and to the conformational change of the nitrogen regulatory protein C receiver domain using an elastic network model. The simplicity of the two-dimensional models allows us to directly compare the efficiency of the WE method to conventional brute force simulations and other path sampling algorithms, while the example of protein conformational change demonstrates how the method can be used to efficiently study transitions in the space of many collective variables.
REFERENCES
1.
C.
Dellago
, P. G.
Bolhuis
, F. S.
Csajka
, and D.
Chandler
, “Transition path sampling and the calculation of rate constants
,” J. Chem. Phys.
108
, 1964
–1977
(1998
).2.
T.
Woolf
, “Path corrected functionals of stochastic trajectories: Towards relative free energy and reaction coordinate calculations
,” Chem. Phys. Lett.
289
, 433
–441
(1998
).3.
T. S.
van Erp
, D.
Moroni
, and P. G.
Bolhuis
, “A novel path sampling method for the calculation of rate constants
,” J. Chem. Phys.
118
, 7762
–7774
(2003
).4.
A. K.
Faradjian
and R.
Elber
, “Computing time scales from reaction coordinates by milestoning
,” J. Chem. Phys.
120
, 10880
–10889
(2004
).5.
R. J.
Allen
, P. B.
Warren
, and P. R.
Ten Wolde
, “Sampling rare switching events in biochemical networks
,” Phys. Rev. Lett.
94
, 018104
(2005
).6.
A.
Warmflash
, P.
Bhimalapuram
, and A. R.
Dinner
, “Umbrella sampling for nonequilibrium processes
,” J. Chem. Phys.
127
, 154112
(2007
).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.
C.
Dellago
and P. G.
Bolhuis
, “Transition path sampling and other advanced simulation techniques for rare events
,” in Advanced Computer Simulation Approaches for Soft Matter Sciences III
, Advances in Polymer Science
Vol. 221
, edited by C.
Holm
and K.
Kremer
(Springer
, Berlin
, 2009
), pp. 167
–233
.9.
M. C.
Zwier
and L. T.
Chong
, “Reaching biological timescales with all-atom molecular dynamics simulations
,” Curr. Opin. Pharmacol.
10
, 745
–752
(2010
).10.
R.
Elber
and M.
Karplus
, “A method for determining reaction paths in large molecules: Application to myoglobin
,” Chem. Phys. Lett.
139
, 375
–380
(1987
).11.
S.
Fischer
and M.
Karplus
, “Conjugate peak refinement: An algorithm for finding reaction paths and accurate transition states in systems with many degrees of freedom
,” Chem. Phys. Lett.
194
, 252
–261
(1992
).12.
G.
Henkelman
and H.
Jónsson
, “Improved tangent estimate in the nudged elastic band method for finding minimum energy paths and saddle points
,” J. Chem. Phys.
113
, 9978
–9985
(2000
).13.
W.
E
, W.
Ren
, and E.
Vanden-Eijnden
, “String method for the study of rare events
,” Phys. Rev. B
66
, 052301
(2002
).14.
L.
Maragliano
, A.
Fischer
, E.
Vanden-Eijnden
, and G.
Ciccotti
, “String method in collective variables: Minimum free energy paths and isocommittor surfaces
,” J. Chem. Phys.
125
, 024106
(2006
).15.
W.
E
, W.
Ren
, and E.
Vanden-Eijnden
, “Finite temperature string method for the study of rare events
,” J. Phys. Chem. B
109
, 6688
–6693
(2005
).16.
A. C.
Pan
, D.
Sezer
, and B.
Roux
, “Finding transition pathways using the string method with swarms of trajectories
,” J. Phys. Chem. B
112
, 3432
–3440
(2008
).17.
A. C.
Pan
and B.
Roux
, “Building Markov state models along pathways to determine free energies and rates of transitions
,” J. Chem. Phys.
129
, 064107
(2008
).18.
E.
Vanden-Eijnden
and M.
Venturoli
, “Revisiting the finite temperature string method for the calculation of reaction tubes and free energies
,” J. Chem. Phys.
130
, 194103
(2009
).19.
A.
Dickson
, A.
Warmflash
, and A. R.
Dinner
, “Nonequilibrium umbrella sampling in spaces of many order parameters
,” J. Chem. Phys.
130
, 074104
(2009
).20.
J.
Rogal
, W.
Lechner
, J.
Juraszek
, B.
Ensing
, and P. G.
Bolhuis
, “The reweighted path ensemble
,” J. Chem. Phys.
133
, 174109
(2010
).21.
M. E.
Johnson
and G.
Hummer
, “Characterization of a dynamic string method for the construction of transition pathways in molecular reactions
,” J. Phys. Chem. B
116
, 8573
–8583
(2012
).22.
B. M.
Dickson
, H.
Huang
, and C. B.
Post
, “Unrestrained computation of free energy along a path
,” J. Phys. Chem. B
116
, 11046
–11055
(2012
).23.
W.
Ren
, E.
Vanden-Eijnden
, P.
Maragakis
, and W.
E
, “Transition pathways in complex systems: Application of the finite-temperature string method to the alanine dipeptide
,” J. Chem. Phys.
123
, 134109
(2005
).24.
T. F.
Miller
III, E.
Vanden-Eijnden
, and D.
Chandler
, “Solvent coarse-graining and the string method applied to the hydrophobic collapse of a hydrated chain
,” Proc. Natl. Acad. Sci. U.S.A.
104
, 14559
–14564
(2007
).25.
V.
Ovchinnikov
, M.
Cecchini
, E.
Vanden-Eijnden
, and M.
Karplus
, “A conformational transition in the myosin VI converter contributes to the variable step size
,” Biophys. J.
101
, 2436
–2444
(2011
).26.
D.
Bhatt
and D. M.
Zuckerman
, “Heterogeneous path ensembles for conformational transitions in semi-atomistic models of adenylate kinase
,” J. Chem. Theory Comput.
6
, 3527
–3539
(2010
).27.
J. L.
Adelman
, A. L.
Dale
, M. C.
Zwier
, D.
Bhatt
, L. T.
Chong
, D. M.
Zuckerman
, and M.
Grabe
, “Simulations of the alternating access mechanism of the sodium symporter Mhp1
,” Biophys. J.
101
, 2399
–2407
(2011
).28.
G. A.
Huber
and S.
Kim
, “Weighted-ensemble Brownian dynamics simulations for protein association reactions
,” Biophys. J.
70
, 97
–110
(1996
).29.
B. W.
Zhang
, D.
Jasnow
, and D. M.
Zuckerman
, “Efficient and verified simulation of a path ensemble for conformational change in a united-residue model of calmodulin
,” Proc. Natl. Acad. Sci. U.S.A.
104
, 18043
–18048
(2007
).30.
B. W.
Zhang
, D.
Jasnow
, and D. M.
Zuckerman
, “The ‘weighted ensemble’ path sampling method is statistically exact for a broad class of stochastic processes and binning procedures
,” J. Chem. Phys.
132
, 054107
(2010
).31.
D.
Bhatt
, B. W.
Zhang
, and D. M.
Zuckerman
, “Steady-state simulations using weighted ensemble path sampling
,” J. Chem. Phys.
133
, 014110
(2010
).32.
A.
Rojnuckarin
, D. R.
Livesay
, and S.
Subramaniam
, “Bimolecular reaction simulation using weighted ensemble Brownian dynamics and the University of Houston Brownian Dynamics program
,” Biophys. J.
79
, 686
–693
(2000
).33.
A.
Rojnuckarin
, S.
Kim
, and S.
Subramaniam
, “Brownian dynamics simulations of protein folding: access to milliseconds time scale and beyond
,” Proc. Natl. Acad. Sci. U.S.A.
95
, 4288
–4292
(1998
).34.
M.
Zwier
, J.
Kaus
, and L.
Chong
, “Efficient explicit-solvent molecular dynamics simulations of molecular association kinetics: Methane/methane, Na+/Cl−, methane/benzene, and K+/18-crown-6 ether
,” J. Chem. Theory Comput.
7
, 1189
–1197
(2011
).35.
A.
Dickson
, A.
Warmflash
, and A. R.
Dinner
, “Separating forward and backward pathways in nonequilibrium umbrella sampling
,” J. Chem. Phys.
131
, 154104
(2009
).36.
F.
Zhu
and G.
Hummer
, “Pore opening and closing of a pentameric ligand-gated ion channel
,” Proc. Natl. Acad. Sci. U.S.A.
107
, 19814
–19819
(2010
).37.
S.
Lettieri
, M. C.
Zwier
, C. A.
Stringer
, E.
Suarez
, L. T.
Chong
, and D. M.
Zuckerman
, “Simultaneous computation of dynamical and equilibrium information using a weighted ensemble of trajectories
,” preprint arXiv:1210.3094 [physics.bio-ph] (2012
).38.
E.
Vanden-Eijnden
and M.
Venturoli
, “Exact rate calculations by trajectory parallelization and tilting
,” J. Chem. Phys.
131
, 044120
(2009
).39.
A.
Dickson
, M.
Maienschein-Cline
, A.
Tovo-Dwyer
, J.
Hammond
, and A.
Dinner
, “Flow-dependent unfolding and refolding of an RNA by nonequilibrium umbrella sampling
,” J. Chem. Theory Comput.
7
, 2710
–2720
(2011
).40.
See http://chong.chem.pitt.edu/WESTPA for the WESTPA software package.
41.
A.
Dickson
, A.
Warmflash
, and A. R.
Dinner
, “Erratum: ‘Nonequilibrium umbrella sampling in spaces of many order parameters’ [J. Chem. Phys. 130, 074104 (2009)]
,” J. Chem. Phys.
136
, 229901
(2012
).42.
A.
Dickson
, A.
Warmflash
, and A. R.
Dinner
, “Erratum: ‘Separating forward and backward pathways in nonequilibrium umbrella sampling’ [J. Chem. Phys. 131, 154104 (2009)]
,” J. Chem. Phys.
136
, 239901
(2012
).43.
A.
Damjanović
, B.
García-Moreno E
, and B. R.
Brooks
, “Self-guided Langevin dynamics study of regulatory interactions in NtrC
,” Proteins
76
, 1007
–1019
(2009
).44.
M.
Lei
, J.
Velos
, A.
Gardino
, A.
Kivenson
, M.
Karplus
, and D.
Kern
, “Segmented transition pathway of the signaling protein nitrogen regulatory protein C
,” J. Mol. Biol.
392
, 823
–836
(2009
).45.
J. L.
Adelman
, J. D.
Chodera
, I.-F. W.
Kuo
, T. F.
Miller
III, and D.
Barsky
, “The mechanical properties of PCNA: Implications for the loading and function of a DNA sliding clamp
,” Biophys. J.
98
, 3062
–3069
(2010
).46.
E.
Lyman
, J.
Pfaendtner
, and G. A.
Voth
, “Systematic multiscale parameterization of heterogeneous elastic network models of proteins
,” Biophys. J.
95
, 4183
–4192
(2008
).47.
L.
Yang
, G.
Song
, A.
Carriquiry
, and R. L.
Jernigan
, “Close correspondence between the motions from principal component analysis of multiple HIV-1 protease structures and elastic network modes
,” Structure
16
, 321
–330
(2008
).48.
D.
Kern
, B. F.
Volkman
, P.
Luginbühl
, M. J.
Nohaile
, S.
Kustu
, and D. E.
Wemmer
, “Structure of a transiently phosphorylated switch in bacterial signal transduction
,” Nature (London)
402
, 894
–898
(1999
).49.
D. L.
Theobald
, “Rapid calculation of RMSDs using a quaternion-based characteristic polynomial
,” Acta Crystallogr. A
61
, 478
–480
(2005
).50.
J.
Chodera
, W.
Swope
, J.
Pitera
, C.
Seok
, and K.
Dill
, “Use of the weighted histogram analysis method for the analysis of simulated and parallel tempering simulations
,” J. Chem. Theory Comput.
3
, 26
–41
(2007
).51.
M. R.
Shirts
and J. D.
Chodera
, “Statistically optimal analysis of samples from multiple equilibrium states
,” J. Chem. Phys.
129
, 124105
(2008
).52.
R.
Du
, V.
Pande
, A.
Grosberg
, T.
Tanaka
, and E.
Shakhnovich
, “On the transition coordinate for protein folding
,” J. Chem. Phys.
108
, 334
(1998
).53.
P.
Geissler
, C.
Dellago
, and D.
Chandler
, “Kinetic pathways of ion pair dissociation in water
,” J. Phys. Chem. B
103
, 3706
–3710
(1999
).54.
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
).55.
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
).56.
L.
Maragliano
, E.
Vanden-Eijnden
, and B.
Roux
, “Free energy and kinetics of conformational transitions from Voronoi tessellated milestoning with restraining potentials
,” J. Chem. Theory Comput.
5
, 2589
–2594
(2009
).57.
B.
Peters
, A.
Heyden
, A. T.
Bell
, and A.
Chakraborty
, “A growing string method for determining transition states: Comparison to the nudged elastic band and string methods
,” J. Chem. Phys.
120
, 7877
–7886
(2004
).© 2013 American Institute of Physics.
2013
American Institute of Physics
You do not currently have access to this content.