Weighted ensemble (WE) is an enhanced sampling method based on periodically replicating and pruning trajectories generated in parallel. WE has grown increasingly popular for computational biochemistry problems due, in part, to improved hardware and accessible software implementations. Algorithmic and analytical improvements have played an important role, and progress has accelerated in recent years. Here, we discuss and elaborate on the WE method from a mathematical perspective, highlighting recent results that enhance the computational efficiency. The mathematical theory reveals a new strategy for optimizing trajectory management that approaches the best possible variance while generalizing to systems of arbitrary dimension.
REFERENCES
1.
G. A.
Huber
and S.
Kim
, “Weighted-ensemble Brownian dynamics simulations for protein association reactions
,” Biophys. J.
70
, 97
–110
(1996
).2.
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
).3.
R. M.
Donovan
, A. J.
Sedgewick
, J. R.
Faeder
, and D. M.
Zuckerman
, “Efficient stochastic simulation of chemical kinetics networks using a weighted ensemble of trajectories
,” J. Chem. Phys.
139
, 115105
(2013
).4.
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
).5.
M. C.
Zwier
, J. W.
Kaus
, and L. T.
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
).6.
E.
Suárez
, S.
Lettieri
, M. C.
Zwier
, C. A.
Stringer
, S. R.
Subramanian
, L. T.
Chong
, and D. M.
Zuckerman
, “Simultaneous computation of dynamical and equilibrium information using a weighted ensemble of trajectories
,” J. Chem. Theory Comput.
10
, 2658
–2667
(2014
).7.
S. D.
Lotz
and A.
Dickson
, “Unbiased molecular dynamics of 11 min timescale drug unbinding reveals transition state stabilizing interactions
,” J. Am. Chem. Soc.
140
, 618
–628
(2018
).8.
T.
Sztain
, S.-H.
Ahn
, A. T.
Bogetti
, L.
Casalino
, J. A.
Goldsmith
, E.
Seitz
, R. S.
McCool
, F. L.
Kearns
, F.
Acosta-Reyes
, S.
Maji
et al, “A glycan gate controls opening of the SARS-CoV-2 spike protein
,” Nat. Chem.
13
, 963
–968
(2021
).9.
H.
Kahn
and T. E.
Harris
, Estimation of Particle Transmission by Random Sampling
(National Bureau of Standards Applied Mathematics Series
, 1951
), Vol. 12, pp. 27
–30
.10.
D.
Aristoff
, “An ergodic theorem for the weighted ensemble method
,” J. Appl. Probab.
59
, 152
–166
(2022
).11.
R. J.
Webber
, D.
Aristoff
, and G.
Simpson
, “A splitting method to reduce MCMC variance
,” arXiv:2011.13899 (2020
).12.
M. C.
Zwier
, J. L.
Adelman
, J. W.
Kaus
, A. J.
Pratt
, K. F.
Wong
, N. B.
Rego
, E.
Suárez
, S.
Lettieri
, D. W.
Wang
, M.
Grabe
, D. M.
Zuckerman
, and L. T.
Chong
, “WESTPA: An interoperable, highly scalable software package for weighted ensemble simulation and analysis
,” J. Chem. Theory Comput.
11
, 800
–809
(2015
).13.
J. D.
Russo
, S.
Zhang
, J. M.
Leung
, A. T.
Bogetti
, J. P.
Thompson
, A. J.
DeGrave
, P. A.
Torrillo
, A.
Pratt
, K. F.
Wong
, J.
Xia
et al, “WESTPA 2.0: High-performance upgrades for weighted ensemble simulations and analysis of longer-timescale applications
,” J. Chem. Theory Comput.
18
, 638
(2021
).14.
S. D.
Lotz
and A.
Dickson
, “Wepy: A flexible software framework for simulating rare events with weighted ensemble resampling
,” ACS Omega
5
, 31608
–31623
(2020
).15.
D.
Aristoff
, F. G.
Jones
, R. J.
Webber
, G.
Simpson
, and D. M.
Zuckerman
, Weightedensemble.jl, Julia package, 2020
.16.
A.
Dickson
and C. L.
Brooks
III, “WExplore: Hierarchical exploration of high-dimensional spaces using the weighted ensemble algorithm
,” J. Phys. Chem. B
118
, 3532
–3542
(2014
).17.
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
).18.
B.
Abdul-Wahid
, H.
Feng
, D.
Rajan
, R.
Costaouec
, E.
Darve
, D.
Thain
, and J. A.
Izaguirre
, “AWE-WQ: Fast-forwarding molecular dynamics using the accelerated weighted ensemble
,” J. Chem. Inf. Model.
54
, 3033
–3043
(2014
).19.
S.-H.
Ahn
, A. A.
Ojha
, R. E.
Amaro
, and J. A.
McCammon
, “Gaussian-Accelerated molecular dynamics with the weighted ensemble method: A hybrid method improves thermodynamic and kinetic sampling
,” J. Chem. Theory Comput.
17
, 7938
–7951
(2021
).20.
D.
Ray
and I.
Andricioaei
, “Weighted ensemble milestoning (WEM): A combined approach for rare event simulations
,” J. Chem. Phys.
152
, 234114
(2020
).21.
D.
Ray
, S. E.
Stone
, and I.
Andricioaei
, “Markovian weighted ensemble milestoning (M-WEM): Long-time kinetics from short trajectories
,” J. Chem. Theory Comput.
18
, 79
(2021
).22.
A.
Ojha
, S.
Thakur
, S.-H.
Ahn
, and R.
Amaro
, “DeepWEST: Deep learning of kinetic models with the weighted ensemble simulation toolkit for enhanced kinetic and thermodynamic sampling
,” ChemRxiv (2022
), https://chemrxiv.org/engage/chemrxiv/article-details/6238beac13d478049d96d707.23.
M. J.
Tse
, B. K.
Chu
, C. P.
Gallivan
, and E. L.
Read
, “Rare-event sampling of epigenetic landscapes and phenotype transitions
,” PLoS Comput. Biol.
14
, e1006336
(2018
).24.
R.
Taylor
, J. F.
Allard
, and E.
Read
, “Weighted ensemble simulation of kinetic segregation shows role of oligomers and close contacts
,” Biophys. J.
121
, 149a
–150a
(2022
).25.
D. M.
Zuckerman
and L. T.
Chong
, “Weighted ensemble simulation: Review of methodology, applications, and software
,” Annu. Rev. Biophys.
46
, 43
–57
(2017
).26.
D.
Bhatt
and D. M.
Zuckerman
, “Beyond microscopic reversibility: Are observable nonequilibrium processes precisely reversible?
” J. Chem. Theory Comput.
7
, 2520
–2527
(2011
).27.
A. S.
Saglam
and L. T.
Chong
, “Protein–protein binding pathways and calculations of rate constants using fully-continuous, explicit-solvent simulations
,” Chem. Sci.
10
, 2360
–2372
(2019
).28.
D.
Bhatt
, B. W.
Zhang
, and D. M.
Zuckerman
, “Steady-state simulations using weighted ensemble path sampling
,” J. Chem. Phys.
133
, 014110
(2010
).29.
W. C.
Swope
, J. W.
Pitera
, and F.
Suits
, “Describing protein folding kinetics by molecular dynamics simulations. 1. Theory
,” J. Phys. Chem. B
108
, 6571
–6581
(2004
).30.
R. J.
Allen
, P. B.
Warren
, and P. R.
Ten Wolde
, “Sampling rare switching events in biochemical networks
,” Phys. Rev. Lett.
94
, 018104
(2005
).31.
F.
Cérou
and A.
Guyader
, “Adaptive multilevel splitting for rare event analysis
,” Stochastic Anal. Appl.
25
, 417
–443
(2007
).32.
F.
Cérou
, A.
Guyader
, T.
Lelièvre
, and D.
Pommier
, “A multiple replica approach to simulate reactive trajectories
,” J. Chem. Phys.
134
, 054108
(2011
).33.
C.-E.
Bréhier
, M.
Gazeau
, L.
Goudenege
, T.
Lelièvre
, and M.
Rousset
, “Unbiasedness of some generalized adaptive multilevel splitting algorithms
,” Ann. Appl. Prob.
26
, 3559
–3601
(2016
).34.
I.
Teo
, C. G.
Mayne
, K.
Schulten
, and T.
Lelièvre
, “Adaptive multilevel splitting method for molecular dynamics calculation of benzamidine-trypsin dissociation time
,” J. Chem. Theory Comput.
12
, 2983
–2989
(2016
).35.
C.
Giardina
, J.
Kurchan
, V.
Lecomte
, and J.
Tailleur
, “Simulating rare events in dynamical processes
,” J. Stat. Phys.
145
, 787
–811
(2011
).36.
E.
Guevara Hidalgo
, T.
Nemoto
, and V.
Lecomte
, “Finite-time and finite-size scalings in the evaluation of large-deviation functions: Numerical approach in continuous time
,” Phys. Rev. E
95
, 062134
(2017
).37.
U.
Ray
, G. K.-L.
Chan
, and D. T.
Limmer
, “Importance sampling large deviations in nonequilibrium steady states. I
,” J. Chem. Phys.
148
, 124120
(2018
).38.
J. M.
Bello-Rivas
and R.
Elber
, “Exact milestoning
,” J. Chem. Phys.
142
, 094102
(2015
).39.
D.
Aristoff
, J. M.
Bello-Rivas
, and R.
Elber
, “A mathematical framework for exact milestoning
,” Multiscale Model. Simul.
14
, 301
–322
(2016
).40.
A.
Warmflash
, P.
Bhimalapuram
, and A. R.
Dinner
, “Umbrella sampling for nonequilibrium processes
,” J. Chem. Phys.
127
, 154112
(2007
).41.
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
).42.
D. S.
Abbot
, R. J.
Webber
, S.
Hadden
, D.
Seligman
, and J.
Weare
, “Rare event sampling improves mercury instability statistics
,” Astrophys. J.
923
, 236
(2021
).43.
J.
Finkel
, R. J.
Webber
, E. P.
Gerber
, D. S.
Abbot
, and J.
Weare
, “Learning forecasts of rare stratospheric transitions from short simulations
,” Mon. Weather Rev.
149
, 3647
–3669
(2021
).44.
J.
Finkel
, R. J.
Webber
, E. P.
Gerber
, D. S.
Abbot
, and J.
Weare
, “Exploring stratospheric rare events with transition path theory and short simulations
,” arXiv:2108.12727 (2021
).45.
R. M.
Donovan
, J.-J.
Tapia
, D. P.
Sullivan
, J. R.
Faeder
, R. F.
Murphy
, M.
Dittrich
, and D. M.
Zuckerman
, “Unbiased rare event sampling in spatial stochastic systems biology models using a weighted ensemble of trajectories
,” PLoS Comput. Biol.
12
, e1004611
(2016
).46.
E.
Darve
and E.
Ryu
, “Computing reaction rates in bio-molecular systems using discrete macro-states
,” in Innovations in Biomolecular Modeling and Simulations
(Royal Society of Chemistry
, 2012
), pp. 138
–206
.47.
H.
Feng
, R.
Costaouec
, E.
Darve
, and J. A.
Izaguirre
, “A comparison of weighted ensemble and Markov state model methodologies
,” J. Chem. Phys.
142
, 214113
(2015
).48.
J. L.
Adelman
and M.
Grabe
, “Simulating rare events using a weighted ensemble-based string method
,” J. Chem. Phys.
138
, 044105
(2013
).49.
D.
Aristoff
, “Analysis and optimization of weighted ensemble sampling
,” ESAIM: Math. Modell. Numer. Anal.
52
, 1219
–1238
(2018
).50.
D.
Aristoff
and D. M.
Zuckerman
, “Optimizing weighted ensemble sampling of steady states
,” Multiscale Model. Simul.
18
, 646
–673
(2020
).51.
M.
Baudel
, A.
Guyader
, and T.
Lelièvre
, “On the Hill relation and the mean reaction time for metastable processes
,” arXiv:2008.09790 (2020
).52.
T.
Lelièvre
, M.
Ramil
, and J.
Reygner
, “Estimation of statistics of transitions and hill relation for Langevin dynamics
,” arXiv:2206.13264 (2022
).53.
T. L.
Hill
, Free Energy Transduction and Biochemical Cycle Kinetics
(Courier Corporation
, 2005
).54.
I.
Ben-Ari
and R. G.
Pinsky
, “Spectral analysis of a family of second-order elliptic operators with nonlocal boundary condition indexed by a probability measure
,” J. Funct. Anal.
251
, 122
–140
(2007
).55.
A. J.
DeGrave
, A. T.
Bogetti
, and L. T.
Chong
, “The red scheme: Rate-constant estimation from pre-steady state weighted ensemble simulations
,” J. Chem. Phys.
154
, 114111
(2021
).56.
U.
Adhikari
, B.
Mostofian
, J.
Copperman
, S. R.
Subramanian
, A. A.
Petersen
, and D. M.
Zuckerman
, “Computational estimation of microsecond to second atomistic folding times
,” J. Am. Chem. Soc.
141
, 6519
–6526
(2019
).57.
J.
Copperman
and D. M.
Zuckerman
, “Accelerated estimation of long-timescale kinetics from weighted ensemble simulation via non-Markovian “microbin” analysis
,” J. Chem. Theory Comput.
16
, 6763
–6775
(2020
).58.
J.
Lu
and J.
Nolen
, “Reactive trajectories and the transition path process
,” Prob. Theory Relat. Fields
161
, 195
–244
(2015
).59.
C. W.
Gardiner
et al, Handbook of Stochastic Methods
, 2nd ed. (Springer
, Berlin
, 1985
).60.
P.
Ma
, R.
Elber
, and D. E.
Makarov
, “The value of temporal information when analyzing reaction coordinates
,” J. Chem. Theory Comput.
16
, 6077
–6090
(2020
).61.
A. J.
DeGrave
, J.-H.
Ha
, S. N.
Loh
, and L. T.
Chong
, “Large enhancement of response times of a protein conformational switch by computational design
,” Nat. Commun.
9
, 1013
–1019
(2018
).62.
W.
Feller
, An Introduction to Probability Theory and its Applications
(John Wiley and Sons
, 2008
), Vol. 2.63.
C. M.
Bender
and S. A.
Orszag
, Advanced Mathematical Methods for Scientists and Engineers I
(Springer
, New York, NY
, 1999
).© 2023 Author(s). Published under an exclusive license by AIP Publishing.
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