Ion channels are proteins that play a significant role in physiological processes, including neuronal excitability and signal transduction. However, the precise mechanisms by which these proteins facilitate ion diffusion through cell membranes are not well understood. This is because experimental techniques to characterize ion channel activity operate on a time scale too large to understand the role of the various protein conformations on diffusion. Meanwhile, computational approaches operate on a time scale too short to rationalize the observed behavior at the microscopic scale. In this paper, we present a continuous-time random walk model that aims to bridge the scales between the atomistic models of ion channels and the experimental measurement of their conductance. We show how diffusion slows down in complex systems by using 3D lattices that map out the pore geometry of two channels: Nav1.7 and gramicidin. We also introduce spatial and dynamic site disorder to account for system heterogeneity beyond the mean field approximation. Computed diffusion coefficients show that an increase in spatial disorder slows down diffusion kinetics, while dynamic disorder has the opposite effect. Our results imply that microscopic or phenomenological models based on the potential of mean force data overlook the functional importance of protein dynamics on ion diffusion through channels.
REFERENCES
1.
D. W.
Deamer
and J.
Bramhall
, “Permeability of lipid bilayers to water and ionic solutes
,” Chem. Phys. Lipids
40
, 167
–188
(1986
).2.
I.
Bernhardt
, J. C.
Ellory
, J. S.
Gibson
, and R. J.
Wilkins
, “Chapter 1—Ion permeability of membranes: From lipid bilayers to biological membranes
,” in Advances in Planar Lipid Bilayers and Liposomes
(Academic Press
, 2009
), Vol. 10
, pp. 1
–27
.3.
D. C.
Gadsby
, “Ion channels versus ion pumps: The principal difference, in principle
,” Nat. Rev. Mol. Cell Biol.
10
, 344
–352
(2009
).4.
A. B.
Mamonov
, M. G.
Kurnikova
, and R. D.
Coalson
, “Diffusion constant of K+ inside gramicidin A: A comparative study of four computational methods
,” Biophys. Chem.
124
, 268
–278
(2006
).5.
C.
Lau
, M. J.
Hunter
, A.
Stewart
, E.
Perozo
, and J. I.
Vandenberg
, “Never at rest: Insights into the conformational dynamics of ion channels from cryo-electron microscopy
,” J. Physiol.
596
, 1107
–1119
(2018
).6.
W. A.
Catterall
, “From ionic currents to molecular mechanisms: The structure and function of voltage-gated sodium channels
,” Neuron
26
, 13
–25
(2000
).7.
Y.
Muroi
, F.
Ru
, M.
Kollarik
, B. J.
Canning
, S. A.
Hughes
, S.
Walsh
, M.
Sigg
, M. J.
Carr
, and B. J.
Undem
, “Selective silencing of NaV1.7 decreases excitability and conduction in vagal sensory neurons
,” J. Physiol.
589
, 5663
–5676
(2011
).8.
S. D.
Dib-Hajj
, Y.
Yang
, J. A.
Black
, and S. G.
Waxman
, “The NaV1.7 sodium channel: From molecule to man
,” Nat. Rev. Neurosci.
14
, 49
–62
(2013
).9.
J.
Payandeh
, T.
Scheuer
, N.
Zheng
, and W. A.
Catterall
, “The crystal structure of a voltage-gated sodium channel
,” Nature
475
, 353
–358
(2011
).10.
E.
Flood
, C.
Boiteux
, B.
Lev
, I.
Vorobyov
, and T. W.
Allen
, “Atomistic simulations of membrane ion channel conduction, gating, and modulation
,” Chem. Rev.
119
, 7737
–7832
(2019
).11.
W. A.
Catterall
, A. L.
Goldin
, and S. G.
Waxman
, “International Union of Pharmacology. XLVII. Nomenclature and structure-function relationships of voltage-gated sodium channels
,” Pharmacol. Rev.
57
, 397
–409
(2005
).12.
L.
Djouhri
, R.
Newton
, S. R.
Levinson
, C. M.
Berry
, B.
Carruthers
, and S. N.
Lawson
, “Sensory and electrophysiological properties of guinea-pig sensory neurones expressing Nav1.7 (PN1) Na+ channel α subunit protein
,” J. Physiol.
546
, 565
–576
(2003
).13.
R.-R.
Ji
and C. J.
Woolf
, “Neuronal plasticity and signal transduction in nociceptive neurons: Implications for the initiation and maintenance of pathological pain
,” Neurobiol. Dis.
8
, 1
–10
(2001
).14.
W. E.
Huckabee
, “Control of concentration gradients of pyruvate and lactate across cell membranes in blood
,” J. Appl. Physiol.
9
, 163
–170
(1956
).15.
M.
Cahalan
and E.
Neher
, “Patch clamp techniques: An overview
,” Methods Enzymol.
207
, 3
–14
(1992
).16.
J. R.
Wilson
, R. B.
Clark
, U.
Banderali
, and W. R.
Giles
, “Measurement of the membrane potential in small cells using patch clamp methods
,” Channels
5
, 530
–537
(2011
).17.
P. J.
White
, B.
Biskup
, J. T. M.
Elzenga
, U.
Homann
, G.
Thiel
, F.
Wissing
, and F. J.
Maathuis
, “Advanced patch-clamp techniques and single-channel analysis
,” J. Exp. Bot.
50
, 1037
–1054
(1999
).18.
F.
Qin
, A.
Auerbach
, and F.
Sachs
, “Estimating single-channel kinetic parameters from idealized patch-clamp data containing missed events
,” Biophys. J.
70
, 264
–280
(1996
).19.
A. B.
Mamonov
, R. D.
Coalson
, A.
Nitzan
, and M. G.
Kurnikova
, “The role of the dielectric barrier in narrow biological channels: A novel composite approach to modeling single-channel currents
,” Biophys. J.
84
, 3646
–3661
(2003
).20.
R.
Lynden-Bell
and J. C.
Rasaiah
, “Mobility and solvation of ions in channels
,” J. Chem. Phys.
105
, 9266
–9280
(1996
).21.
B.
Roux
and M.
Karplus
, “Ion transport in a gramicidin-like channel: Dynamics and mobility
,” J. Phys. Chem.
95
, 4856
–4868
(1991
).22.
S.-H.
Chung
, T. W.
Allen
, M.
Hoyles
, and S.
Kuyucak
, “Permeation of ions across the potassium channel: Brownian dynamics studies
,” Biophys. J.
77
, 2517
–2533
(1999
).23.
C.
Berti
, S.
Furini
, D.
Gillespie
, D.
Boda
, R. S.
Eisenberg
, E.
Sangiorgi
, and C.
Fiegna
, “Three-dimensional Brownian dynamics simulator for the study of ion permeation through membrane pores
,” J. Chem. Theory Comput.
10
, 2911
–2926
(2014
).24.
S.-W.
Chiu
, J. A.
Novotny
, and E.
Jakobsson
, “The nature of ion and water barrier crossings in a simulated ion channel
,” Biophys. J.
64
, 98
–109
(1993
).25.
T.
Schirmer
and P. S.
Phale
, “Brownian dynamics simulation of ion flow through porin channels
,” J. Mol. Biol.
294
, 1159
–1167
(1999
).26.
D.
Gordon
, V.
Krishnamurthy
, and S.-H.
Chung
, “A generalized Langevin algorithm for studying permeation across biological ion channels
,” Mol. Phys.
106
, 1353
–1361
(2008
).27.
S.
Berneche
and B.
Roux
, “A microscopic view of ion conduction through the K+ channel
,” Proc. Natl. Acad. Sci. U. S. A.
100
, 8644
–8648
(2003
).28.
S. Y.
Noskov
, W.
Im
, and B.
Roux
, “Ion permeation through the α-hemolysin channel: Theoretical studies based on Brownian dynamics and Poisson-Nernst-Plank electrodiffusion theory
,” Biophys. J.
87
, 2299
–2309
(2004
).29.
W.
Im
and B.
Roux
, “Brownian dynamics simulations of ions channels: A general treatment of electrostatic reaction fields for molecular pores of arbitrary geometry
,” J. Chem. Phys.
115
, 4850
–4861
(2001
).30.
V.
Krishnamurthy
and S.-H.
Chung
, “Adaptive Brownian dynamics simulation for estimating potential mean force in ion channel permeation
,” IEEE Trans. NanoBiosci.
5
, 126
–138
(2006
).31.
K. I.
Lee
, H.
Rui
, R. W.
Pastor
, and W.
Im
, “Brownian dynamics simulations of ion transport through the VDAC
,” Biophys. J.
100
, 611
–619
(2011
).32.
W.
Im
and B.
Roux
, “Ion permeation and selectivity of OmpF porin: A theoretical study based on molecular dynamics, Brownian dynamics, and continuum electrodiffusion theory
,” J. Mol. Biol.
322
, 851
–869
(2002
).33.
W.
Im
, S.
Seefeld
, and B.
Roux
, “A grand canonical Monte Carlo–Brownian dynamics algorithm for simulating ion channels
,” Biophys. J.
79
, 788
–801
(2000
).34.
M. E. A.
Hodgson
and P. J.
Green
, “Bayesian choice among Markov models of ion channels using Markov chain Monte Carlo
,” Proc. R. Soc. London, Ser. A
455
, 3425
–3448
(1999
).35.
G.
Moy
, B.
Corry
, S.
Kuyucak
, and S.-H.
Chung
, “Tests of continuum theories as models of ion channels. I. Poisson–Boltzmann theory versus Brownian dynamics
,” Biophys. J.
78
, 2349
–2363
(2000
).36.
B.
Corry
, S.
Kuyucak
, and S.-H.
Chung
, “Tests of continuum theories as models of ion channels. II. Poisson–Nernst–Planck theory versus Brownian dynamics
,” Biophys. J.
78
, 2364
–2381
(2000
).37.
B.
Corry
, S.
Kuyucak
, and S.-H.
Chung
, “Dielectric self-energy in Poisson-Boltzmann and Poisson-Nernst-Planck models of ion channels
,” Biophys. J.
84
, 3594
–3606
(2003
).38.
B.
Eisenberg
, “Multiple scales in the simulation of ion channels and proteins
,” J. Phys. Chem. C
114
, 20719
–20733
(2010
).39.
C.
Maffeo
, S.
Bhattacharya
, J.
Yoo
, D.
Wells
, and A.
Aksimentiev
, “Modeling and simulation of ion channels
,” Chem. Rev.
112
, 6250
–6284
(2012
).40.
B.
Eisenberg
, Y.
Hyon
, and C.
Liu
, “Energy variational analysis of ions in water and channels: Field theory for primitive models of complex ionic fluids
,” J. Chem. Phys.
133
, 104104
(2010
).41.
A.
Aksimentiev
and K.
Schulten
, “Imaging α-hemolysin with molecular dynamics: Ionic conductance, osmotic permeability, and the electrostatic potential map
,” Biophys. J.
88
, 3745
–3761
(2005
).42.
E. H.
Lee
, J.
Hsin
, M.
Sotomayor
, G.
Comellas
, and K.
Schulten
, “Discovery through the computational microscope
,” Structure
17
, 1295
–1306
(2009
).43.
J.
Li
, R.
Shen
, Y.
Hoon Koh
, E.
Perozo
, and B.
Roux
, “Molecular determinants of C-type inactivation for the hERG channel and its disease-associated mutants
,” Biophys. J.
118
, 111a
(2020
).44.
D. A.
Köpfer
, C.
Song
, T.
Gruene
, G. M.
Sheldrick
, U.
Zachariae
, and B. L.
de Groot
, “Ion permeation in K+ channels occurs by direct Coulomb knock-on
,” Science
346
, 352
–355
(2014
).45.
H. W.
De Haan
, I. S.
Tolokh
, C.
Gray
, and S.
Goldman
, “Nonequilibrium molecular dynamics calculation of the conductance of the KcsA potassium ion channel
,” Phys. Rev. E
74
, 030905
(2006
).46.
M.
Bachmann
, Thermodynamics and Statistical Mechanics of Macromolecular Systems
(Cambridge University Press
, 2014
).47.
L.
Delemotte
, “Bridging the gap between atomistic molecular dynamics simulations and wet-lab experimental techniques: Applications to membrane proteins
,” in Computational Techniques for Analytical Chemistry and Bioanalysis
(Royal Chemical Society
, 2020
), pp. 247
–286
.48.
I.
Bertini
, A.
Giachetti
, C.
Luchinat
, G.
Parigi
, M. V.
Petoukhov
, R.
Pierattelli
, E.
Ravera
, and D. I.
Svergun
, “Conformational space of flexible biological macromolecules from average data
,” J. Am. Chem. Soc.
132
, 13553
–13558
(2010
).49.
S. H. W.
Scheres
, H.
Gao
, M.
Valle
, G. T.
Herman
, P. P. B.
Eggermont
, J.
Frank
, and J.-M.
Carazo
, “Disentangling conformational states of macromolecules in 3D-EM through likelihood optimization
,” in Single-Particle Cryo-Electron Microscopy: The Path Toward Atomic Resolution: Selected Papers of Joachim Frank with Commentaries
(World Scientific
, 2018
), pp. 413
–415
.50.
A. J.
Pak
and G. A.
Voth
, “Advances in coarse-grained modeling of macromolecular complexes
,” Curr. Opin. Struct. Biol.
52
, 119
–126
(2018
).51.
S.
Jonić
, “Computational methods for analyzing conformational variability of macromolecular complexes from cryo-electron microscopy images
,” Curr. Opin. Struct. Biol.
43
, 114
–121
(2017
).52.
K.
Wu
, S.
Xu
, B.
Wan
, P.
Xiu
, and X.
Zhou
, “A novel multiscale scheme to accelerate atomistic simulations of bio-macromolecules by adaptively driving coarse-grained coordinates
,” J. Chem. Phys.
152
, 114115
(2020
).53.
R. E.
Amaro
and A. J.
Mulholland
, “Multiscale methods in drug design bridge chemical and biological complexity in the search for cures
,” Nat. Rev. Chem.
2
, 0148
(2018
).54.
Q.
Shao
, “Enhanced conformational sampling technique provides an energy landscape view of large-scale protein conformational transitions
,” Phys. Chem. Chem. Phys.
18
, 29170
–29182
(2016
).55.
F.
Noé
and S.
Fischer
, “Transition networks for modeling the kinetics of conformational change in macromolecules
,” Curr. Opin. Struct. Biol.
18
, 154
–162
(2008
).56.
C.
Guardiani
, F.
Cecconi
, L.
Chiodo
, G.
Cottone
, P.
Malgaretti
, L.
Maragliano
, M. L.
Barabash
, G.
Camisasca
, M.
Ceccarelli
, B.
Corry
, R.
Roth
, A.
Giacomello
, and B.
Roux
, “Computational methods and theory for ion channel research
,” Adv. Phys.: X
7
, 2080587
(2022
).57.
R.
Horn
, B.
Roux
, and J.
Åqvist
, “Permeation redux: Thermodynamics and kinetics of ion movement through potassium channels
,” Biophys. J.
106
, 1859
–1863
(2014
).58.
B.
Roux
, “Ion channels and ion selectivity
,” Essays Biochem.
61
, 201
–209
(2017
).59.
A. E.
Cardenas
, R. D.
Coalson
, and M. G.
Kurnikova
, “Three-dimensional Poisson-Nernst-Planck theory studies: Influence of membrane electrostatics on gramicidin A channel conductance
,” Biophys. J.
79
, 80
–93
(2000
).60.
P.
Graf
, A.
Nitzan
, M. G.
Kurnikova
, and R. D.
Coalson
, “A dynamic lattice Monte Carlo model of ion transport in inhomogeneous dielectric environments: Method and implementation
,” J. Phys. Chem. B
104
, 12324
–12338
(2000
).61.
M. G.
Kurnikova
, R. D.
Coalson
, P.
Graf
, and A.
Nitzan
, “A lattice relaxation algorithm for three-dimensional Poisson-Nernst-Planck theory with application to ion transport through the gramicidin A channel
,” Biophys. J.
76
, 642
–656
(1999
).62.
J.
Klafter
and R.
Silbey
, “Derivation of the continuous-time random-walk equation
,” Phys. Rev. Lett.
44
, 55
(1980
).63.
D. T.
Gillespie
, “A general method for numerically simulating the stochastic time evolution of coupled chemical reactions
,” J. Comput. Phys.
22
, 403
–434
(1976
).64.
B.
Wallace
and K.
Ravikumar
, “The gramicidin pore: Crystal structure of a cesium complex
,” Science
241
, 182
–187
(1988
).65.
A.
Finkelstein
and O. S.
Andersen
, “The gramicidin A channel: A review of its permeability characteristics with special reference to the single-file aspect of transport
,” J. Membr. Biol.
59
, 155
–171
(1981
).66.
H.
Shen
, D.
Liu
, K.
Wu
, J.
Lei
, and N.
Yan
, “Structures of human Nav1.7 channel in complex with auxiliary subunits and animal toxins
,” Science
363
, 1303
–1308
(2019
).67.
W.
Park
and G. S.
Chirikjian
, “Interconversion between truncated Cartesian and polar expansions of images
,” IEEE Trans. Image Process.
16
, 1946
–1955
(2007
).68.
R. D.
Shannon
, “Revised effective ionic radii and systematic studies of interatomic distances in halides and chalcogenides
,” Acta Crystallogr., Sect. A: Found. Adv.
32
, 751
–767
(1976
).69.
A. F.
Voter
, “Introduction to the kinetic Monte Carlo method
,” in Radiation Effects in Solids
(Springer
, 2007
), pp. 1
–23
.70.
C. R.
Harris
, K. J.
Millman
, S. J.
van der Walt
, R.
Gommers
, P.
Virtanen
, D.
Cournapeau
, E.
Wieser
, J.
Taylor
, S.
Berg
, N. J.
Smith
, R.
Kern
, M.
Picus
, S.
Hoyer
, M. H.
van Kerkwijk
, M.
Brett
, A.
Haldane
, J. F.
del Río
, M.
Wiebe
, P.
Peterson
, P.
Gérard-Marchant
, K.
Sheppard
, T.
Reddy
, W.
Weckesser
, H.
Abbasi
, C.
Gohlke
, and T. E.
Oliphant
, “Array programming with NumPy
,” Nature
585
, 357
–362
(2020
).71.
72.
M.
Andersen
, C.
Panosetti
, and K.
Reuter
, “A practical guide to surface kinetic Monte Carlo simulations
,” Front. Chem.
7
, 202
(2019
).73.
M.
Suenson
, D.
Richmond
, and J.
Bassingthwaighte
, “Diffusion of sucrose, sodium, and water in ventricular myocardium
,” Am. J. Physiol.
227
, 1116
–1123
(1974
).© 2024 Author(s). Published under an exclusive license by AIP Publishing.
2024
Author(s)
You do not currently have access to this content.
