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Brownian Dynamics Study of Ion Transport in the Vestibule of Membrane Channels
Brownian Dynamics Study of Ion Transport in the Vestibule of
Membrane Channels
Siu Cheung Li,* Matthew Hoyles,*
#
Serdar Kuyucak,
#
and Shin-Ho Chung*
*Protein Dynamics Unit, Department of Chemistry, and
#
Department of Theoretical Physics, Research School of Physical Sciences,
Australian National University, Canberra, A.C.T. 0200, Australia
ABSTRACT
Brownian dynamics simulations have been carried out to study the transport of ions in a vestibular geometry,
which offers a more realistic shape for membrane channels than cylindrical tubes. Specifically, we consider a torus-shaped
channel, for which the analytical solution of Poissons equation is possible. The system is composed of the toroidal channel,
with length and radius of the constricted region of 80 Å and 4 Å, respectively, and two reservoirs containing 50 sodium ions
and 50 chloride ions. The positions of each of these ions executing Brownian motion under the influence of a stochastic force
and a systematic electric force are determined at discrete time steps of 50 fs for up to 2.5 ns. All of the systematic forces
acting on an ion due to the other ions, an external electric field, fixed charges in the channel protein, and the image charges
induced at the water-protein boundary are explicitly included in the calculations. We find that the repulsive dielectric force
arising from the induced surface charges plays a dominant role in channel dynamics. It expels an ion from the vestibule when
it is deliberately put in it. Even in the presence of an applied electric potential of 100 mV, an ion cannot overcome this repulsive
force and permeate the channel. Only when dipoles of a favorable orientation are placed along the sides of the transmem-
brane segment can an ion traverse the channel under the influence of a membrane potential. When the strength of the dipoles
is further increased, an ion becomes detained in a potential well, and the driving force provided by the applied field is not
sufficient to drive the ion out of the well. The trajectory of an ion navigating across the channel mostly remains close to the
central axis of the pore lumen. Finally, we discuss the implications of these findings for the transport of ions across the
membrane.
INTRODUCTION
Models of ion transport in membrane channels have been
dominated by two competing approaches based on macro-
scopic continuum and reaction-rate theories (Hille, 1992).
The first involves the solution of two coupled differential
equations, the Nernst-Planck equation for the diffusion of
ions and Poissons equation for the mean electric potential.
In the second approach, popularly known as Eyring rate
theory, the diffusion of ions is modeled as a hopping pro-
cess. That is, ions move by jumping from site to site, the
probability of a hop being determined by the energy differ-
ence between the sites and the available thermal energy.
Both approaches have their shortcomings when applied to
ion channels, and we refer to Cooper et al. (1985) for a
critical review. Cooper et al. (1985) made a case for use of
Brownian dynamics (BD) in describing ionic motion in
membrane channels. In BD, one simulates the motion of
ions by solving the Langevin equation. Collisions of an ion
with the surrounding water molecules are mimicked by a
random fluctuating force plus an average frictional force.
The systematic electric forces acting on ions due to various
sources can be determined, in principle, by solving Pois-
sons equation for the system of ions and the protein bound-
ary. Then by iterating the Langevin equation at small time
steps, it is possible to follow the trajectories of ions. Thus,
as long as one can neglect the dynamics of water molecules
in a channel, BD provides a direct solution for the many-
body problem of ions confined within an arbitrary boundary.
Despite the strong plea made by Cooper et al. (1985),
there had been only a few BD studies of ion channels,
mainly by Jakobsson and collaborators (Jakobsson and
Chiu, 1987; Chiu and Jakobsson, 1989; Bek and Jakobsson,
1994). In all of these studies, model channels were taken as
narrow cylindrical tubes that confined the ionic motion to
one dimension, and long time steps were used to save in
computation time. Nevertheless, in their BD simulations of
a potassium-selective channel, Bek and Jakobsson (1994)
were able to capture some of the salient features of this
channel. Among these are the magnitude of conductance,
the shape of the current-voltage relationship, and the satu-
ration of currents with an increasing ionic concentration.
Here we report the results of a study aimed at elucidating
the permeation of ions through a model ion channel. The
shape of the channel is approximated as a toroid, and
cylindrical reservoirs containing 25 chloride and 25 sodium
ions are placed at each end of the channel. The choice of a
toroidal boundary stems from the desire to carry out the BD
simulations in a realistic vestibular geometry, with all of the
electric forces acting on the ions properly included. A
numerical solution of Poissons equation for an arbitrary
boundary is possible, but in a BD simulation it would be
computationally prohibitive. Thus one requires a coordinate
system in which Poissons equation can be solved analyti-
Received for publication 15 April 1997 and in final form 21 October 1997.
Address reprint requests to Dr. Shin-Ho Chung, Department of Chemistry,
Australian National University, Canberra, A.C.T. 0200, Australia. Tel.:
61-2-6249-2024; Fax: 61-2-6247-2792; E-mail: shin-ho.chung@anu.
edu.au.
© 1998 by the Biophysical Society
0006-3495/98/01/37/11
$2.00
37
Biophysical Journal
Volume 74
January 1998
3747 cally. Toroidal coordinates are the only system satisfying
this dual requirement of vestibular geometry and analytical
solutions (Kuyucak et al., 1997). The importance of vesti-
bules in a channel is amply demonstrated in our simulations.
For example, the repulsive dielectric force arising from the
induced surface charges is strong enough to expel an ion
when it is deliberately placed inside the vestibule, and a
driving force provided by a membrane potential of 100 mV
is not sufficient to overcome this force. Only when dipoles
with a total moment of 100
10
30
Cm are placed at each
end of the transmembrane segment in a favorable direction
can ions drift across the channel under such a driving force.
If the strengths of dipoles are increased, the average drift
velocity of an ion decreases, rather than increases, owing to
the resulting potential well in which the ion becomes
trapped.
METHODS
Water-protein boundary and reservoirs
A toroidal channel was generated by rotating around the z
axis two identical circles with radii of 40 Å and centers
located on the x axis at 44 and
44 Å (see Fig. 1). Thus the
narrowest section of the channel formed in this way had a
radius of 4 Å, from which the vestibules extended to 40 Å.
On each side of the vestibule, a cylindrical reservoir with
radius 30 Å and length 90 Å was placed. Each reservoir
contained 25 sodium and 25 chloride ions, corresponding to
an ionic concentration of 150 mM. Simulations under var-
ious conditions, each lasting either 30,000 or 50,000 time
steps, were repeated many times (usually nine times). For
successive trials, the positions of the ions in the last time
step were used as the initial starting positions of the follow-
ing trial, except that a test particle was taken out and placed
at a desired position.
The ion-ion potential was computed from Coulombs
law. To prevent two ions from coming too close to each
other, a repulsive short-range potential, varying as 1/r
9
, was
included (Pauling, 1942). This steeply rising potential imi-
tates the repulsive force produced when the electron shells
of two ions begin to overlap. An external electric field
across the channel was created by applying a potential
difference across the two reservoirs such that the strength of
the field
would be 10
7
V/m in the absence of the dielectric
boundary.
In some simulations, a set of four dipoles was placed
inside the protein boundary at z
5 Å, and another set of
four dipoles was placed at z
5 Å. Their orientations
were perpendicular to the central axis of the lumen (z axis).
For each dipole, the negative pole, placed 2 Å inside the
water-protein boundary, was separated from the positive
pole by 5 Å. Thus, if 5/16 of an elementary charge is placed
on each pole, then the total moments of four such dipoles is
100
10
30
Coulomb-meter, or
30 Debyes (1 Debye
3.33
10
30
Coulomb-meter). (Hereafter, Coulomb-meter
will be abbreviated as Cm.) The same configuration of
dipoles was used in all of the simulations, giving rise to an
attractive potential for a cation and a repulsive force for an
anion in the channel. We changed the total moment in the
ring of four dipoles by varying the electric charges on the
poles, and quote only their total strength below.
All of the above electric fields are significantly modified
in the presence of the dielectric boundary. Especially when
an ion is in the vestibule, induced surface charges can alter
the local fields around the ion drastically, leading to an
insurmountable potential barrier. The effect of the induced
surface charges on the motion of ions is properly taken into
account by solving Poissons equation in toroidal coordi-
nates, as described by Kuyucak et al. (1997).
The steep repulsive force at the dielectric boundary du