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Study of Ionic Currents across a Model Membrane Channel Using Brownian Dynamics
Study of Ionic Currents across a Model Membrane Channel Using
Brownian Dynamics
Shin-Ho Chung,* Matthew Hoyles,*
#
Toby Allen,* and Serdar Kuyucak
#
*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 ionic currents flowing across a model mem-
brane channel under various conditions. The model channel we use has a cylindrical transmembrane segment that is joined
to a catenary vestibule at each side. Two cylindrical reservoirs connected to the channel contain a fixed number of sodium
and chloride ions. Under a driving force of 100 mV, the channel is virtually impermeable to sodium ions, owing to the repulsive
dielectric force presented to ions by the vestibular wall. When two rings of dipoles, with their negative poles facing the pore
lumen, are placed just above and below the constricted channel segment, sodium ions cross the channel. The conductance
increases with increasing dipole strength and reaches its maximum rapidly; a further increase in dipole strength does not
increase the channel conductance further. When only those ions that acquire a kinetic energy large enough to surmount a
barrier are allowed to enter the narrow transmembrane segment, the channel conductance decreases monotonically with the
barrier height. This barrier represents those interactions between an ion, water molecules, and the protein wall in the
transmembrane segment that are not treated explicitly in the simulation. The conductance obtained from simulations closely
matches that obtained from ACh channels when a step potential barrier of 23 kT
r
is placed at the channel neck. The
current-voltage relationship obtained with symmetrical solutions is ohmic in the absence of a barrier. The current-voltage
curve becomes nonlinear when the 3 kT
r
barrier is in place. With asymmetrical solutions, the relationship approximates the
Goldman equation, with the reversal potential close to that predicted by the Nernst equation. The conductance first increases
linearly with concentration and then begins to rise at a slower rate with higher ionic concentration. We discuss the implications
of these findings for the transport of ions across the membrane and the structure of ion channels.
INTRODUCTION
Theoretical studies of the biological ion channel have been
hampered by a lack of detailed structural knowledge. The
exact shape of any biological channel, either ligand-gated or
voltage-activated, is unknown, as are the positions, densi-
ties, and types of dipoles and charge moieties on the protein
wall. These details are needed to compute the intermolecu-
lar potential operating between water molecules, ions, and
the protein wall, which is the essential ingredient for mi-
croscopic studies of channels using molecular dynamics.
Even if such information were available, it would not be
feasible at present to carry out molecular dynamics calcu-
lations for all of the water molecules and ions in a biological
channel and its surroundings for a period long enough to
deduce any of its macroscopically observable properties.
For these and other reasons, it has not been possible to
compare the results obtained from microscopic models with
the real data obtained from patch-clamp recordings. There is
a need to develop models that can relate the structural
parameters of channels to experimental measurements and
to build a theoretical framework that interlaces all of the
disparate sets of observations into a connected whole. The
theoretical model described in this paper was produced in
the hope of furthering this aim.
It is possible to make the computations tractable by
making several simplifications of and idealizations about
the channel and electrolyte solutions, and to examine the
magnitude of currents flowing through the model conduit
under various conditions, using Brownian dynamics simu-
lations (Cooper et al., 1985). Brownian dynamics provides
one of the simplest methods for following the trajectories of
idealized particles in a fluid interacting with a dielectric
boundary. Here the water is treated as a continuum, and the
motions of individual ions are assumed to be governed by
electrostatic forces emanating from other ions, fixed charges
in the proteins, the applied electric field and the dielectric
boundary. The effects of solvation and the structure of water
are taken into account by frictional and random forces
acting on ions. Brownian dynamics simulations have al-
ready been fruitfully utilized to investigate the movement of
ions across a model cylindrical tube (Jakobsson and Chiu,
1987; Chiu and Jakobsson, 1989; Bek and Jakobsson, 1994)
and a toroidal channel (Li et al., 1998). With these simula-
tions, it was possible to capture some of the salient features
of biological ion channels and reveal the importance of the
vestibules in influencing the permeability properties of ions
through the pore.
To deduce the conductance of a model channel with
Brownian dynamics simulations, the number of ions placed
in the reservoirs that mimic the extracellular and intracel-
Received for publication 5 December 1997 and in final form 5 May 1998.
Address reprint requests to Dr. S. H. Chung, Protein Dynamics Unit,
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
0006-3495/98/08/793/17
$2.00
793
Biophysical Journal
Volume 75
August 1998
793809 lular media must be relatively large, and the simulation
period needs to be sufficiently long that reliable statistics for
currents flowing across the channel can be collected. With
this requirement in mind, we have devised a method of
reducing the amount of computational effort involved in
simulating a system of charged particles interacting with a
protein wall. Instead of computing the force acting on an ion
at each position and at each time step, we precalculated the
values of the electric potential and field for a grid of
positions and stored them in a set of lookup tables. Using a
multidimensional interpolation algorithm (Press et al.,
1989), the field and potential experienced by an ion at any
position could be deduced from the information stored in
the lookup tables. Using this method, we were able to study
ionic currents flowing across a realistic model channel
under various conditions.
Brownian dynamics simulations do not adequately cap-
ture the transport process of ions in the narrow, constricted
channel region. This is because water is treated as contin-
uum, ions are idealized as point particles, and the protein
wall is represented as a structureless, rigid, and smooth
dielectric surface. In this narrow cylindrical region, polar
groups on the protein wall are likely to interact with the
primary or secondary hydration shell of an ion as it drifts
across the conduit, possibly replacing several water mole-
cules in the shell and forming temporary hydrogen bonds
with the ion. Elucidation of the permeation processes taking
place in the transmembrane segment will require molecular
dynamics calculations, such as the ones carried out for the
gramicidin pore (Roux and Karplus, 1991a). In the absence
of a detailed knowledge of the location and types of polar
groups lining the channel wall and the precise geometry of
the transmembrane segment, we have represented the inter-
molecular interactions taking place between ions, water
molecules, and the protein wall in this region as a step
potential barrier of variable height. The barrier is con-
structed such that it rises gently to the desired height in 1 Å,
and its first and second derivative are zero at the end points.
Thus the energy must be paid to enter the neck and is
returned when the ion exits.
Here we describe a model channel and report the results
of Brownian dynamics calculations aimed at elucidating the
permeation of ions through it. The shape of the channel is
made approximately the same as that of the ACh channel
(Toyoshima and Unwin, 1988), and cylindrical reservoirs
containing sodium and chloride ions are placed at each end
of the channel. We show that by placing an appropriate
strength of dipoles in the channel wall and erecting a small
energy barrier at the entrance of the narrow transmembrane
segment, we can replicate some of the macroscopically
observable properties of biological ion channels. Among
these are the channel conductance, an ohmic current-voltage
relationship, inward rectification as predicted by the
Goldman equatio