From the Department of Physiology, University of Minnesota, Minneapolis, Minnesota 55455
Modeling of biological ion channels has a long history,
going back more then 100 yr (Hille 1984 Exact Molecular Dynamic Solution
This approach is made possible by the availability of
channel structures that are known to atomic resolution. Until recently, the only channel that met this condition was the relatively simple gramicidin channel.
However, with structures now available for the porin
channel (Kreusch and Schulz, 1994 Unfortunately (or, fortunately, if you want to justify
the use of other models), this direct approach is still beyond our current computational limits. The simulation
must be run for a time period that is at least long
enough to observe an ion crossing the channel. In fact,
since the ion can take a large variety of paths through
the channel and since the specific interactions with the
channel and the other ions will vary randomly during the crossing, one should sample a minimum of 10 such
crossings to estimate the channel conductance. For a
channel with a conductance of 50 pS and an applied
voltage of 100 mV, 10 crossings would take about 0.4 µs. For comparison, a recent simulation of the porin
trimer channel OmpF (consisting of 1,020 amino acid
residues, 300 phosphatidyl choline molecules, 12,992 water molecules, and 27 sodium atoms) required 2 h of
computer time for 1 ps of real time (Tieleman and Berendsen, 1998 These numerical calculations also have some other
limitations, beyond that of the computer time. The
most serious problem is a fundamental limitation in
the accuracy of the atomic force constants. For example, if one wants to estimate an ion channel conductance accurate to within a factor of 7, one needs to be able to determine the interaction energy of the ion and
channel to within a factor of ~2 kT. This is less than
half the energy in a single hydrogen bond! Second and
third order effects (polarization, etc.) that are usually
neglected in molecular dynamic calculations can easily
lead to errors many times this size. In addition, although the models of the water molecules that are used
in these calculations have been carefully refined and
tested, they still are not likely to be accurate enough for
the purposes of calculating ion channel flux. These
fundamental limitations are clearly illustrated by the recent MD studies of gramicidin. Because of its small size and known atomic structure, gramicidin has, historically, been the proving ground for ion channel models.
It is the channel for which any new MD approach or extension is first tested. It presents a particularly rigorous
test of these models because the small uniform channel
radius ( Despite these current limitations, MD simulations
still have an important function in modeling ion channels. From a cursory survey of the current literature,
they are now the most popular approach to the theoretical investigation of ion channels. Simulations for short
times (10 ps) can be used to obtain information about the local potential energy and diffusion coefficient of
the ion as a function of position in the channel (Smith
and Sansom, 1998 Three-Dimensional Brownian Dynamics Approach
The above discussion makes it clear that some simplifying assumptions are required. In the approach described in this section, it is assumed that the protein
structure is held fixed and the water molecules are replaced by a continuum. With these assumptions, the
three-dimensional (3-D) movement of ion i can be described by the following simple equation:
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References
). In the old,
premolecular biology era, interpretation of these simple models provided the primary source of information
about channel structure. As molecular biology and,
now, x-ray diffraction have provided direct information about structure, modeling has been reduced to playing
more of a secondary role in channel biochemistry. The
early channel models were always understood to be
highly simplified approximations that, hopefully, captured the "essential" features of the channel. The combination of atomic resolution channel structures plus
highly sophisticated computer routines raises the possibility of obtaining "exact" (molecular dynamic) solutions for channel flux
something that would have
seemed a wild dream only a few years ago. Such solutions have the purpose of interpreting and relating the
channel structure to its function. As will be discussed
below, although this exact solution is still beyond our
reach, a combination of currently available approaches
now make it possible to obtain solutions that approach
this ideal. In this review I will attempt to illustrate the
various approaches and approximations that are currently being used to model ion channels by starting
with this exact solution, and then describing the series of
assumptions that are made as one progresses to simpler
and simpler models. No attempt is made to provide a
comprehensive review of each of the approximations
and the few specific examples that are referred to were
chosen simply to provide illustrations of the different approaches (see Jakobsson, 1998
, for a recent review).
), potassium channel (Doyle et al., 1998
), iron transporting channel (Ferguson et al., 1998
), and a mechanosensitive channel
(Chang et al., 1998
) (all from bacteria), this approach
now has the potential to be applied to these biologically interesting channels. The basic idea is quite simple. First, one assembles an initial atomic model of the
channel protein, the channel water, a nearby region of
the channel lipid membrane, and samples of the bulk
water at both ends of the channel. One then places
some ions in the water, sets the temperature, and applies a voltage or concentration gradient and directly
measures the ion flux as a function of time as the exact
atomic dynamics of the model are simulated on the
computer. At each time step in the simulation, all the
forces on each atom are calculated (covalent, fixed
charges, dipolar, etc.) and the atom moves under this
force for a time period short enough that the forces
should remain constant.
). Thus, the 0.4 µs of real time needed to
simulate the 10 ion crossings would require ~100 yr of
computer time. This calculation is for the ideal case
where the entire channel and part of the lipid are included in the simulation. Computer times could be reduced by simulating only the protein residues lining
the channel and replacing the rest of the protein and
lipid by some continuum approximation. For example,
in the molecular dynamic (MD) simulation of the acetylcholine channel (Sansom et al., 1998
), only the M2
helix bundles were directly simulated and the computer time was reduced to ~1 yr for a single ion crossing. Because porin has such a high conductance,
Suenaga et al. (1998)
were able to observe a single Na+
ion crossing the channel during a 1.3-ns MD simulation
of a reduced porin model. Given the projected improvements in computational speed, such direct MD
simulations should become possible within a few years.
3 Å) means that the ion directly interacts with
the channel wall along its entire length (
30 Å). Most
importantly, there is direct experimental NMR information about the specific localized binding sites of ions
in the channel. Woolf and Roux (1997)
attempted to
predict these NMR results using MD simulations. They
used a fully solvated gramicidin-dimyristoyl phosphatidylcholine model and modified the standard
CHARMM (Brooks et al., 1983
) force field to include
first and second order polarization effects of the ion on
the peptide. Despite the fact that this MD model was optimized to fit gramicidin, the theoretical predictions
of the ion binding sites still differed significantly from
the experimental results. Accurate predictions of the
ion binding locations require that the ion free energy
as a function of position be calculated to an accuracy of
a few tenths of a kilocalorie per mole (Woolf and Roux,
1997
), and this is beyond the limits of current MD calculations.
; Tieleman and Berendsen, 1998
;
Zhong et al., 1998
). This information can then be used
in combination with Brownian dynamics or Poisson-Nernst-Planck theory (see below) to estimate the channel flux. Molecular dynamics is becoming a relatively
routine procedure thanks to the availability of extremely sophisticated software (AMBER, GROMOS,
CHARMM, XPLOR). Although MD calculations of flux
in ion channels now require some special modifications of these routines, it is likely that some user-friendly interface will soon be developed to bring these calculations to the masses.
(1)
where mi, vi, qi and fi are the mass, velocity, charge, and frictional coefficient on the ith ion, respectively. FR is a random thermal force representing the effects of collisions with the water and channel wall. Ei is the total electrical field on the ion, including the partial charges in the protein, all the other ions in the system, and the induced charges from the variation in the dielectric constant at the boundaries between the protein, water, and lipid. It is assumed in Eq. 1 that the only significant forces on the ion are long range electrical forces plus some kind of short range scattering condition when the ion contacts the hard sphere radius of the channel wall. Although one could always add other short-range specific force terms, this would, in effect, be adding an empirical term that did not arise directly from the known protein structure. The solution for this approach proceeds as in the above molecular dynamics method. The channel boundaries are defined, all the ions in the channel and attached bulk reservoirs are positioned, and then, for each ion i, Eq. 1 is integrated in discrete time steps (Brownian dynamics). Because the dynamics of the water and protein are no longer included and relatively long time steps can be taken for the ion motion, this approach is many orders of magnitude faster than the exact molecular dynamics approach (see below for a specific example).
The ability to accurately account for the interaction
between ions in the channel system is one of the most
difficult and critical aspects of modeling ion channels.
In the absence of such interactions, the channel conductance will vary linearly with the ion concentration.
Since almost all ion channels show some nonlinearity
(e.g., saturation) in the physiological concentration range, it is essential that this interaction be accurately
modeled. A major advantage of this Brownian dynamic
approach is that it allows a direct simulation of this
ion-ion interaction. At each step in the dynamics, the
position of all the ions in the channel system are determined and their interaction energy (Ei) is calculated for the next time step. One difficulty with this approach is that, because of the induced charges at the
membrane and channel water interface, the calculation of the electrostatic energy (Ei) at each step requires an involved, time-consuming calculation. S.H. Chung's group has recently described two three-dimensional Brownian dynamics (BD) simulations that use
exact 3-D electrostatic potentials (Chung et al., 1998; Li
et al., 1998
). In the first calculation, Li et al. (1998)
used an idealized, unrealistic channel model geometry
that allowed an analytical solution for Ei. It provides an
interesting analysis of the effects of allowing the ion to wander over the entire mouth and pore region of the
channel area, rather then be constrained to the channel axis as in the standard 1-D solutions. The second
calculation was for an idealized acetylcholine receptor
channel (ACHR) (Chung et al., 1998
). For this case, it was necessary to first use a numerical procedure to
solve for the electric field and potential on a grid. This
data was then stored in a lookup table that was used
during the BD simulation of a channel that was in contact with reservoirs large enough to hold 52 ions. A 1-µs
real time simulation of this ACHR channel required
only 18 h of computer time, making this approach four to five orders of magnitude faster than exact MD simulations.
It is now generally recognized that a combination of MD and BD provides the best available approach to modeling the ion conductance of channels whose atomic structure is known. In this combination approach, local MD simulations are carried out for different positions of the ion. These calculations are then used to determine the local diffusion coefficient and the perturbation of the local channel structure (e.g., main chain and carbonyl shifts) induced by the ion. This local value for the diffusion coefficient is then used to determine the frictional coefficient (fi, Eq. 1) and the perturbed structure is used for the fixed channel structure and both are varied as a function of the position of the ion in the BD calculations.
Three-Dimensional Poisson-Nernst-Planck Approach
The next simplifying approximation is to keep Eq. 1,
but replace the exact expression for Ei by a mean field
approximation Ei that represents a sort of average over
all the possible positions of the other ions in the system. This E is calculated using Poisson's equation. This
combination of random thermal motion of the ion combined with a Poisson solution for E is referred to
as the Poisson-Nernst-Plank solution (PNP). Although
most PNP solutions have been for the 1-D case (see below), a general 3-D PNP solver has recently been described (Kurnikova et al., 1999). The 3-D steady state
Nernst-Planck equation is given by:
![]() |
(2) |
where R is the 3-D position vector, ci is the concentration of the ith ion, = 1/kT and Vi is total potential energy of the ith ion and is described by:
![]() |
(3) |
where U(R) is the potential due to nonelectrostatic
forces, is the electrostatic potential, zi is the valence of
the ith ion, and e is the electron charge. The value of
is then determined from the solution to the 3-D Poisson equation. Given the concentration and potential
on the boundaries in the bulk solution, these equations have a unique interior solution for ci from which the
channel flux of ion i can be obtained. PNP is much
faster than BD because it replaces the simulation of the
ion movement at a sequence of time steps by a global
numerical solution of a differential equation. A unique
feature of the approach of Kurnikova et al. (1999)
is that the solution to Eqs. 2 and 3 is combined with the
standard Poisson-Boltzmann Equation solver Delphi
(Nicholls et al., 1990
), which is routinely used to solve
for the electrostatic potentials in proteins.
This solver was tested using the gramicidin channel
(Kurnikova et al., 1999). A long standing question
about gramicidin has been the origin of its cation selectivity. Since the channel is uncharged, this selectivity
presumably arises from partial dipolar charges in the
channel. This is now a classical problem that has been
studied by a large number of investigators and it has
been a challenge to obtain theoretical results that are
in good agreement with experiment without imposing
some arbitrary adjustable parameters. For example, in
the molecular dynamics study of Roux et al. (1995)
and
the reduced dynamic model of Dorman et al. (1996)
,
special care and adjustments had to be made to get the
models to agree with experimental results. Both of
these models are significantly more complicated than
the PNP model of Kurnikova et al. (1999)
, which was
applied to the gramicidin channel without any modifications. (Kurnikova et al., 1999
, did arbitrarily assume a
channel diffusion coefficient of 1.27 × 10
6 cm2/s to fit
the data). In general, the results of this 3-D PNP solution for gramicidin were in remarkably good agreement with the experimental results except, possibly, for
the location of one of the high affinity sites in the channel. This PNP result, which uses just the fixed, ion-free
gramicidin structure, is surprising because the molecular dynamic calculations indicated that it was essential to include local ion-induced peptide (carbonyl) perturbations in channel structure to explain the qualitative
features of the gramicidin channel. In any case, this
PNP result is very impressive considering that it represents a direct application of this general "off the shelf" model.
There are clearly some limitations to this 3-D PNP approach. The use of a mean field approximation (Poisson equation) is clearly a major simplification. It reduces the specific ion-ion interaction to an interaction
between the ion and this mean field. It is difficult to
quantitate the accuracy of this approximation, and its
range of validity is the subject of considerable debate
(Jakobsson, 1998). Mean field theories have a habit of
working better than one might expect (e.g., the Debye-Huckel theory), and the only way to test them is by
comparison with theoretical models of higher accuracy;
e.g., BD. Despite its obvious limitations, the 3-D PNP solver of Kurnikova et al. (1999)
has the potential to become the routine, quick first approach for estimating
the conductance of a channel whose atomic structure is
known or can be estimated. All the user of this software
would need to do is to input the atom coordinates, the
bulk concentrations and applied potential, and the
program would then output the fluxes of the different ions.
One-Dimensional Brownian Dynamic and Poisson-Nernst-Plank Approaches
The next major simplification is to replace the 3-D concentration function used above by a 1-D concentration averaged over the radial cross section of the channel. As with the use of the mean field approximation, it is difficult to quantitate this 1-D assumption, and its range of validity is not known. In general, with each additional assumption that has been described here, the model results become more empirical and less related to the detailed knowledge of the channel structure. However, if the channel structure is not known to atomic resolution, which is still the usual case, then the error introduced by these approximations is probably of the same order as the uncertainty in the structure, and there is no advantage in using more complicated models.
As with the 3-D case, the 1-D BD approach has the
major advantage that it allows for accurate modeling of
ion-ion interactions. Despite this apparent theoretical
advantage, there are not many examples of the use of
this approach. One of the most detailed is the simulation of a multiply occupied cation channel by Bek and
Jakobsson (1994). These authors were only interested
in the qualitative features of the solution, and did not
try to model the real channel electrostatic potential
terms. One problem that limits the accuracy of this approach is the difficulty of simulating the 3-D bulk solutions by an equivalent 1-D model.
The most complete application of the 1-D PNP
model to ion channels is the modeling of the acetylcholine receptor channel by Levitt (1991a,b). In this solution, realistic channel geometry and electrostatics were
used and the cation and anion flux as a function of
varying channel partial charges was obtained. This solution is appealing because just the fundamental structural features of the channel are used, and there is a
minimum of adjustable parameters. The resulting ion
flux was in good agreement with the experimental data.
Recently, Nonner and Eisenberg (1998) have used
this approach to model L-type calcium channels. Their
solution does not attempt to rigorously model the
channel geometry or electrostatics, but rather is used
to illustrate the general features that are required to fit
the flux data for these calcium channels. It had previously been assumed that calcium channel kinetics required direct interactions between two or more ion
binding sites. The results of Nonner and Eisenberg
(1998)
demonstrated that a simple PNP model could
qualitatively reproduce many of the experimental calcium channel properties. An essential feature of this solution is the addition of a local, nonelectrostatic ion-
channel interaction (Nonner and Eisenberg, 1998
).
For example, to fit the calcium channel data, the channel was assumed to have a relatively high affinity site for
calcium relative to sodium that could not be explained
in terms of the partial charges in the channel. This is a
semi-empirical correction factor and has the disadvantage that the channel behavior can no longer be understood in terms of its well understood long range electrostatic properties. Of course, it is possible that some nonelectrostatic ion-channel interactions are important in determining channel behavior. One potential
problem in using a large local attractive potential is
that there is nothing in the standard PNP theory to prevent the local ion concentration from reaching the unphysical concentration of more than one ion per hard
sphere ion volume. In the approach of Levitt (1991), a
simple modification of the PNP theory was introduced
to correct for this problem.
As with the 3-D PNP approach, the 1-D PNP solution
uses an approximation to the direct ion-ion interaction whose validity is difficult to evaluate. It certainly
can reproduce some aspects of this interaction, as is evidenced by its ability to mimic the concentration dependence of the conductance for the calcium (Nonner and Eisenberg, 1998) and acetylcholine receptor channels (Levitt, 1991). Levitt (1987)
has described a modification of the PNP solution to allow for direct interaction between two ions. Although the approach seems to
provide a more accurate way to treat ion-ion interaction, it does so at the cost of a large increase in mathematical complexity.
Reaction-Rate Approach
In this final simplification, it is assumed that the ions
are localized in specific regions of the channel, and the
kinetics are represented by the rate constants for jumping between these regions and between these regions
and the bulk solution. This approximation obviously
represents a gross simplification that reduces the channel kinetics to its bare fundamentals. The reaction-rate (RR) model should be classified as a simplification of
BD rather than PNP because it does not use the mean
field assumption. In fact, the RR approximation is ideally suited for modeling strong ion-ion interactions.
There has been a long-standing debate (Levitt, 1986)
centered on the issue of reaction-rate "versus" continuum theory (where continuum theory refers either to
1-D NP or PNP approximations). The answer depends
on what one is trying to model. If one wants to interpret the channel conductance in terms of the actual
channel geometry and electrostatics, then some sort of
continuum model is essential since it provides a first order approximation to the actual channel structure. On
the other hand, if one is simply trying to parameterize
the channel or one knows that strong ion-ion interactions are important, then the RR model may be preferable (for example, see Dang and McCleskey, 1998
).
The problem with the RR model occurs when the investigator over-interprets the results and attempts to relate the rate constants to real energy barriers (Levitt,
1986). For example, the rate of going from the bulk solution to a binding site in the channel may simply be
limited by the ion diffusion rate and there may not be
any physical energy barrier. The problems become particularly severe if one tries to explain the experimental
current-voltage curves using the voltage dependence
of the RR energy barriers. For example, most biological
ion channels have relatively linear current-voltage relations, while the voltage dependence of a single RR energy barrier is highly nonlinear (Levitt, 1986
). To fit a linear I-V curve with the RR model, it is necessary to add
multiple energy barriers that probably have no relation
to the actual physical energy barriers in the channel.
Summary and Conclusions
As illustrated in this brief review, there is a hierarchy of
approaches to modeling ion channels ranging from the
exact molecular dynamics simulation down to reaction-rate theory. Each new simplification introduces a new
limitation or uncertainty in the result. In the past, there
was little reason to use the most exact solutions because
of the computational limitations and, more importantly, without atomic resolution channel structures,
these solutions were basically empty exercises. That situation is clearly about to change and we are at the beginning of a new era in ion channel modeling. The obvious
first candidate for this new approach is the Streptomyces lividans potassium channel, whose structure has been recently solved by x-ray diffraction (Doyle et al., 1998). This
channel should become a major testing ground for
checking and calibrating this new generation of ion
channel models. There is no question that strong ion-
ion interactions are important for this channel since at
least two ions are directly observed near the selectivity filter. Thus, an accurate simulation of this ion-ion simulation should require the use of the 3-D Brownian dynamic
approach and rules out some sort of Poisson mean field approximation. One will probably need to use molecular
dynamics to estimate the cation diffusion coefficient and
potential function in the region of the narrow selectivity
filter. A stringent test of the competing models will be
provided by their ability to predict the conductance
changes that occur for specific channel mutations.
For most channels, the structure is not known, and the
old fashioned use of channel models to guide the interpretation of flux measurements in terms of structure is
still relevant. This is illustrated by the current debate
about the selectivity mechanism of the calcium channel
(Dang and McCleskey, 1998; Nonner and Eisenberg, 1998
). Since the two protagonists in this debate are represented by separate articles in this issue, no more will be
added here. Suffice it to say, the excitement that is generated by this issue is the best illustration that these old
fashioned, semi-empirical models still have a role to play
in our understanding of ion channel function.
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FOOTNOTES |
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Original version received 8 February 1999 and accepted version received 23 April 1999.
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