From the * Department of Physiology and Biophysics, University of Miami School of Medicine, Miami, Florida 33101-4819; and Department of Molecular Biophysics and Physiology, Rush Medical College, Chicago, Illinois 60612
Many laboratories study the movement of ions through
single channel molecules every day using the dazzling
techniques of molecular and membrane biology. Atomic
details of a channel are changed and the correlations of
structure and function are observed routinely, but the
correlations have been difficult to understand without a quantitative physical theory that links structure to
function. The inputs to such a theory are the structure
of the channel, the physical properties of ions, and concentrations and electrical potentials in the bath. The
output of the theory is the current observed in a single
open channel for a range of transmembrane potentials
and compositions and concentrations of the surrounding solutions.
Verbal Models and Direct Simulations
Physical laws are written as mathematical equations,
and for good reason. Verbal formulations of physical
laws are not specific and do not permit direct comparison with experiments (which usually have numbers as
their output). Verbal formulations lead different people
to different conclusions, depending on how they subjectively weigh different effects: verbal theories are rarely
objective and often lead to interminable argument.
Mathematical statements of physical laws are less subject
to these human distortions, but they are hardly perfect
either. Mathematical laws lead to mathematical complexities and computational problems quite independent of
their physical meaning, and the necessary approximations are often illogical and can be distorted as well.
These generalities are easily replaced with specifics
when theories of channels are considered. Verbal models of permeation cannot link structure and function
because permeation usually involves competing effects.
If one effect increases current and another decreases it,
the net effect can only be determined quantitatively. It
is natural then to turn to computational models of
channels that promise to be complete and rigorous.
Simulations of molecular dynamics promise to directly compute properties of biological interest, starting with fundamental physical laws and decent representations of the properties of atoms. Unfortunately,
the simulations of molecular dynamics cannot provide
a quantitative theory of ions moving through a channel, driven by a gradient of concentration and electrical
potential, even though the underlying equations of motion of ions and atoms in channels are reasonably well
known. Direct simulations are not possible because the
simulated system needs to be large enough and computed long enough to define the variables that are measured and controlled experimentally; e.g., current, transmembrane potential, and concentration. Direct simulation of atomic motions use a discrete time step of 10 It is not known mathematically (Frenkel and Smit,
1996 Simulated systems of proteins and channels must also
be large enough to define concentrations of ions in the
bath, if those concentrations influence biological phenomena of interest (as they almost always do). Since errors tend to vary with the square root of the number of
particles studied, something like 1,000 ions need be
present in a simulation to define a concentration with
3% precision. Those thousand ions are solvated by a
very much larger number of water molecules, since in
biological systems the mole fraction of ions is small
(ranging from, say, 0.150/55 for typical external sodium concentrations to 10 Short duration simulations of small systems are often
extrapolated to biological sizes and times in a sensible attempt to describe biological phenomena, but
then they are indirect calculations, no better than the
theory used to extrapolate them. Such indirect simulations are not a priori any more reliable than those
made with a theory of lower resolution.
Finally, a technical difficulty is important when considering proteins like channels that act as devices in the engineering sense of the word. Devices and channels almost always function away from equilibrium, but simulations of molecular dynamics of channels are nearly
always done at equilibrium and with treatments of the
electric field that do not allow transmembrane potentials at all (Roux and Karplus, 1994 Averaged Theories: Poisson-Nernst-Planck
Given the difficulties of direct simulations of molecular
dynamics, it seems that some of their resolution must
be abandoned if one wishes to actually fit biologically
relevant data. The choice is only what to give up. The
choice is made by using a series of simplified theories
and picking the simplest that describes the experiments and behavior of interest, while retaining as
much structural meaning as possible.
Here we will show that a simplified model that considers (mostly) the electrical properties of the open
channel (the fixed charge on its walls and the mobile
charge in its contents) does surprisingly well in understanding and predicting the currents that flow through
several open channels, for a range of potential and concentrations of several ions. An electrical model seems a
good place to begin a theory of channels, since the
function of channels directly involves currents and
transmembrane potentials and the structure of channels involves highly charged molecules lining tiny volumes. (One charge in a selectivity filter 7 × 10 Å is
some 5 M.) It does not take much charge to produce
huge forces (see first page of Feynman et al., 1963 The model we will consider forms a working hypothesis, to be tested and then revised as it fails. In its simplest
and original form, the working hypothesis was that a
channel could be described as a one-dimensional distribution of fixed charge, corresponding in a crude way to
the fixed charges on the atoms that line the wall of the
channel's pore. In the crudest form, the effective charge
would be the total charge on the wall of the channel in some cross-sectional slice. In more refined versions of
the model, other estimates of the effective charge would
be used. The only contribution of the channel protein to
the permeation properties of mobile ions was supposed
to be the electric field created by these charges, and the
concomitant distribution of the probability of location
("concentration") of the permeating ion. Single filing effects, and specific chemical interactions (not described
by Coulomb's law) were purposely omitted from the first
version of the model, so the entire force field on the permeating ions could be computed unambiguously (at this
level of resolution) once the charges on the channel
protein were specified (along with the bath concentrations, transmembrane potential, geometry, diffusion, and dielectric coefficients). The electric field in this
model is computed by the Poisson equation, which is the
differential form of Coulomb's law. Details are described
in Eisenberg (1998) PNP depends on the same approximations and representations of underlying atomic variables as the Gouy-Chapman model of an electrified interface, the (nonlinear version of) the Debye-Hückel [DH] theory of
ionic solutions), and the (nonlinear) Poisson-Boltzmann theory of proteins. Although the physics underlying
PNP and these other models are similar, the actual
behavior, mathematics, and computational properties
of the systems are quite different because PNP includes
flux in its every calculation, never requiring equilibrium (Eisenberg, 1998 The Nernst-Planck equations have much more resolution than a crude continuum description of ionic
concentration and movement. Theory and four types
of simulations show that the Nernst-Planck equation
with constrained potential describes the concentration and flux of discrete particles diffusing over barriers of
arbitrary shape and size (Barcilon et al., 1993 The Nernst-Planck equations use one parameter to
describe frictional forces that limit the two components
of flux, diffusion and drift, assuming that the Nernst-Einstein relation between diffusion coefficient and mobility is valid under all conditions of interest. Less elementary versions of PNP do not require that assumption. The diffusion coefficient of the ion in fact also depends on the properties of the channel wall, both because of steric effects and (probably much more importantly) because of dielectric friction produced by motions
of the protein's atoms induced by the electric field of the
permeating ion. But PNP does not explicity invoke this
important physical mechanism (Wolynes, 1980 The PNP model contains much less atomic detail
than most models of channels or proteins. PNP neglects correlation effects of individual atoms. It contains only those effects that are mediated by the mean
field. For example, PNP is rich with binding phenomena because it often predicts localized maxima in the
concentration of permeating ions, but it does not predict the phenomena of single filing that depend on the
correlated motion of ions. We have considered correlation effects to be important for a long time, and so it seemed highly unlikely to us (as we derived the model
and as Duan Chen solved its equations) that PNP would
actually fit data. Indeed, that is why we worked so hard
on higher resolution models. However, the low resolution PNP model does fit a wide range of data, using a
different distribution of fixed charge for each type of
channel (corresponding presumably to the different
structure of each type of channel) and different diffusion coefficients for each type of ion. Note that no parameters are changed as solutions or transmembrane potentials are changed other than the concentrations
and potentials themselves.
PNP Fits Data
Thus far, the simple PNP model fits the highly rectifying (sublinear) I-V relations measured in a wide range
of solutions, symmetrical and asymmetrical, from the
synthetic cation-conducting leucine serine (LS) channel (Chen et al., 1997b The simplest form of PNP fits most of the data but, in
some cases, an extra parameter (a constant) is needed
that describes chemical interaction, as described below.
In most cases, PNP is the only theory that fits the entire
data set, including asymmetrical solutions. The fact that
PNP fits these data sets contradicts the general view,
which we certainly shared until PNP showed us otherwise, that a theory of permeation must explicitly include much more than Coulomb's law. Specifically, it contradicts the view that a successful theory must have separate
components and parameters that describe dehydration/
resolvation, obligatory single filing, or chemical interactions of the permeating ion with the channel protein.
Chemical Effects
Chemical interactions are seen in some cases. Binding
must be included as an extra parameter in PNP when
describing permeation of Li+ and Na+ in the cardiac
CRC channel (Chen et al., 1999 PNP as a Structural Theory
If a theory is to serve as a link between structure and
function of an open channel, it must do more than fit
I-V relations. It must bear a well defined and close relation to the structure of the channel. The crudest form of
PNP is a one-dimensional theory, and so the relation of
its parameters and those of a three-dimensional structure
are not immediately obvious even though the one-dimensional theory was derived by explicit mathematics; e.g.,
averaging (see Appendix of Chen and Eisenberg, 1993 The question remains what is the meaning of the parameters estimated by fitting 1-D PNP to data? The answer to that question is not completely known, but we
note that all of the parameter estimates are reasonable,
and so it is possible that they may be meaningful and reliable estimates of the underlying physical properties,
although that certainly has not been proven to be the
case. For example, typical diffusion coefficients are some 10-50× smaller than in bulk solution; the fixed charge
densities correspond to only one or two fixed charges in
the "selectivity filter" (narrow part) of the channel, although those one or two charges induce a concentration
of mobile charge in the channel of 5-10 M, since they attract one or two counter ions into the channel's pore,
and the pore volume is very small. In fact, the fixed
charge of the CRC channel turns out to be very close to The validity of 1-D PNP can be probed by studying
the effects of mutations on its estimates of fixed charge.
Tang et al. (1997) Three-Dimensional PNP
A three-dimensional version of the PNP theory has recently been developed and calculated (3DPNP), in
which all atoms are assigned the charges used in a standard molecular dynamics program and all atoms are
assigned positions known from nuclear magnetic resonance. Unlike 1DPNP, 3DPNP is a structural theory:
the charges are estimated from the chemical literature,
not from curve fitting. In 3DPNP, only the diffusion coefficient has to be specified by experimental measurements of I-V curves. Only the diffusion coefficient is left unspecified by a priori structural and physical data.
In the first fairly low resolution finite difference calculation (Kurnikova et al., 1999 Why Does PNP Work?
It seems then that for a range of open channels, a simple electrostatic model is able to fit all the I-V data
available in a range of solutions, sometimes with the addition of a single extra parameter to describe the excess
chemical potential of a given ion in a given channel.
There are physical reasons why a mean field theory of a
tiny, highly charged, (nearly) one-dimensional system
is a good approximation. Generally, one would expect
low resolution-averaged theories to be a reasonable approximation because of the long duration of channel
currents (on an atomic time scale). Very little temporal
resolution is needed to describe single channel currents. One would also expect low resolution-spatially averaged theories to be a decent initial description of
(tiny) channels because of the large number of atoms
outside the channel involved in determining concentrations of ions, transmembrane potential, and the reaction field to the ions in the pore. The reaction field
for a permeating ion is largely in the surrounding baths
where a mean field theory is likely to be more than adequate. Specifically, it is known (and not just as a matter
of speculation) that mean field terms dominate correlation terms when charge is distributed along a narrow
cylinder (van den Brink and Sawatzky, 1998 Despite these arguments, a derivation of PNP is
needed to understand its theoretical limitations. Deriving PNP requires a superior theory of higher resolution
that describes single file motion of ions while still computing the electric field from the charges in and near
the channel. Such a Langevin-Poisson theory is being
worked on, by us and others, but is not yet available.
Traditional Barrier Models of Permeation
It seems idle to us to spend much further effort discussing models of permeation that do not fit data, when a
simple model is available that does, particularly given
our discussions in recent papers (e.g., see Appendices
of Chen et al., 1997; Nonner and Eisenberg, 1998 It should be clearly understood that barrier models
rarely are able to actually fit I-V curves measured over a
range of transmembrane potentials and solutions, including asymmetrical solutions, even when using many
adjustable parameters, including an incorrect or adjustable prefactor (see equation in Scheme I). As a rule of
thumb, I-V relations observed in asymmetrical solutions are more linear, often much more linear, than
predicted by barrier models.
Traditional barrier models (Heckmann, 1972 Traditional barrier models also contain two large errors, each factors The correct expression for the rate constant k j = Jf/ The prefactor can be viewed as a measure of the entropy of activation and thus a measure of the effective
volume available for ions to diffuse in the channel compared with the volume available in the surrounding solution (Berry et al., 1980 In this equation, Dj is the diffusion coefficient in the
channel of ion j of valence zj, e is the charge on the proton, kB is the Boltzmann constant, and T is the absolute
temperature. Now, if the barrier is, say, k BT/e high and
1-nm long, and the diffusion coefficient is some 1.3 × 10 The error in barrier models that more or less balances
the error in the prefactor involves the potential barrier.
The Heckmann model does not detail electrical interactions among its internal components. The spatial coordinates needed to specify an electric field are not
present in the model as originally specified, but are put
in by an artificial concept of electrical distance. The model postulates discrete sites when none need be
present. It is not surprising that the Heckmann model
does not predict the same electric field as Poisson's
equation (see below).
In barrier models, and all other models of permeation familiar to us (that actually predict current), except PNP, the potential barrier is assumed, not computed. Potentials in channels arise, however, from the
fixed charges on the protein, the mobile charges inside
the channel's pores, and the charges in the solutions
and electrodes outside the channel. These produce a
potential profile that changes as conditions change;
e.g., as transmembrane potential, bath concentration,
or fixed charge on the channel protein changes. In
general, the resulting potential profiles do not have
large barriers, and so currents are much larger (indeed, exponentially larger) than they would be in otherwise similar theories with large barriers. The currents
predicted are very different from those of traditional
barrier models. Having been raised in the barrier tradition, we are often surprised by the electrostatic effects
predicted by PNP, although they are easy to compute
and to understand once they are computed.
An example may be useful. Nonner and Eisenberg
(1998; see Appendix) compute the electric field of a
barrier model of the L-type calcium channel using Poisson's equation instead of repulsion factors. The energy
profile of Dang and McCleskey (1998) Surprisingly, the PNP calculation using this profile
predicted a reversal potential close to zero for external
calcium concentrations <10 mM. That is, the "calcium"
channel became a nonselective channel when Ca2+
was <10 mM. When Ca2+ in the external solution was
100 mM, the selectivity reversed (i.e., the reversal potential became By explicitly using Poisson's equation, we had found
that binding of calcium dramatically reduces the calcium current, not because of "interference" by single
filing, or an effect on diffusion constant, but because of
an electrostatic effect. Of course, the details of this effect depend on the size of the repulsive potential that
we chose. If we had chosen a repulsive potential smaller in magnitude, the channel would have become a chloride channel at lower external calcium concentration.
If we had chosen a repulsive potential larger in magnitude, the chloride "selectivity" would have not appeared in the 100 mM Ca2+ solution. But the electrostatic effects of the binding would have been profound
in any case. Such effects are absent in traditional barrier models or other models that do not use Poisson's
equation or Coulomb's law to go from charge to potential.
Binding invariably has a large effect on potential because of the accumulation of charge that binding necessarily entails (that is what the word "binding"
means!). That charge changes potential, and the
change in potential is large because the system is so
small (i.e., its capacitance is tiny). Bound ions repel nearby ions and thus have large effects, creating, for example, depletion layers that can dominate channel
properties. These electrostatic effects of binding will be
seen no matter what the details of the calculation or
choice of repulsion potential, and the existence of
these effects is the main point of our discussion.
Barrier models miss the electrostatic effects of binding altogether. Rather than solving Poisson's equation,
barrier models use fixed profiles of free energy to characterize the interactions of permeating ion and channel protein independent of charge; they use repulsion
factors to characterize interactions between permeating ions as if the ions were always separated by a fixed distance, producing a fixed repulsive energy independent
of the fixed charge of the protein and the screening of
this fixed charge by nearby ions (in the pore and in the
baths). Screening has been known to dominate the properties of electrolyte solutions and interfaces since the
work of Debye-Hückel and Gouy and Chapman (Newman, 1991 Using fixed profiles of free energy to characterize the
interactions of permeating ion and channel protein is
inaccurate because the potential profile inside the
channel depends strongly on the concentration of ions
in the bath, as is easily verified in a Poisson Boltzmann
or 3DPNP calculation, because of the long range nature of the electric field: the ionic atmosphere of the
fixed charge lining the channel's wall extends into the
surrounding baths. Or, to put the same thing another
way, the dielectric charge in the protein and lipid and
the mobile charges in the channel's pore do not screen
the fixed charge of the protein from the ions in the
bath. Thus, the interactions of ions within the channel cannot be described by a theory that ignores the concentration of ions in the bath. The ions in the bath
help determine the interaction between ions in the
channel's pore.
Using fixed interionic distances in traditional barrier
models is inaccurate because the distance between ions is
in fact quite variable. The distance depends on screening
that varies with the concentration of ions in the bath, the
transmembrane potential, and the charges and shape of
the channel protein itself. In fact, to maintain a fixed average distance between permeating ions (as conditions
are changed), large amounts of energy would have to be
injected directly into the channel's pore by a deus ex
machina, always the theorists' most helping hand.
We suggest that the fundamentally flawed treatments
of electrostatics in barrier models are likely to produce
a qualitative misunderstanding of the role of occupancy
and quantitative errors of at least one order of magnitude. Our calculation (Nonner et al., 1998 If a channel is lined by a fixed structural charge, electrostatic effects tend to maintain a (nearly equal) number of mobile ions within the pore, thereby maintaining approximate electroneutrality. Wide variations in
ionic occupancy are buffered by the need for approximate electrical neutrality; i.e., wide variations in occupancy can only be produced by large energies not typically available to channels. Even though ionic occupancy
is buffered, screening depends on bath concentration
because the electric field generated by the fixed charge
lining the wall of channels reaches through the protein
and lipid into the surrounding baths: the electric field
is long range. Electroneutrality in the channel is approximate, not exact, and the residual ("unneutralized")
charge is large enough to have profound effects.
Conclusion
We conclude that traditional barrier models overestimate current because they neglect friction and underestimate current because they compute electrostatics
incorrectly. These errors of course do not balance precisely and that is why barrier models fail to fit reasonably large data sets, particularly if the data sets include
I-V relations measured in asymmetrical solutions.
These errors are fundamental to the whole class of barrier models, and so we believe such models must be
abandoned. When a model fails as badly as barrier
models do, its qualitative features cannot be considered
a reliable indicator of underlying mechanism. It is not
useful for our main purpose.
To us, abandoning barrier models seems no great
loss. Those models do not fit the data anyway (if the
data is taken over a reasonable range of conditions).
But abandoning barrier models is a great loss, in a
more human sense, because so many gratifying insights
into mechanism have been developed using them, often at great effort. Abandoning barrier models means
calling these insights into question. It means these insights must be reexamined using theories that fit data
and have some physical basis. It means that much more
experimentation is needed to reexamine issues already
thought to be settled. Reexamination of settled issues is
bound to be unsettling.
Unsettled Questions
The outstanding problems in permeation are to understand the role of electrostatics, chemistry, and geometry in determining the movement of ions (in our view).
So far, the role of electrostatics and geometry seems
approachable by 3DPNP. We suspect, but have not
proven, that the effective one-dimensional profile of
charge we call P(x) will be approximated by derivatives
of the solution Knowledge of permeation is limited by a surprising
lack of published I-V curves in asymmetrical solutions.
These are important because they are often much
more linear than expected from traditional models.
Knowledge is also limited because we have so little
structural information, particularly of the eukaryotic
channels of greatest anthropomorphic interest; e.g., the
voltage-gated Na+ and K+ channels of nerve fibers. As
we turn to these specialized channels, it seems likely
that the simplest version of PNP will need to be supplemented by models containing more explicit chemistry; i.e., binding energies in binding regions. The question
is how to introduce chemistry into a model without requiring analysis of uncomputable trajectories. We are
following the chemists, taking a most successful theory
of ions in bulk solution, namely the MSA and applying
it inside a channel: it is comforting to work with a theory and people who have solved the problem of selectivity in bulk solution. The MSA has a long history (Blum
et al., 1996 The MSA is a mean field theory that in essence reworks Debye-Hückel analysis, now treating ions as
spheres. The distribution of point charges around a
central sphere (as assumed in DH) is quite different
from the distribution of spherical charges around a
sphere (MSA), particularly at concentrations more
than a millimole or so, because finite ion diameters create exclusion zones around ionic charges. The resulting charge distributions produce quite different electric fields, according to Poisson's equation, and this difference allows MSA to fit much data that DH cannot.
Fortunately, the MSA is hardly more complex than
DH because both theories express thermodynamic
functions in terms of one quantity, a characteristic length
of screening Interestingly, most current theories of bulk solution
and narrow spaces (DH, HNC, and MSA) are mean field
theories; they do not try to follow or explicitly average individual ionic trajectories. These theories deal self-consistently with the average effects of excluded volume, including diameter-constrained electrostatic interactions.
To be applied rigorously to channels, these theories need to be reexamined and rederived for the geometry
of channels and the specific properties (e.g., excluded
volume) of the amino acids that line the wall of the
channel, taking note of its high surface charge density.
Even better, density-functional theory (DFT, a generalization of HNC designed to describe inhomogeneous systems; Henderson, 1992 Even before these rigorous treatments are available,
it is already clear that many of the distinguishing characteristics of K+, Na+, and Ca2+ channels can arise naturally from the two main (antagonistic) effects in MSA:
electrostatic attraction between permeating cations and
the groups forming the selectivity filter, and "chemical" repulsion arising from the effects of the finite volume
of ions (Catacuzzeno et al., 1999 Single Filing
Finally, we address the issue of obligatory single filing,
an important property of ionic channels that has concerned us for many years. Nonner et al. (1998) Nonetheless, it is clear that PNP in its several forms
does not predict the ratio of unidirectional fluxes observed in K+ channels, or expected in single file systems. (See citations in Nonner et al., 1998 The paradoxical fact is that PNP predicts net fluxes
(i.e., currents) in a wide range of channels and conditions, while it does not fit the ratio of the component unidirectional fluxes correctly, at least if we assume the flux
ratio of all channels is rather like that of the K+ channel.
Resolution of the paradox requires analysis or simulation of a system with both obligatory single filing and with an
electric field computed from the charges present.
Wolfgang Nonner has constructed a mean field
model of single filing, by extending PNP to include
convection. This Navier-Stokes extension of PNP is
clearly able to predict the appropriate ratios of unidirectional flux and I-V curves that PNP itself has not fit.
We have also done much work (with Schuss and his students, available by anonymous FTP from
in directory /pub/Eisenberg/Schuss) to formulate a
self-consistent model of Brownian motion; i.e., one in
which ions move according to a Langevin equation in
an electric field determined by all charges present,
computed by a Poisson equation updated at each time
step. The mean field arises naturally in the model because of the long range nature of the electric field. The
fixed charge lining the wall of the channel is "neutralized" by ions in the bath (in large measure), and those ions can be described by a mean field theory under biological conditions. That is to say, the reaction field of
the fixed charge on the wall of the channel is the mean
field, even in a model constructed in atomic detail. It is
clear from this work that a well-posed mathematical
model can be constructed, and can predict flux ratios,
but the model has not been solved in general.
Interestingly, the analysis with Schuss shows that nonindependent flux ratios can arise without changing net
flux. In that analysis, a "single file term" is found in
both the influx and efflux, and so the net flux is not
changed by single filing, but the flux ratio is. Perhaps
this is how PNP manages to fit net flux data so well in
the K+ channel, while it gives ratios of unidirectional
flux not expected in single file systems.
We, along with others, are also trying to simulate
such a single file Langevin-Poisson system. Until this
simulation is actually performed, it is wise to be prudent and not guess its outcome. Rather, we will trust
the work, particularly the resulting numbers.
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References
16 s
to resolve atomic vibrations. Calculations with (substantially) longer time steps than 10
16 s are no longer direct
simulations, but depend on theory and assumptions to
fill their gaps. A single ion takes some 100 ns to cross a
channel (109 time steps), and calculations of thousands
to billions of crossings must be made (Barcilon et al.,
1993
) to estimate measured currents. Direct integration
of differential equations of molecular dynamics (i.e., the
differential equations that are Newton's laws) is not reliable over times larger than several picoseconds because
the calculated trajectories exponentially diverge (Ott,
1997
) and are exquisitely sensitive to initial conditions
and round-off error (Frenkel and Smit, 1996
; Rapoport,
1997
) as can easily be verified by running any of the
codes widely available, some without cost.
) whether calculated trajectories (after a few picoseconds of calculation) are reliable, unbiased, or even
useful estimates of the trajectories of real atoms ("unbiased estimate" here is defined as in probability theory
and statistics). Simulated trajectories may not sample some of the phenomena of biological interest at all, particularly those phenomena that appear slowly (i.e., after
microseconds). Trajectories of chaotic systems, starting
from particular initial locations, are often confined to
limited regions of "phase space" and do not explore other
regions at all. Even within an accessible region of phase
space, simulated trajectories may not be reliable or unbiased estimates of systems. After all, while the trajectories of real atoms are not subject to round off error, the computation is, and how the round off error influences the
computation is not known (Beck and Schlögl, 1995
).
5/55 for typical internal calcium concentrations). Simulations need to involve
some 3 × 105 molecules (nearly one million atoms) to
define a typical extracellular Na+ concentration in the
solutions surrounding a channel and 2 × 109 atoms to
define a typical intracellular Ca2+ concentration.
). The mathematical
difficulties involved in computing nonequilibrium biological and chemical systems with transmembrane potentials have not been overcome (Eisenberg, 1998
), and
the methods used by physicists to perform such calculations have not, to the best of our knowledge, been tried
in channels, proteins, or ionic solutions. The best portal
to these methods seems to be the DACMOCLES website
().
).
, and the code that solves these equations is available in various forms at an anonymous ftp
site ( in directory /users/Eisenberg). In the
initial working hypothesis, the electric field is supposed to influence current only according to electrodiffusion
as described by the Nernst-Planck equation. The model
is specified by the Poisson-Nernst-Planck (PNP) equations, which, as it happens, are nearly identical to the
drift diffusion equations that have been used for a very
long time to describe the motion of charged quasi-particles like the holes, electrons, and superparticles of semiconductors (Lundstrom, 1992
) or the hydrated ions of
electrolyte solutions (Newman, 1991
). We have suggested
that the correlated motion of an ion, water, and atoms of
the channel protein might act as a quasi-particle or super-particle, a permion (Elber et al., 1995
), with the permion rather than the individual ions satisfying the conservation laws of PNP.
).
; Eisenberg et al., 1995
) from a bath of one concentration and
potential to another.
).
). It fits most (but not all) of the
highly rectifying, superlinear I-V relations in the neuronal background anion (NBAC) channel (Chen et al., 1995
). It fits the rectifying sublinear I-V relations from
porin (Tang et al., 1997
), a channel of known structure,
and its mutant, G119D, also of known structure. It fits
I-V relations from both cardiac and skeletal forms of
the calcium release channel (CRC) of sarcoplasmic
reticulum (Chen et al., 1997a
, 1999
), all of which are
quite linear; and it fits I-V relations from a number of
other channels (most notably gramicidin, see below).
So far, PNP has fit the I-V relations of every channel for
which we have data, but we continually expect it to fail
on the next attempt.
), or the anomalous mole fraction property of K+ channels (Nonner et al.,
1998
) or the interactions of Na+, Ca2+, and pH in the
L-type calcium channel (Nonner and Eisenberg, 1998
).
It is enough (at the present level of experimental and structural resolution) to add a single constant for each
ion that describes the excess chemical potential of that
ion in the channel. That constant is the same for a
given type of ion in a given type of channel and does
not vary with transmembrane potential or concentration of any of the ions. This constant is the only representation of dehydration/resolvation, nonelectrostatic
binding, and single filing needed to fit these data sets.
Indeed, it is possible that this constant arises entirely
from Coulomb's law, as a correction for effects of the
three-dimensional field not correctly described in the
one dimensional model. As we consider more complex
channels and mixed solutions containing many different ionic species, it seems likely that more specific
chemical effects will be seen. We propose to deal with
these (in the first place) in the Mean Spherical Approximation (MSA) that has proven so successful in bulk solution, as discussed later in this paper.
)
of the structurally based three-dimensional (3-D) theory.
1 (Chen et al., 1997a
, 1999
), and this is an invariant of
the curve fitting procedure (changing all sorts of parameters in the curve fitting does not change the estimate of
the total fixed charge, although it changes the estimates
of the diffusion coefficient), suggesting that the selectivity filter of this channel (in the form studied here) is
dominated by a single fully ionized acidic amino acid, although of course the PNP results do not prove this.
compared the estimates of charge in
a wild-type porin and its aspartate-substituted mutant,
G119D, which has one additional negative charge and
known crystallographic structure. The estimated difference
0.93, found in those and many subsequent experiments, is surprisingly close to the value expected.
), 3DPNP shows qualitative agreement between the theory and experimental
data. A high resolution analysis (Hollerbach et al., 1999
)
using spectral elements shows quantitative agreement.
) or when
systems are highly charged (Henderson et al., 1979
).
; see
also Eisenberg, 1998
; Nonner et al., 1998
). Nonetheless, it is necessary, given the purpose and arena of this
paper, to reiterate some of the things we have said previously about traditional barrier models of open channels that have been so widely used to study permeation.
; Hille,
1992
) are not very useful for relating structure and
function, because they involve states and transitions
only vaguely related to the actual structure. The models
do not contain spatial coordinates as variables. The positions of individual components like ions are not within
the scope of the model and so the Heckmann level of
description is a priori incapable of relating ion flow to
the geometrical structure of a channel.
104, that act in opposite directions
and so more or less balance each other in the limited
sense that they allow the model to fit the current measured at one transmembrane potential in one symmetrical solution. Most glaring is the choice of prefactor in
the expression for the current over a high barrier. For
historical reasons, the prefactor used in nearly all barrier
models of open channels is k BT/h, even though that
prefactor leaves out friction altogether (i.e., it contains
no diffusion coefficient or variable to describe friction).
Friction is a dominant determinant of atomic movement in condensed phases (like ionic solutions, proteins, and
channels) because condensed phases contain (almost)
no empty space. Workers on channels pointed out this
problem some time ago (Cooper et al., 1985
, 1988a
,b;
Chiu and Jakobsson, 1989
; Läuger, 1991
; Roux and Karplus, 1991
; Andersen and Koeppe, 1992
; Barcilon et al., 1993
; Crouzy et al., 1994
; Eisenberg et al., 1995
) and, indeed, Eyring clearly states that k BT/h is not to be used by
itself as a prefactor in condensed phases (Wynne-Jones
and Eyring, 1935
). There is general agreement among
workers on condensed phases that the expression k BT/h
is inappropriate (Fleming and Hänggi, 1993
; Hänggi et
al., 1990
, which cites some 700 papers relating to this
matter). When reading these classic papers, it is important to be aware that the value of the transmission factor
(Frauenfelder and Wolynes, 1985
) is now known in condensed phases dominated by friction (Fleming and
Hänggi, 1993
; Pollak et al., 1994
).
C l
in a condensed phase for one-dimensional (unidirectional) flux Jf from a solution of concentration C l over a
high barrier was apparently first published by Kramers,
1940
. The rate constant and unidirectional flux can be
written in a particularly neat form if the potential profile is a large symmetrical parabolic barrier spanning
the whole length
of the channel, with peak height
max(xmax), at location xmax =
/2, much larger than the transmembrane potential and k BT/e.
). It seems natural that the effective volume should involve the length and height of
the potential barrier, and Hill discusses the role of diffusion velocity in the prefactor (Hill, 1976
).
6 cm2 /s, the numerical value of the prefactor is
~2.8 × 108 s
1. The numerical value of the usual prefactor in barrier theory is k BT/h, which is ~2.2 × 104
times larger, ~6.3 × 1012 s
1. As one might expect, ions
hopping over barriers experience much less friction
than ions diffusing over them, and the amount of the
friction will depend on the identity of the ion. Numerical errors of this size have serious qualitative as well as
quantitative consequences, as was pointed out some
time ago (Cooper et al., 1988a
).
was included as
a spatially varying excess chemical potential µ0Ca(x) to
describe the binding properties of the channel. Cl
was
excluded from their calculation, so, to be fair and comparable, we also excluded Cl
by applying a large repulsive energy, specifically ~12 kBT/e, just for Cl
, which
we thought would reduce Cl
occupancy by ~e12
1.6 × 105. The repulsive potential did not act on the
cations, and diffusion coefficients of all ions were set
equal for illustrative purposes in this calculation.
20 mV). The calcium channel had become a chloride channel, if we use common lab jargon.
The change in selectivity of the channel was produced
without invoking any specific chemical effects at all, it
was produced by electrostatic repulsion, computed
from PNP, just as the anomalous mole fraction effect
(studied in Nonner et al., 1998
) was a purely electrostatic effect. Neither single filing nor definite occupancy were involved. When placed in a 100 mM Ca2+
solution, 0.2 Ca ion was found in each pore in the calculation. This bound calcium produced a (small positive) local excess in net charge and that produced a
large positive potential of nearly 110 mV (see Figure 12 of Nonner and Eisenberg, 1998
). Of course, that potential was a severe barrier for cation movement. This potential barrier (produced by calcium binding) reduced
cation movement (particularly divalent cation movement like calcium flux) so effectively that the residual
conductance was dominated by chloride, even though
chloride was subject to a repulsive potential of 12 kBT/e.
) and it seems unwise to ignore it in channels.
) shows occupancy predicted by Poisson's equation is very different
from that predicted with barrier models. Our calculations show that if a channel is to hold a certain number
of mobile charges, it must have balancing structural
charges, and the interactions of these mobile and fixed
charges cannot be described by a fixed free energy. The
wide variations in ionic occupancy that typically occur
in barrier models are likely to be in severe conflict with the electrostatics of Poisson's equation customarily used
to describe the electric field. That is to say, if such variations in occupancy actually occurred, the electrical potential would vary wildly because of the severe violations of local electrical neutrality.
3DPNP(x,r,
) of the three-dimensional
PNP equation (specifically, the divergence of the three-dimensional electric field in the radial direction and
equivalently its derivative along the path of permeation, with the concentration of permeating ions subtracted off). Further analysis will tell whether the excess
free energy needed to fit some of our data sets is an expression of the three-dimensional field, of actual chemical interactions, or of some other effect.
), and recent versions have been remarkably
successful at predicting the properties of solutions
from infinite dilution to saturation, even molten salts
(Simonin et al., 1999
).
1 that is given by algebraic formulas, albeit more complex formulas in the MSA than in DH.
MSA in its primitive form treats water as a continuous
ideal dielectric, whereas "nonprimitive" versions include the solvent as a polar molecular species (up to
octopolar, as is needed to model hydrogen bonding).
More accurate theories are available (e.g., the hyper-netted chain [HNC]; Henderson, 1983
), but they are
more elaborate to compute and do not lead to algebraic expressions for activity coefficients.
) should be applied to channels
and proteins. We are trying, thanks to Laura Frink of
Sandia National Laboratory (Albuquerque, NM).
). In the analysis of
these three types of channels, no other specific chemical interactions are needed to describe selectivity (beyond those resulting from finite volume of ions as described by the MSA).
shows
that one of the experimental phenomena (the anomalous mole fraction effect) thought to require an explanation involving obligatory single filing can in fact be explained without single filing. The effect appears (in a
mean field theory without obligatory single filing) as a
necessary consequence of (a small amount of) localized
excess chemical potential; i.e., binding or repulsion.
. Measurements are not available of flux ratios in most channels.
Measurements of flux ratios independent of gating are
not available at all, to the best of our knowledge.)
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FOOTNOTES |
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Original version received 9 February 1999 and accepted version received 23 April 1999.
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