From the Department of Biochemistry, Howard Hughes Medical Institute, Brandeis University, Waltham, Massachusetts 02254-9110
In the whirlwind of cloning, mutagenesis, and, suddenly, structure that the ion channel field has been
riding for the past 15 yr, it is easy to forget that we still
don't have a satisfactory view of that most basic task carried out by these proteins: ion permeation. The diffusion of ions through the aqueous pores of ion channels
(a process much simpler than gating) is being treated in two different ways by two increasingly polarized
schools of thought. For want of terms that are both
precise and concise, I refer to these as the "chemical-
kinetic" and "continuum" descriptions of channel-
mediated electrodiffusion For the present discussion, I assume a qualitative understanding of the basic ideas behind the two competing views of ion conduction (but not necessarily the
details of their implementation) and will offer some
reasons why I consider the chemical-kinetic approach
to be of greater practical utility. Rather than making general arguments about electrodiffusion to defend
this position, I will illustrate the current clash of these
theories by examining one particular case.
An Experimental Example
The subject we will look upon is calcium channel permeation, models of which have been described in lucid
detail (Almers and McCleskey, 1984 Chemical Kinetic Viewpoint: Multiple Occupancy on
Discrete Sites
According to the canonical model, the observed AMFE
is a direct reflection of the binding of two Ca2+ ions in
a single-filing pore. The idea is simple, proceeding from
the postulate that the channel is designed to coordinate Ca2+ at specific anionic sites. In the absence of
Ca2+, when these are electrostatically hungry, the pore
is merely charge selective, allowing virtually any monovalent cation to permeate as long as it is physically small
enough to squeak through. Thus, at low [Ca2+], the
Na+ conductance is high. In the presence of micromolar Ca2+ concentrations, the pore's selectivity region
now becomes occupied by a single Ca2+ a significant
fraction of the time (which varies according to the bath
concentration). Because of its intimate coordination by
protein groups, this bound ion's dissociation rate from
the channel is low, ~103 s To quantify these effects, the chemical-kinetic approach makes an explicit distinction between four different occupancy forms of the channel: a Na+-conducting form with no bound Ca2+ [O, O], two nonconducting forms, each with one Ca2+ bound on either side
[O, X] and [X, O], and a Ca2+-conducting form with two
Ca2+ bound [X, X]. The average current and Ca2+/Na+
selectivity are given by the kinetic transitions among
these various forms of the channel; i.e., by a set of rate
constants between explicit chemical intermediates, exactly as in any conventional chemical kinetic problem.
The values of the rate constants cannot be estimated
from first principles, but must be derived by fitting experimental data to a kinetic model, a straightforward
but rarely unambiguous procedure.
Continuum Viewpoint: A Nanoscale Ion Exchanger
Much of the present controversy centers on the use of
Poisson-Nernst-Planck electrodiffusion models in biological channels. Such models have been in use for a
long time, both as qualitative handles for the classical
squid axon channels and as more intricate frameworks
for permeation in ion-selective channels with firm structural foundations (Levitt, 1982 In modeling the AMFE, PNP2 is a theory of ionic
cleansing. With no Ca2+ present, the pore conducts
well because Na+ ions dwell there at high concentration, this being the only cation available for electroneutrality. But when a little Ca2+ is added to the bath, these
divalent intruders, with their heavy artillery in the form
of a +2 valence, take over, displacing the numerous but
poorly armed Na+ ions. In electrostatics, divalents
always beat monovalents. Thus, as Ca2+ is increased,
the pore, initially Na+ rich, becomes loaded with Ca2+,
and the conductance goes down because the Ca2+ diffusion coefficient is assumed to be lower than that of
Na+. By the time bath Ca2+ concentration reaches, say,
100 µM, all the Na+ has been expelled from the pore,
and the current is carried solely by Ca2+. Thus, the falling phase of the AMFE.
But why does the channel conductance rise again as
Ca2+ is raised further? The surprising answer provided
by the PNP2 treatment is that it doesn't! The conductance is predicted to remain essentially flat as Ca2+ rises
to high levels because electroneutrality forbids admittance to additional Ca2+ over and above the fixed negative charge; but the current measured at a given voltage
(e.g., Evaluation and Conclusions
So here we have two very different ways of interpreting
a fundamental set of facts about ion permeation in calcium channels. I will state my opinion bluntly. First, no
theory, however mathematically sophisticated, that rejects specific ionic coordination by protein moieties,
dismisses the finite size of ions, and ignores the single-filing effects necessarily arising from the small spaces in
the molecular structures of ion channels can have
much worthwhile to say about selective ion permeation.
Second, a ubiquitous feature of continuum theory (1) The continuum approach ignores ion channel chemistry.
For many years, indirect experiments have suggested that ions permeate selective channels by binding to localized sites at which protein functional groups
replace waters of hydration, and that ion selectivity
mainly reflects the energetics of the switch from water
solvation to protein coordination (Hille, 1991
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both of which treat ions as stumbling through one-dimensional random walks
along the pore. In chemical-kinetic descriptions, ions
hop along a small number of binding sites (Hille, 1991
);
in continuum theories, they diffuse in continuous space
along the pore under the influence of local electrochemical gradients (Sten-Knudsen, 1978). This distinction may seem the stuff of academic hairsplitting, but it
is not
it is fundamental, and the vituperation spent
on these models in recent years attests to their current irreconcilability.
; Hess and Tsien,
1984
; McCleskey, 1997
; Nonner and Eisenberg, 1998
).
This case serves to bring out essential differences in a
palpable and physiologically important context. I deal
with just one small corner of this subject
the key observation that launched calcium channel permeation as
a rich area of investigation. This is the remarkable fact
that under physiological conditions the channel is
strongly selective for Ca2+, but when bath [Ca2+] is reduced below 1 µM this selectivity is lost and monovalent cations easily permeate. In the well-known classical
experiment, inward current through calcium channels
is measured as a function of external Ca2+ concentration in a conventional physiological bath medium at a
fixed holding voltage of, say,
30 mV. At very low Ca2+
(<0.1 µM) there is a large inward current carried by
Na+. As [Ca2+] is raised up to ~100 µM, the current
decreases to near zero. But then, as [Ca2+] increases
farther into the 1-10 mM range, inward current rises again, with Ca2+ as the charge carrier, and the reversal
potential shifts positive towards ECa. This nonmonotonic
variation in current with external [Ca2+], sometimes
termed the "anomalous mole-fraction effect" (AMFE), is explained in vastly different ways by the two opposing viewpoints.
1, some three to four orders
of magnitude slower than the throughput of Na+ ions.
Under these conditions, Na+ roars through the pore
when Ca2+ is absent; but whenever a Ca2+ binds, the
flow of Na+ current is fully blocked. This block lasts on
the order of 0.1-1 ms, and it is due directly to the single-filing property: the impossibility of a Na+ ion diffusing "around" a bound Ca2+. Only after the Ca2+ vacates
the binding site can the flow of Na+ through the channel resume. This effect, averaged over many channels in a macroscopic experiment (or over time in a single-channel experiment), leads to the "falling phase" of
the AMFE; i.e., the decrease of inward current as Ca2+
increases through the micromolar range. If Ca2+ concentration is pushed up into the millimolar range, a
new phenomenon appears. Now a second Ca2+ can
bind and, as a result of this double occupancy, the exit rate of Ca2+ from the pore increases ~1,000-fold. This
huge increase in Ca2+ off rate is usually explained by
invoking electrostatic repulsion between the two ions,
but other mechanisms could be involved (McCleskey, 1997
). In any case, as a result of this double occupancy,
Ca2+ now flows through the channel at rates high enough
to show up as current, which increases with Ca2+ concentration to produce the "rising phase" of the AMFE.
, 1986
). For this discussion, I will focus on a recent application to calcium
channels termed PNP2 (Nonner and Eisenberg, 1998
),
which views the pore as a continuum containing several
negatively charged groups smeared out over a reasonable pore volume; this represents a very high concentration of fixed charge (~10 M). Both Ca2+ and Na+
have free access to this forest of negative charge, where
they act as gegenions, cations that are not chemically
coordinated by the fixed charge, but rather are held
nonspecifically by the demands of electroneutrality (or,
more properly, the Poisson equation), as in an ion-
exchange resin. The system is described by the simultaneous solutions of three equations in which the three
crucial variables, ion concentration, electrical potential, and distance along the pore, are nonlinearly entwined. The solutions lead to mathematically self-consistent predictions of ionic current as a function of transmembrane voltage and bath ion concentrations.
40 mV) does increase to produce the AMFE for
a simple reason: the reversal potential keeps moving positive as external Ca2+ is increased. It's the driving
force that goes up, not the conductance. In other words,
this analysis asserts, everyone in the field has been
dunderheaded all these years on a most elementary
point, having apparently forgotten that current equals
the product of conductance and driving force! (I am
oversimplifying a little here; a small rise in conductance
with [Ca2+] is predicted by PNP2, but this is a second-
order effect having to do with surface polarization.)
the mean-field assumption
invalidates, or at least
greatly vitiates, its application to channels in which only
a small number of ions reside at any one time. Third,
PNP2 is inadequate to understand the particular calcium channel problem under examination here. Fourth,
the undoubted quantitative weaknesses of the chemical-kinetic approach do not undercut its value in capturing the mechanistic essence of permeation in ion-
selective channels.
). In soluble proteins, it is hardly a radical notion that binding
of dehydrated inorganic ions lies at the basis of a multitude of functions (Falke et al., 1994
), and now, with
the structure of KcsA (Doyle et al., 1998
), this idea has
been confirmed directly for a strongly selective ion
channel. The KcsA structure dramatically confirms for
a K+ channel the multi-ion single-filing assumption,
long known also to be valid for the peptide channel
gramicidin A (Finkelstein and Andersen, 1981
). For
permeation, the qualitative consequences of localized,
structured binding sites and single-filing are profound; they lead naturally and necessarily to familiar phenomena seen in many channels: strong, concentration-
dependent selectivity, discrete ionic block of permeation, and anomalously high ratios of unidirectional
ionic fluxes.
)
or by assigning to each ion its own diffusion coefficient.
These are worthy additions to otherwise "featureless" electrodiffusion theories, but they are simply not
enough; nobody has yet figured out how to weld single-file arrangements of binding sites to continuum theories in a general manner, although Levitt (1982
, 1986
,
1991a
,b) has achieved impressive success in incorporating these features in particular cases, and Nonner et
al. (1998)
similarly have modeled a subset of K+ channel behaviors with a selective binding "region." Conquest of this analytical impediment would represent
a major advance in modeling permeation; in such a
case, the entire channel field would unhesitatingly
embrace continuum electrodiffusion as the preferred
approach to the problem.
(2) The mean-field assumption is inapplicable in small
spaces.
To obtain solutions for the ionic fluxes, continuum treatments must use concentration and electrical
potential as continuous spatial variables in the coupled
differential equations. The concentration at a given position determines the net charge density, which in turn
influences the value of potential at that and nearby positions. Concentration is an intrinsically probabilistic
quantitythe average number of ions per unit volume.
For a macroscopic object such as a worm of ion-exchange
gel, the number of ions present is sufficiently large that
this average can be taken within each cross-sectional slab of the object at each moment in time. The concentration at each position will fluctuate with time, but, if
the object is large enough, these fluctuations will be
negligible and the concentration, and therefore the potential, will be a time-invariant spatial average. This is
the "mean-field" assumption: this average potential may be validly used in the three crucial equations. In
an object of molecular dimensions, however, a huge
problem arises. For something the size of a calcium
channel selectivity filter, a concentration of 10 M represents on average only one or two ions in the entire volume. "Concentration" is still defined as a statistical
average, but in this case the average must be taken over
time; i.e., by sitting at a given position in the pore and
asking what fraction of the time an ion is present. This
is a perfectly good stochastic definition of concentration, but when you try to use it to relate concentration
to potential via the Poisson equation, a fundamental difficulty asserts itself. A channel containing, say, one
Ca2+ on average will be fluctuating in occupancy among
0, 1, or 2 ions (0, 5, and 10 M concentration). The
mathematics represents the channel as having a time-invariant potential equivalent to the average situation:
single Ca2+ occupancy. But this is a severe misrepresentation of the potential that a Ca2+ approaching an
empty channel, or a Ca2+ about to leave a doubly occupied channel, actually sees, and these events are often
rate determining for permeation. It is as if the I.R.S. applied to every taxpayer a uniform exemption calculated
for 2.6 children, the mean number of children per
American family. Described another way, a Ca2+ aspiring to enter an empty channel at a given moment is
treated by the electrodiffusion equations as though it
experiences the repulsive electric field that existed, say,
a microsecond before this moment, when the channel
had one ion in residence; since occupancy-dependent
changes in field are enormous and are established instantaneously, large errors in predicted behavior will
arise from using an average potential. Thus, while valid
for macroscopic objects and large, wide channels, the
mean-field assumption applied to physically small channels yields solutions to the electrodiffusion equations that
are mathematically chaste but physically debauched.
(3) PNP2 misrepresents calcium channel behavior.
The
single example in the literature of a continuum treatment of calcium channel behavior, PNP2 (Nonner and
Eisenberg, 1998), does not achieve the goal it sets for
itself. The analysis is very similar to that of the classical
macroscopic ion-exchange membrane, where electrodiffusion is well understood (Teorell, 1953
). Emphasized in the analysis is that only the channel current at a
fixed voltage, and not the conductance, is expected to
show AMFE. Nonner and Eisenberg (1998)
claim that
published calcium channel experiments have demonstrated an AMFE only in current at fixed voltage, and
that proponents of the standard view have merely assumed without evidence that the AMFE also applies to conductance, as chemical-kinetic theory says it must.
This claim, if correct, would be a deadly criticism of the
chemical-kinetic approach.
(4) Chemical kinetics preserves the basics.
As for the weaknesses of the chemical-kinetic view, they are certainly
prominent and well-appreciated (Cooper et al., 1985; Levitt, 1986
; Dani and Levitt, 1990
). It is impossible to
predict a priori what the absolute values of the rate
constants should be or how to relate rate constants to
transition-state free energies. Likewise, the use of Eyring-like exponential voltage dependence to the rate constants is theoretically unjustified and always leads to incorrect I-V curve shapes. And physical space inside of
channels is in fact continuous, not a lattice of sites.
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FOOTNOTES |
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Original version received 4 February 1999 and accepted version received 9 April 1999.
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REFERENCES |
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