From the Department of Physiology and Department of Anesthesiology, School of Medicine, University of California, Los Angeles, Los Angeles, California 90095
One measure of the voltage dependence of ion channel conductance is the amount of gating charge that moves during activation and vice versa. The limiting slope method, introduced by Almers (Almers, W. 1978. Rev. Physiol. Biochem. Pharmacol. 82:96-190), exploits the relationship of charge movement and voltage sensitivity, yielding a lower limit to the range of single channel gating charge displacement. In practice, the technique is plagued by low experimental resolution due to the requirement that the logarithmic voltage sensitivity of activation be measured at very low probabilities of opening. In addition, the linear sequential models to which the original theory was restricted needed to be expanded to accommodate the complexity of mechanisms available for the activation of channels. In this communication, we refine the theory by developing a relationship between the mean activation charge displacement (a measure of the voltage sensitivity of activation) and the gating charge displacement (the integral of gating current). We demonstrate that recording the equilibrium gating charge displacement as an adjunct to the limiting slope technique greatly improves accuracy under conditions where the plots of mean activation charge displacement and gross gating charge displacement versus voltage can be superimposed. We explore this relationship for a wide variety of channel models, which include those having a continuous density of states, nonsequential activation pathways, and subconductance states. We introduce new criteria for the appropriate use of the limiting slope procedure and provide a practical example of the theory applied to low resolution simulation data.
Key words: limiting slope; thermodynamics; Monte Carlo; gating current; voltage dependenceThe sine qua non of voltage-dependent ion channels is
their ability to alter ion permeability of membranes in
response to changes in the transmembrane potential.
Hodgkin and Huxley (1952) accounted for the voltage
sensitivity of Na+ and K+ conductance in the squid giant
axon by postulating charge movement between kinetically distinct states of hypothetical "activating particles". With the advances that have been made in the
biochemistry and molecular biology of ion channels
since then, we now recognize that the embodiment of
their "activating particle" resides in the ion channel itself, as a structural component, though the molecular
details of the voltage sensitive "gating" of the pore remains largely unknown. The first verification of gating
charge movement was by Armstrong and Bezanilla
(1973)
, who recorded nonlinear capacitive current
transients from tetrodotoxin (TTX) blocked Na channels in the squid axon. An important quantity that
characterizes the voltage sensitivity of the channel is
the range of gating charge displacement
q energetically linked to channel activation. The most significant
advancement in measuring
q that followed the work
of Hodgkin and Huxley was due to Almers (1978)
, who
treated the case of a linear sequence of discrete closed
states followed by a single open state (Almers' criterion, see Fig. 1 A). Almers stated that for such a model,
the following relation holds:
![]() |
(1) |
where Po is the fraction of time the channel is open at
equilibrium (open probability), V is the membrane potential in millivolts, and kT is the characteristic thermal
energy (Boltzmann constant times absolute temperature 25 milli-electron volts at room temperature). The experimental use of Eq. 1, often referred to as the
limiting slope procedure, has been used by many investigators to estimate total gating charge movement of activation (e.g., Stimers et al., 1985
; Liman et al., 1991
;
Papazian et al., 1991
; Schoppa et al., 1992
; Zagotta et
al., 1994
).
The Q/N technique is an alternative to the limiting
slope method and is a direct determination of gating
charge movement per channel. Independent measurements of the total charge Q and the number of channels N are made in a single preparation, and the ratio,
qQ/N, is then calculated. The first reliable value of
qQ/N was published by Schoppa et al. (1992)
, who recorded from a single patch containing Shaker K+ channels expressed in Xenopus oocytes. They obtained a
qQ/N of 12.3 eu (electronic unit of charge
1.602 × 10
19 Coulumbs)1 per channel, which has since been
verified by our lab and others (Aggarwal and MacKinnon, 1996
; Noceti et al., 1996
; Seoh et al., 1996
). A similar value was obtained from a mutant channel in which the hydrophobic residue Leu370, located in the putative
voltage sensing domain of Shaker (the S4 transmembrane segment), was replaced with valine. Their finding of no change in
qQ/N in the mutant compared to
the unmutated channel was expected since the net
charge of the protein was unchanged. However, use of
the limiting slope procedure on the two channels yielded values that were significantly lower than
qQ/N:
9.5 eu for the unmutated channel and 5.5 eu for the
mutant (Schoppa et al., 1992
). The discrepancy between the results from the Q/N technique and the limiting slope procedure might be explained by the existence of gating charge movement which is dissociated
from the activation process, making no contribution to
voltage sensitivity of opening. This noncontributing
charge component may take on two forms: gating charge movement occurring after the channel has
opened (latent charge movement) and charge movement that is independent of the activation process (peripheral charge movement). The limiting slope procedure, in contrast to the Q/N technique, measures only the range of charge movement which is energetically
linked to channel opening (activation charge movement), ignoring the latent and peripheral charge
movements. In their conclusion, Schoppa et al. were
careful to point out that, despite the possibility that latent and/or peripheral charge movement exists in
Shaker, the most likely source for the reduced limiting
slope values was lack of experimental resolution. Indeed, there has been sharp criticism of the reliability of
the limiting slope procedure from various authors
(Anderson and Koeppe, 1992; Bezanilla and Stefani, 1994
; Zagotta et al., 1994
). The chief reasons for the
skepticism in the validity of the technique are: (a) the
inability to accurately determine where the limiting
value of the slope occurs on the voltage axis and (b) the
difficulty of measuring the low values Po required to
reach that value (typically <10
3).
In this paper, we address the problem of poor resolution in the limiting slope procedure by introducing a
measure of the progress of activation named the activation charge displacement, qa. Plotting mean activation
charge displacement qa
versus voltage is a more reliable way of estimating limiting slope, and, furthermore,
there is a relation between
qa
and mean gating charge
displacement,
q
, which makes the additional measurement of gating currents a valuable tool in interpreting
limiting slope data (for example, the gating current recordings by Schoppa et al. (1992)
can be used to predict the amount by which they underestimated
q with the limiting slope technique). We develop the theory
to include models containing an arbitrary number and
arrangement of states (including continuum models)
and allow for multiple open states with different conductances. We follow with a series of examples illustrating the theory, ending with a sample analysis of low resolution simulated data. We conclude with a summary of
results and an approach for applying the limiting slope
method in practice.
Theory
Nature of the model.We preface the discussion of specific models by stating assumptions regarding their
general nature. We assume that after a sudden step in
membrane potential, the occupancy distribution of the
channel propagates through a network of conformational states representing the activation pathway. The
reaction coordinate is the gating charge displacement
(q). To place the quantity q in its proper context, we
note that the mean gating current for N channels Ig
is
Nd
q
/dt and the plot of N
q
vs. voltage is the Q -V
curve. There may be more than one state for a given
value of q, which allows for loops and parallel pathways
in the activation sequence (see Fig. 1 B). Since the
mean
q
is an extensive state variable, q has a specific
value for every state in the model. In addition, states
are assigned a potential of mean force F(q,
,V) and a
fractional conductance f(q,
). Both are assumed to be
defined by the value of q and other degrees of freedom
indicated by
, and, in the case of F, also the membrane
potential V. In terms of the variable
, we will be pri-marily concerned with degrees of freedom produced
by states that are degenerate with respect to q. An example is a branching of the activation path, producing a duplication of states. Accordingly, in some cases it is
useful to include a degeneracy factor
, which for a
nondegenerate state has the value of one. We assume
for the purpose of derivations that all gating charge is
energetically coupled to the activation process (essential charge), since only essential charge contributes to
the voltage sensitivity of activation. We shall consider
the effect of peripheral charge in Fig. 4. The fractional
conductance is defined as the ratio of the state conductance g(q) to the maximal value at the given potential,
go. We assume a linear voltage-dependence of F(q,V)
(e.g., Tsien and Noble, 1969
):
![]() |
(2) |
where G(q) = F(q,0) and depends on thermodynamic
variables other than voltage such as temperature and
pressure. By assigning a set of variables to each state
rather than a difference of values between states (as is
often done in describing kinetic models), we assure
that the model automatically obeys detailed balance and conservation of charge density around any loops in
the kinetic model. In this paper we consider only thermodynamic quantities, so we ignore the nature of pathways between states that determine kinetics. We require
only that equilibration occurs between states on a time
scale which is faster than that of our method of measurement (see Fig. 9 and corresponding text).
Definition of terms.
The word activation refers to the
process of opening the channel pore. Technically,
then, a blocked channel does not activate, but in such
cases we alter the definition to include any conformational changes in the channel protein that produces a
change in gating charge displacement. Without loss in
generality, we assume that conducting channels activate with increasing membrane potential, since most
classes of ion channels conform to this behavior. However, the theory outlined below is easily applied to a
channel that opens with hyperpolarization by making
trivial changes in sign. We formally define gating
charge displacement q as any nonlinear capacitive
charge movement in a single channel, regardless of whether it makes an energetic contribution to opening
(activation coupling) or not. The variable q serves as a
convenient reaction coordinate of activation. The value
of q in the limit V
is referenced to zero. The
maximum value of q at depolarizing potentials is by definition the total gating charge movement per channel
q, which is measurable experimentally with the Q/N
procedure, i.e.
q =
qQ/N. The component of q which
is activation coupled is the essential charge displacement (qe) and the remainder is the peripheral charge
displacement (qp). The latter does not contribute to
the voltage sensitivity of the channel. The open probability (Po) is defined for a population of channels as the
ratio of the mean conductance to the maximum conductance. We introduce the activation potential (Wa =
kT ln[Po]), which is a measure of the electrical energy
needed to open the channel. Finally, we define the activation charge displacement (qa) through its mean value, which equals the negative gradient of the activation potential:
qa
=
dWa/dV. Note that if Po is multiplied by a constant it does not affect the value of
qa
, i.e.,
![]() |
(3) |
This is useful experimentally since the value of Po obtained from analysis of macroscopic ionic current traces is usually defined only within a multiplicative constant. In the next two sections we will derive a relationship between the mean activation charge and gating charge displacements that will serve as the foundation for developing a practical approach in measuring essential gating charge movement.
Finite number of states.We start with models that contain a finite number of discrete states, by which we
mean that the time spent by the channel during the
transition from one state to another is negligible compared to the dwell times in the states themselves. For
the purpose of this derivation, and the one following
which deals with continuum models, all gating charge
movement is essential (i.e., q = qe). Each state i is completely specified by the state variables (qi, Gi, fi) as well
as an optional degeneracy factor i. Subconductance
states are produced when 0 < fi < 1. The open probability of a single channel at equilibrium is Po =
g
/go, where go is the open pore conductance, and
g
is the
average conductance.
![]() |
(4) |
The pi are the equilibrium state probabilities which are found using the normalized Boltzmann distribution:
![]() |
(5) |
The denominator in Eq. 5 has the form of a coarse-grained partition function. From Eqs. 4 and 5 we obtain an expression for the mean open probability:
![]() |
(6) |
Then, using the linear relation of Eq. 2, which for a discrete model has the form:
![]() |
(7) |
we evaluate the derivative of ln(Po) using Eqs. 6 and 7:
![]() |
(8) |
The two terms on the right side of Eq. 8 are averages
over q. The first term uses a reduced partition function
where each term is weighted by the fractional conductance of the corresponding state. The second term,
which sums over all states, is the average gating charge
displacement q
, which, in the absence of peripheral charge movement, is proportional to the experimentally derived Q-V curve. We rewrite Eq. 8 as:
![]() |
(9) |
The second term on the right of Eq. 9, which sums over
states that have at least partial conductance, represents
the mean latent charge displacement (ql
). In the expression for
ql
, the quantity that is being averaged
over is
q
q, which has its origin where the gating
charge displacement reaches its largest value (i.e., q =
q) and increases with decreasing value of q. Thus, for
example, the open state in the model of Fig. 1 A, although it conducts (f
0), contributes nothing towards the value of
ql
because it lies to the right of all
the other states, at the value of maximum gating charge
displacement.
Finally, after moving ql
to the left side of the equation, we have:
![]() |
(10) |
The relationship between the activation and essential
charge displacements is now clear. The sum of the activation and latent charge displacement curves (qa
+
ql
) superimposes onto that of the mean gating charge
displacement
q
by simply inverting and shifting upwards by
q. In the event that
ql
vanishes or remains
constant for all potentials,
qa
can be superimposed onto
q
. In such cases, we say that the channel is saturated with activation charge, since the total range of activation charge displacement (
qa) matches the charge
movement per channel,
q. In a saturated channel,
knowledge of the shape of
q
versus voltage, obtainable
from a Q-V measurement, allows one to predict the
range in potentials where
qa
reaches its limiting value.
A model that satisfies Almers' criterion (Fig. 1 A) is obviously saturated. However, so are kinetic models with a
more complicated arrangement (loops and parallel
pathways) of closed states that converge onto a single
or degenerate set of open state(s). The requirement
for saturation is twofold: (a) the activation sequence
has only one open state, or, if there is a cluster of open
states, they must all have the same value of q ; (b) there
is no peripheral charge movement. If the first requirement is satisfied, but not the second, then we can speak
of saturation of the essential activation sequence. However, the presence of peripheral charge movement may
contaminate the shape of the Q -V, making it less useful
for limiting slope analysis. In any case, Eq. (10) can be
made generally valid even when there is peripheral
charge movement by replacing q with qe.
A similar analysis can be applied to a continuous system. Here, the partition function becomes an integral over a continuum of states. We maintain the condition that the only variable to be summed over is the gating charge q. The meaning of f(q) in a continuum model may be best thought of as a conditional probability relating q to the likelihood that the pore is open. With these considerations in mind, the derivation follows exactly that of the discrete state model. The continuum analogue of Eq. 6 is:
![]() |
(11) |
From Eq. 11, we immediately obtain the counterpart to Eq. 9:
![]() |
(12) |
As we did in the case of discrete models, we identify the
second term on the right of Eq. 12 as the latent charge
position ql
. Obviously,
ql
cannot be made to vanish
or even maintain a constant value with respect to voltage without rendering the channel permanently
closed. This is because, in order for the channel to conduct at all, f(q) must have a nonvanishing value across
some range of the continuous variable q. In other
words, it is meaningless to speak of a single open state
in a continuum model, and so the channel will never
be saturated with activation charge. However, one can
come close (within experimental resolution) by allowing the channel to be open only for a range of values of
q near its maximum value (e.g., see Fig. 7).
Numerical Computation of Activation Curves
Activation curves of discrete state models were calculated using
Eqs. 5-7, 9, and 10 from the array of state variables (qi, Gi, fi)
and degeneracy factors i that define a particular model. Continuum models were evaluated using the same algorithm by increasing the number of "states" to 2,000. It was found that as few as 50 states were needed for the numerical solution to converge within 3% of the continuum solution.
Ramp Simulations
Simulation of ramp experiments were performed with a Monte
Carlo algorithm. The long period (>2 × 1018) random number
generator of L'Ecuyer with Bays-Durham shuffle (Press et al.,
1992) was used to generate random numbers rn with values between 0 and 1. Waiting times
ab of transitions from state a to
state b were calculated from transformation of rn. Unlike the situation where the membrane potential is held constant over the
time course of simulation, the distribution of waiting times for an
applied voltage ramp is not exponential since the value of the
unidirectional rate constant
ab changes with time. According to
the theory of failure rate analysis (Papoulis, 1991
), the probability density function f(
ab) of the waiting time is:
![]() |
(13) |
We used an Arrhenius expression for the rate constant: ab(t) =
oab exp{qabxabV(t)/kT }. V(t) is given by Vo + mt, where Vo is the
ramp potential at time zero (i.e., the time at which the channel entered state a) and m is the ramp speed in units of voltage per unit time. The variable qab is the transition gating charge movement, and xab is the fractional position of the activation barrier
between states a and b. By integrating the expression for conservation of probability, f(
ab)d
ab = f(rn)drn (Papoulis, 1991
), we
obtain the following relationship between the random variables
ab and rn:
![]() |
(14) |
In the event of competing transitions out of state a (multiple b
states possible), selection was made by generating a waiting time
for each one and choosing the smallest value. The single channel
gating current record ig(t) was constructed by binning transition
impulses weighted by qab onto the discretized time axis. For the
single channel ionic current record i(t), we used the formula i = g(V Vx), where g is the conductance of the resident state, and
Vx is the reversal potential of the permeant ion(s). The single
channel traces were digitally filtered using a Gaussian filter with
cutoff frequency fcand delay 1/2fc (which approximates a multipole Bessel filter; see Crouzy and Sigworth, 1993
) and accumulated
to produce macroscopic currents. To obtain a quantity proportional to the open probability, the mean ionic current was divided by the driving force, (V + 100) mV, for all ramp voltages V.
Eq. 3 was used to obtain the sample estimate of the mean activation charge displacement, q_ a, except that the expression {d Po/
dV }/Po was evaluated instead of the equivalent d(lnPo)/dV in order to reduce round-off error. The ramp Q -V curve was obtained
from the integral of the mean gating current. The software used
in numerical calculations and the Monte Carlo simulation was
written in the C programming language and was run on a 60 mHz Pentium processor.
We now demonstrate the theory with a series of numerical examples illustrated by Figs. 2-8. Starting
with the simple discrete state models (Figs. 2-4), one or more schematic state diagrams are shown in
increasing value of q on the left side of each figure. Following standard kinetic notation, allowable transitions
between discrete states are indicated by a line labeled
with the corresponding gating charge movement. This
is done mainly because a kinetic model is more familiar
looking than a random assortment of disconnected states, but it must be kept in mind that the nature and
arrangement of connections between states are irrelevant for calculations of equilibrium quantities (such as
Po and qa
), so long as a path exists which connects all
states, and the time required for equilibration is much
shorter than the length of the experiment. It is easy to see
that the theoretical equations derived earlier are in accordance with this general rule, and they hold for models with continuous as well as discrete densities of states.
For all models we considered, whether discrete or
continuous, we set the total essential charge movement
qe equal to 4 eu to facilitate comparisons between
models. Associated with each example are numerically
derived plots of the equilibrium values of the gating
charge displacement Q (normalized to one, as is normally done in practice) and the mean open probability
Po (actual value). The mean activation charge displacement
qa
and the value of
q -
q
, also numerically derived, are featured in Fig. 2 C. Recall that the quantities
Q and
q
are related by a factor of N, so they both have
the same shape when normalized. Although we sometimes appear to use them interchangeably, we use Q
when referring to experimental gating charge measurements where N is not known, whereas
q
is the mean of
the single channel gating charge displacement, which
is not directly measurable.
The discrete state models in Figs. 2-4 share the following features: the zero voltage potential Giis the same for each state, though this is not explicitly shown in the figures. This tends to center the activation curves around V = 0. The closed states (f = 0) are labeled "C." Similarly, "O" stands for open (f = 1).
Two-state Model
Model 2a (Fig. 2) is a simple two state model, in which
the total charge movement of 4 eu occurs in a single
transition. A property of any two-state model is that all
equilibrium properties lie on the same Boltzmann
curve after normalization (Lumry et al., 1966). Thus
the Po-V and Q -V curves superimpose, and, because the
two states are equipotential, the plots are centered
around V = 0. Since there is only one open state and
no peripheral charge movement, the model represents
a saturated channel (see Theory). It is also the simplest
example which satisfies Almers' criterion. Fig. 2 B
shows the superposition of the plots of
qa
and
q -
q
, which demonstrates graphically that the limiting value
of
qa
is reached only at potentials where the Q -V curve
is close to saturating.
Three-state Nonsequential Models
We relax the restriction to linear sequential models
with models b and c of Fig. 2. In both cases we see a parallel movement of 4 eu that either converges to a single
open state (model 2b) or diverges into two open states
(model 2c). The total range of gating charge movement for each model is not 8 eu, but rather 4 eu, since
state probability is split equally among the two pathways. As a rule, the maximal charge movement of a
model containing communicating parallel paths is simply the largest net charge movement of any of the possible pathways. Both models 2b and 2c represent saturated channels. In the case of model 2c, although there
are two open states, they are degenerate with respect to
q ((open) = 2), producing no latent charge movement. The addition of an extra state merely shifts each
Boltzmann plot to one side, without a change in the steepness of the curve at midpoint. This is because a
state degeneracy produces an increase in entropy in
the respective state. For example, in model 2b, setting
V = 0 distributes 2/3 of the system probability to the
two closed states, which lowers
q
and Po. The same
reasoning applied to other potentials explains why the
equilibrium curves for model 2b are shifted to the right
on the voltage axis. Similarly, the degenerate open
state of model 2c produces a shift to the left.
As mentioned earlier, a different arrangement of allowable transitions in a particular model does not change the equilibrium values of state variables. For example, in model 2c, a line could be drawn between the two open states, and one of the existing lines from the closed state deleted, and the results shown in Fig. 2, A and B, would be unchanged. However, the kinetics of activation, which are not of concern here, but are an important determinant of channel function, would be drastically altered by such a change.
Three-state Linear Sequential Models
In channels with three or more nondegenerate states,
we do not expect activation curves representing different thermodynamic quantities to coincide. By inserting
an extra closed state into the activation pathway of
model 2a, we displace the Po-V to the right of center
and slightly broaden the Q -V (Fig. 3 A). However, this
model remains saturated since it satisfies Almers' criterion of a linear sequence of closed states followed by
one open state. Thus, qa
and
q -
q
superimpose (Fig.
3 B), and the limiting slope procedure could be accurately used to measure total charge movement per
channel.
Switching the positions of the open state and its
neighboring closed state results in model 3b, which
now has a closed state at the end of the activation pathway. The Q -V curve is unchanged but the Po-V now approaches zero at positive as well as negative potentials.
An example of this behavior may be seen in a channel experiencing voltage-dependent block (Bezanilla and
Armstrong, 1972). Here we have the first example in
which the open state appears in the middle of the activation sequence, and so we expect
ql
to have a finite
(but constant) value. Indeed, we see that the plot of
qa
vs. V is displaced downward by a constant value of 2 eu
(Fig. 3 D). Consequently,
qa
becomes negative at depolarizing potentials, which comes about because the
open probability decreases with increasing positive potential. Note that the total range of
qa
, which we will
refer to as
qa
, is still 4 eu, though the limiting slope
value in the hyperpolarizing direction (which we will
refer to as
qa+, see Figs. 1 B and 3 D) is only 2 eu and
would by itself give an underestimate of
q. Thus, in order to correctly estimate the entire range of gating
charge movement in this case, one would also need to
apply the limiting slope procedure to depolarized potentials to obtain the negative component of the mean
activation charge displacement (
qa
). Model 3b remains an example of a saturated channel, since there is
only one open state and no peripheral charge movement.
In the last model of Fig. 3 (model 3c), there are two
open states in sequence at the end of activation. The Q -V
curve is still unchanged, but now the Po-V is shifted to
the left of center. This is due to gating charge movement between open states, leading to a positive value
for ql
, which is now a function of voltage. This channel is not saturated. The range in mean activation
charge displacement
qa
reflects only the initial transition from the closed to the first open state. Hence the
limiting slope procedure applied to this particular
model underestimates
q by one half its value. In contrast, the Q/N procedure will give the correct value of
q
for this model since it measures all essential charge movement. Although model 3c is displayed in Fig. 3 as a linear sequential model, the same outcome arises from a
branching model in which a closed state makes transitions
to two open states with respective gating charge movements, 2 eu and 4 eu (compare to model 2c, but with unequal charge movement per leg). Branching models, although useful in studying kinetics, are no different than
linear sequential models in their equilibrium properties.
Models with Peripheral Charge
In Fig. 4 we demonstrate the effects of charge movement energetically uncoupled to the main activation
pathway (peripheral charge movement). Model 4a represents a saturated channel with a looped network of
four closed states converging to an open state. The
range in gating charge displacement q is 4 eu and is
entirely essential. By severing the links between the upper and lower charge relay systems, we obtain model
4b, which has two independent charge systems, with
only the lower one linked to activation. If one were to
measure the total range of
q
in this fragmented
model, one would find it to be 6 eu, not 4 (if this is not
immediately obvious, consider that we routinely sum
the charge movements of N independent channels to
obtain the gating current; also, the sum of all state
probabilities in model 4b gives two, rather than one as
in model 4a). The range in essential charge movement
is unchanged (
qe = 4 eu), but now, due to the additional peripheral charge movement
qp of 2 eu, there is
a mismatch between the limiting value of
qa
, which
correctly estimates
qe, and that of
q =
qQ/N, the
value obtained if one were using the Q/N procedure. An example of peripheral charge movement present in
neuronal
1E Ca2+ channels has recently been demonstrated (Noceti et al., 1996
)
in absence of the
2a regulatory subunit, there appears to be a parallel charge system that, on the time scale of the observations, is kinetically isolated from the main activation sequence of the channel, leading to a
qQ/N value of 14 eu, but a value
of only 9 eu for
qe. With addition of the
2a subunit to
the system, the Q/N value dropped to the value of the
limiting slope, which remained at 9 eu, suggesting that
there was a lowering of the energy barrier which had
separated the 5 eu of peripheral charge movement
from the remainder, resulting in faster equilibration between the two charge systems. The
2a subunit has no
significant effect (<10 mV) on the half activation potential of the mean gating charge displacement or the
initial rise of the open probability (Olcese et al., 1996
),
meaning that it is unlikely that the limiting slope value
before addition of subunit was being selectively underestimated due to a larger voltage separation between
the Q -V and the Po-V plots.
Models with Continuous Distribution of States
In Figs. 5-7, we shift our focus to continuous systems, where we will confirm some of the observations made above with discrete state systems, and encounter some new behavior. In these figures, the left side shows the potential profile G(q) as well as the fractional conductance f(q).
We begin with a flat free energy landscape for Fig. 5,
which is implicitly the same assumption we have been
making for all of the discrete state models described so
far. The charge density is confined to within the
boundaries q = 0 and q = q = 4 eu. The function f(q)
is zero (corresponding to closed states) for lower values
of q, and then undergoes an abrupt change at q = a
q
to a value of one (open state), where it remains until it reaches the upper boundary of q. Thus, this model has
a varying amount of saturation which increases with the
value of a. Of the discrete models we have considered
so far, the most analogous to this model would be the
model 3c of Fig. 3, since both are linear sequential (no
degeneracies) with a finite amount of charge movement between open states. We concluded earlier in the
discussion of Fig. 3 that the limiting value of
qa
could
never reach
q due to latent charge movement. This is
shown to be the case as well with the continuum model,
where it is apparent from Fig. 5 B that
qa
reaches the
maximum value of a
q. Note also the crossing of Po-V
with the normalized Q -V in Fig. 5 A, indicating the presence of multiple open states.
In Fig. 6 we demonstrate the effect of broadening
f(q) around the gating charge position q = 0.9q. We
notice immediately a striking difference between the
plots of
qa
in Fig. 6 and those we have seen so far. The
value of
qa
plateaus without reaching the maximum
value
q, and then appears to shrink to zero with increasing negative potential. The effect is more pronounced for broad distributions of f(q) than for very
sharp ones. It would be quite interesting to see experimental evidence of the fall-off of
qa
at negative potentials, since it would support the notion of subconductance states along the activation pathway. Incidently, a
similar effect occurs in a discrete model similar to
model 3c, if the last transition (between open states)
moves significantly more charge than the first, leading
to an early plateau in
qa
, which can be significantly
larger in magnitude than the eventual limiting value
seen at hyperpolarized potentials.
We choose for our final continuum model (Fig. 7)
one which approximates a two-state model. Instead of a
flat potential profile, we now introduce an inverted parabolic peak 4 kT high with a slight flattening at the edges
and bounded on the q-axis at q = 0 and q = q = 4. The channel is fully open for q values greater then 0.9 ·
q. The result is a slightly broader equilibrium curve
than for the true two state model (Fig. 2), and the Po-V
curve is a bit shifted to the right of the Q -V. However,
the
qa
and
q-
q
plots coincide enough to make the
difference in their limiting values (as could be obtained from limiting slope and Q/N methods, respectively) difficult to resolve experimentally, if this were a
real channel.
Multi-state Model with Subconductance States
Our concluding model (Fig. 8) shows that bizarre behavior of the equilibrium curves is not limited to continuum models. This model is discrete with five states.
In contrast to the earlier discrete models we considered, we assign a different potential to each state, and
we add subconductance states. Due to the small number of states, the plots of Po and Q follow a somewhat
erratic pathway. In a continuum model with a similar
free energy landscape and the same approximate increase in f(q), the Po vs. V and Q-V would broaden out
and appear smoother. The voltage dependence of qa
shows some interesting behavior, increasing from zero at positive potentials to a local peak of about 1/4
q,
then dipping nearly to zero at more negative potentials, only to rise again to a maximum value slightly
higher than that of the initial peak. Clearly, a plateau in
the value of
qa
may be misleading since it does not always represent the limiting slope value. The only way to
be sure one has reached the final plateau is through
comparison with the Q-V, which must be saturated in
the range of potentials from which the limiting slope is
recorded.
With the recent advances in molecular biology it has
become possible to neutralize amino acids that are suspected to be voltage-sensing residues in ion channels
and study the effects on gating. An important measure
of channel responsiveness to voltage is the total charge movement of activation q. A significant reduction of
q has been found after removal of charged residues in
the S2 and S4 transmembrane domains of the Shaker
potassium channel, indicating that these residues play
an active role as components of the gating apparatus of
the channel (Aggarwal and MacKinnon, 1996
; Seoh et
al., 1996
). Almers (1978)
summarized two methods by
which one can obtain
q : (a) the Q/N procedure,
which requires that one measure the maximum range
of gating charge movement and the number of channels in the same preparation, and (b) the limiting value
of the logarithmic potential sensitivity (similar to our
mean activation charge displacement
qa
, except that
Po/[1
Po] replaces Po as the argument of the logarithmic function), which equals the essential component
of
q in the absence of latent charge movement.
Using a statistical mechanics approach, we have expanded on Almers' theory by explicitly expressing
qa
= kTd(lnPo)/dV as a function of gating charge displacement, and by generalizing the result to include
parallel activation pathways, multiple open states, subconductance states, and continuum models. The outcome of the derivation can be summarized by the following relation:
![]() |
(15) |
.
In cases where the ion channel is saturated and ql
is
zero, Eq. 15 reduces to Almers' original expression for
the limiting slope (Eq. 1) at hyperpolarizing potentials
where the Q -V curve saturates (
q
= 0).
A major obstacle in correctly estimating q using the
limiting slope method is the accurate determination of
qa
at potentials where the open probability is vanishingly small. This problem has been discussed by several
authors (Andersen and Koeppe, 1992
; Bezanilla and
Stefani, 1994
; Zagotta et al., 1994
), but no criterion has
been given to accept or reject the estimate except to
take the value as a lower limit of the actual
q. A useful
guideline is provided by plotting
qa
as a function of V
or Po (Zagotta et al., 1994
) where it is possible to visualize whether the estimate shows signs of reaching a limiting value. Such limiting values have been observed in a
few cases through single channel analysis in sodium
channels (Hirschberg et al., 1996
) or with macroscopic currents in neutralization mutants of Shaker K+ channels
(Noceti et al. 1996
; Seoh et al., 1996
). Often, however, poor experimental resolution prohibits the measurement of a sufficiently low value of Po, making a prediction of where the value of
qa
saturates extremely useful. This is possible, based on Eq. (16), if the Q-V curve
is known and the channel is saturated. What needs to
be done is to invert the Q -V curve and scale it so that it superimposes onto the incomplete plot of
qa
vs. V. It is
important that a large enough range of voltage is used
to insure good overlap between the two curves so that
proper scaling is achieved. The value of
q is then read
off from the scaled Q -V curve (Seoh et al., 1996
).
An example of estimating q with limited resolution
data is shown in Fig. 9. The data was obtained from a
Monte Carlo simulation of gating and ionic currents
following a ramp voltage protocol (see METHODS). The
model is shown in the lower right corner of Fig. 9 and
has five closed states followed by an open state. All five
transitions move an identical amount of gating charge
for a total of
q = 10 eu. The ramp protocol is a useful
experimental tool since it allows measurements of the
signal for a large number of voltages in a single record.
This is important when one is required to differentiate
a measured quantity with respect to voltage, as we do
when evaluating Eq. 3. The drawback in the method is
that the ramp speed must be infinitely slow (quasi-static) for the channel to remain close to equilibrium
during the ramp. Nevertheless, the alternative procedure of obtaining equilibrium values of ionic and gating currents from an activation series of pulses can be
prohibitively difficult if one desires very fine resolution
in voltage, making the ramp protocol the method of choice in certain cases (e.g., the presence of rundown
in the preparation). In our simulation the midpoint of
the integral of the mean gating current was shifted
+2.5 mV with respect to the midpoint of the numerically derived Q -V, indicating that the quasistatic condition was not rigorously adhered to. This error will be
evident in the final result. Fig. 9 B contains three plots. The estimate q- a and the ramp Q -V were calculated from
the averaged ionic and gating currents plotted in Fig. 9
A. The numerical Q -V was derived from the model (experimentally, it can be accurately obtained from a voltage series of long pulses, Stefani et al., 1994
). It is apparent that the scatter in q- a increases significantly with
low Po due to the increasing rarity of open events. The
minimum value of Po obtained in the plot of q- a was 5 × 10
3, which is insufficient to reach the expected limiting value of 10 eu. Because the model satisfies Almers'
criterion, we can use the inverted Q -V curves to "predict" the value of
q by scaling them so they superimpose onto the foot of the plot of q- a vs. V. We see that,
due to the insufficiently slow ramp speed, the predictions from the ramp Q -V curve and true Q -V curve are both 10% in error, with the real value of
q lying approximately in between the two values. Thus, experimentally, it is important to use as slow a ramp speed as
possible. This procedure was applied to the Shaker B K+
channel and charge neutralization mutants by Seoh et
al. (1996)
. Their results show that for many of the mutations on the S2 and S4 putative transmembrane domains, the inverted and scaled Q -V curve superimposes
on the q- a vs V curve. The predicted value of the limiting
slope matched the value of
q obtained by the Q/N
procedure, providing an internal check for consistency
and at the same time showing that all charge that moves
in Shaker is energetically coupled to the open state.
We conclude our discussion by suggesting an approach for measuring total gating charge movement in
a generalized ion channel. A useful analysis should separate the gating charge into its components of peripheral, latent, and activation charge displacements. Latent charge movement comes about when gating
charge moves during a transition between two or more
discrete open or subconductance states, or along a continuum. Analysis of single channel records may uncover kinetically distinct discrete open states and/or
subconductance states. Rate constants between successive open states may be voltage dependent, suggesting
latent charge movement. Another useful test for the
presence of latent charge movement in a channel that
opens with depolarization is performed by comparing
Po-V and Q -V traces gathered from macroscopic currents. If the two plots cross, or if the Po is shifted to the
left of the Q -V, latent charge movement exists and the
limiting slope procedure will underestimate the range
of essential charge movement (e.g., Fig. 3 F ). Oftentimes, noise analysis of macroscopic ionic currents is
easier to carry out than single channel analysis. In such
cases, a nonparabolic variance vs. mean plot of nonstationary ionic currents (Steffan and Heinemann, 1996)
points at the presence of subconductance states. Also,
autocovariance analysis of stationary macroscopic ionic
currents can detect multiple open states (Sigworth,
1981
). If the channel has latent charge movement, it is
not saturated with activation charge, and the range of
activation charge
qa is necessarily less than the range
of essential charge movement
qe. It must be kept in
mind that the complete range in
qa
spans from maximally negative to maximally positive potentials, and in
cases where the channel closes with extreme potential
on either side of the voltage axis, the limiting slope procedure must be carried out at positive as well as negative potentials (e.g., Fig. 3 D). In a saturated channel,
the limiting slope procedure can be used to estimate
qe, but lack of experimental resolution often makes it
difficult to determine at which potential a limiting value of the activation charge displacement is reached.
This is where the Q -V plot becomes invaluable, since
the Q -V and the
qa
-V plots plateau at the same voltage
range in saturated channels. The value of Q is easier to
measure than
qa
(assuming ionic currents can be effectively eliminated) near the region of limiting slope,
since the relation of Q with gating charge displacement is linear, whereas that of
qa
to Po is obtained through
differentiation, which amplifies the measurement error. Thus, the Q -V is a much more sensitive tool at determining the voltage at which
qa
will saturate and can
even be used to predict the value of
qe from low-resolution data (Fig. 9). There may be instances when a Q -V is unobtainable (inadequate expression of channels, incomplete blocking of the pore, etc.). In such cases, an
alternate method (Cole-Moore shift) for determining
whether
qa
has reached its final value may prove useful as a last resort. The Cole-Moore shift is defined as
the time delay of opening of the channel as a function
of holding potential (Cole and Moore, 1960
; Stefani et
al., 1994
)). At those potentials where
qa
has reached
saturation, the delay should also have reached its maximum value. It should be kept in mind, however, that
the Cole-Moore shift is a kinetic measurement, even
though its voltage dependence arises from the equilibrium probability distribution at the holding potential. Thus, unlike the Q -V, the plot of opening delay vs. voltage has no functional relationship with
qa
vs. voltage.
Nevertheless, the initial probability distribution does
not change beyond the voltage at which the gating
charge saturates, so in principle the Cole-Moore shift, if
accurately measured, could serve as a poor man's Q -V
in limiting slope experiments.
A separate issue is whether or not there is contamination of the gating charge movement with a peripheral
(independent) charge system. Peripheral charge movement shows up in the Q/N estimate of total gating
charge movement, so qQ/N will be larger than
qe in
the presence of peripheral charge. Combining this result and the earlier one, which set the upper limit of
qa as
qe, we obtain the inequality:
![]() |
(16) |
If the values of q a and
qQ/N are the same, it is a good
indicator that there is neither latent nor peripheral
charge movement, and the estimate of
qe is an accurate
one. If both latent and peripheral charge are associated
with the same channel, it may be difficult to determine
the contribution of each. In such a case, kinetic modeling might be required in order to estimate the amount
of charge movement between open states through the voltage dependence of the transition rate constants.
In summary, when measuring qe, the range in essential charge movement in a single channel, it is desirable
to obtain the full activation curves of the open probability and gating charge displacement. For channels without
latent charge movement, the limiting slope procedure,
in conjunction with information from the Q-V plot, will
produce an accurate estimate of
qe. In the general
case, the value of
qe will lie between the range of activation charge displacement,
qa, and the total charge
movement per channel,
qQ/N, obtained from the limiting slope method and Q/N procedure, respectively.
Original version received 21 June 1996 and accepted version received 16 September 1996.
Address correspondence to Dr. F. Bezanilla or D. Sigg, Department of Physiology, UCLA School of Medicine, Los Angeles, CA 90095. FAX: 310-794-9612; E-mail: pancho{at}cvmail.anes.ucla.edu
We thank Drs. Dorine Starace, David E. Patton, Francesca Noceti, and Enrico Stefani for reading and commenting on the manuscript.
This work was supported by United States Public Health Service grant GM30376 and a Dissertation Fellowship from the Graduate Division at UCLA to D. Sigg.
eu, electronic unit of charge 1.602 × 10-19 Coulombs.