From the Department of Molecular and Cellular Physiology, and Howard Hughes Medical Institute, Stanford University, Stanford, California 94305
Large-conductance Ca-activated potassium channels (BK channels) are uniquely sensitive to both membrane potential and intracellular Ca2+. Recent work has demonstrated that in the gating of these channels there are voltage-sensitive steps that are separate from Ca2+ binding steps. Based on this result and the macroscopic steady state and kinetic properties of the cloned BK channel mslo, we have recently proposed a general kinetic scheme to describe the interaction between voltage and Ca2+ in the gating of the mslo channel (Cui, J., D.H. Cox, and R.W. Aldrich. 1997. J. Gen. Physiol. In press.). This scheme supposes that the channel exists in two main conformations, closed and open. The conformational change between closed and open is voltage dependent. Ca2+ binds to both the closed and open conformations, but on average binds more tightly to the open conformation and thereby promotes channel opening. Here we describe the basic properties of models of this form and test their ability to mimic mslo macroscopic steady state and kinetic behavior. The simplest form of this scheme corresponds to a voltage-dependent version of the Monod-Wyman-Changeux (MWC) model of allosteric proteins. The success of voltage-dependent MWC models in describing many aspects of mslo gating suggests that these channels may share a common molecular mechanism with other allosteric proteins whose behaviors have been modeled using the MWC formalism. We also demonstrate how this scheme can arise as a simplification of a more complex scheme that is based on the premise that the channel is a homotetramer with a single Ca2+ binding site and a single voltage sensor in each subunit. Aspects of the mslo data not well fitted by the simplified scheme will likely be better accounted for by this more general scheme. The kinetic schemes discussed in this paper may be useful in interpreting the effects of BK channel modifications or mutations.
Key words: potassium channel; BK channel; Monod-Wyman-Changeux model; relaxation kinetics; msloLarge conductance Ca-activated potassium channels
(BK channels) sense changes in both intracellular Ca2+
concentration and membrane voltage. They therefore
can act as a point of communication between cellular
processes that involve these two common means of signaling. Although progress has been made in understanding the Ca-dependent properties of BK channel gating (McManus and Magleby, 1991; for review see
McManus, 1991
), the relationship between the mechanism by which Ca2+ activates the channel, and that by
which voltage activates the channel is less clear. To understand this relationship further, we have recently examined the macroscopic steady state and kinetic properties of the BK channel clone mslo over a wide range of
Ca2+ concentrations and membrane voltages (Cox et
al., 1997
; Cui et al., 1997
). From these studies, and
work from other laboratories, several conclusions can
be drawn. (a) There are charges intrinsic to the channel protein that sense changes in the transmembrane electric field, and thereby confer voltage dependence
on mslo gating (Methfessel and Boheim, 1982
; Blair and
Dionne, 1985
; Pallotta, 1985
; Singer and Walsh, 1987
;
Wei et al., 1994
; Meera et al., 1996
; Cui et al., 1997
). (b)
The mslo channel can be nearly maximally activated
without binding Ca2+ (Cui et al., 1997
). (c) At most
Ca2+ concentrations and membrane voltages, Ca2+
binding steps equilibrate rapidly relative to those that
limit the channel's kinetic behavior (Methfessel and
Boheim, 1982
; Moczydlowski and Latorre, 1983
; Cui et
al., 1997
). (d) At least 3 Ca2+ binding sites are involved
in activation by Ca2+ (Barrett et al., 1982
; McManus et
al., 1985
; Golowasch et al., 1986
; Cornejo et al., 1987
;
Oberhauser et al., 1988
; Carl and Sanders, 1989
; Reinhart et al., 1989
; Mayer et al., 1990
; McManus and Magleby, 1991
; Markwardt and Isenberg, 1992
; Perez et
al., 1994
; Art et al., 1995
; Cui et al., 1997
). (e) Gating
steps that do not involve Ca2+ binding are highly cooperative (Cui et al., 1997
).
To test these conclusions, as well as to gain new insights into BK channel gating, we have attempted to develop a physically reasonable kinetic model that is consistent with these conclusions and can mimic the kinetic and steady state behavior of mslo macroscopic currents. In previous work, we proposed a general kinetic scheme for the gating of the mslo channel (Cui et
al., 1997). Here we demonstrate how this scheme Is a
simplification of a more complex scheme based on simple assumptions about the structure of the channel. We
describe the basic properties of models of the simplified form, and we examine their ability to mimic mslo
macroscopic currents.
Electrophysiology
All experiments were performed with the mbr5 clone of the
mouse homologue of the slo gene (mslo), which was kindly provided by Dr. Larry Salkoff (Washington University School of Medicine, St. Louis, MO). mbr5 cRNAs were expressed in Xenopus laevis
oocytes and recordings were made in the inside-out patch clamp
configuration from excised membrane patches as described previously (Cox et al., 1997). The solution in the recording pipette
contained the following (mM): 140 KMeSO3, 20 HEPES, 2 KCl,
and 2 MgCl2, pH 7.20. Internal solutions were prepared as described previously (Cox et al., 1997
) and contained the following
(mM): 140 KMeSO3, 20 HEPES, 2 KCl, 1 HEDTA, and CaCl2 to
give the indicated free Ca2+ concentrations ([Ca]), pH 7.20. Final free Ca2+ concentrations ([Ca]) were determined with a Ca-sensitive electrode. HEDTA was not included in the internal solutions, which contained 490 and 1,000 µM [Ca]. 1 mM EGTA was
substituted for HEDTA and no CaCl2 was added to the internal
solution containing ~2 nM [Ca]. At this [Ca], mslo channels can
be activated by depolarization without binding Ca2+ (Meera et
al., 1996
; Cui et al., 1997
). Solutions bathing the cytoplasmic face
of the excised patch were exchanged using a sewer pipe flow system (DAD 12; Adams and List Associates Ltd., Westbury, NY) as
described previously (Cox et al., 1997
). Unless otherwise indicated, all currents were filtered at 10 kHz with a four pole low
pass Bessel filter. Experiments were performed at 23°C. G-V relations were determined from the amplitude of tail currents 200 µs after repolarization to a fixed membrane potential (
80 mV)
after 20-ms voltage steps to the indicated test voltages. Each mslo
G-V relation was fitted with a Boltzmann function (G = Gmax/(1 + ezF(V1/2
V)/RT)) and normalized to the peak of the fit. The procedures used for the recording and analysis of mslo macroscopic
currents minimized problems with permeation-related artifacts
(see Cox et al., 1997
).
Modeling
Voltage-dependent Monod-Wyman-Changeux (MWC)1 models were fitted to mslo conductance voltage (G-V) relations at several [Ca] simultaneously using Eq. 5 and Table-Curve 3D software that employs an 80-bit Levenburg-Marquardt algorithm (Jandel Scientific, San Rafael, CA). Paramater standard errors were estimated by the curve-fitting algorithm. Similar fits were also found by eye. Parameters from fits to the steady state data constrained the rate constants used in fitting the kinetic data. The best over all steady state parameters were chosen as those that allowed for the best subsequent fit to the kinetic data. Four free parameters were involved in fitting voltage-dependent MWC models to the mslo G-V data. mslo G-V data were fitted to the general form of scheme II using Eqs. 2 and 3 and to two-tiered Koshland-Nemethy-Filmer (KNF) models using the following equation:
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(1) |
To fit the macrosopic relaxation kinetics of mslo with voltage-dependent MWC models, simulated currents were generated using "Big Channel" kinetic modeling software written by Dr. Toshi
Hoshi (University of Iowa, Iowa City, IA) and adapted to be compatible with "Igor Pro" curve fitting software (Wavemetrics Inc.,
Lake Osewgo, OR) by Dorothy Perkins (Howard Hughes Medical
Institute, Stanford University). The activation and deactivation
time courses of both real and model currents were then fitted with
single exponential functions, and the time constants of these fits
were compared. In both cases, fitting started 200 µs after initiation of the voltage step. Specific rate constants for fitting the
model time constants to those of the mslo data were found by eye,
but were aided by fitting to the relaxation time constants predicted by Eq. 9, which approximates the kinetic behavior of voltage-dependent MWC models under the conditions employed.
Many of the rate constants of the models were constrained by the
parameters determined from fitting the steady state data. To
make the kinetic behavior of the models as exponential as possible, high Ca2+ binding rate constants were used. For each subunit, the Ca2+ binding rate constant was fixed at 109 M1s
1 in
both open and closed conformations. With this restriction, fitting
the mslo kinetic data with voltage-dependent MWC models involved 10 free parameters. The general form of scheme II (see Fig. 3 A) was fit to the kinetic data of patch 1 in the same manner as was the voltage-dependent MWC model. For each subunit, the Ca2+ binding rate constant was fixed at 109 M
1s
1 in both closed
and open conformations. Ca2+ unbinding rate constants were
then determined from fits to the G-V data, and vertical rate constants were found by eye with the aid of Eq. 9. Fitting the mslo kinetic data to the general form of scheme II (see Fig. 3 A) involved 16 free parameters.
Statistical comparisons between voltage-dependent MWC, two-tiered KNF and general 10-state models were performed as described by Horn (1987). The statistic T, defined as
![]() |
was calculated and compared with the F distribution with kf and n kg degrees of freedom at the desired level of significance. Here,
SSEf represents the residual sum-of-squares for the fit with model
f, which is a subhypothesis of model g; SSEg represents the residual sum-of-squares for the parent model g, n represents the number of data points, and kf and kg represent the number of free parameters in models f and g, respectively.
The data from three representative patches were fitted independently (see Tables I, II, and III). For each of theses patchs, steady state and activation kinetic data were recorded at 8 or 9 [Ca] and at a large number of membrane voltages. Data from 19 additional patches recorded over more limited ranges of [Ca] and voltage behaved similarly. To increase the signal to noise ratio, typically four current families were recorded consecutively under identical conditions and averaged before analysis and display. Due to the long times necessary to acquire these data from a single patch deactivation, voltage families at a sufficiently large number of [Ca] were recorded only from patch 1. In fitting patches 2 and 3, the deactivation kinetics of patch 1 were used to constrain the kinetic fitting.
Table I. Voltage-dependent MWC Model Parameters |
Table II. General 10-State Model Parameters |
Table III. Voltage-dependent MWC Model Parameters |
For simplicity, in modeling mslo macroscopic currents we have made the following assumptions about the structure of the channel.
(a) When expressed alone, msloThe work of Shen et al. (1994) supports this assumption. When they coexpressed mutant
and wild-type dslo RNAs in Xenopus oocytes, they observed four different channel phenotypes at the single channel level that might reasonably correspond to channels with either zero, one, two, or three mutant subunits
if each of four subunits contribute equally to the free
energy of TEA binding. Also, over the first ~1/3 of their
sequence, a region that includes the putative pore region, slo channels are structurally homologous to homotetrameric K+ channels (Atkinson et al., 1991
; Adelman et al., 1992
; Butler et al., 1993
; Tseng-Crank et al.,
1994
; McCobb et al., 1995
).
There are many reports of Hill coefficients for
Ca2+ activation >2.0 for both native and cloned BK
channels (see McManus, 1991 and references given in
introduction) indicating that at least three Ca2+
binding sites contribute to channel activation. We have
observed Hill coefficients >2.0 for the mslo channel as
well (Cui et al., 1997
). Given that the Hill coefficient in
all but the most extremely cooperative case represents
an underestimate of the true number of binding sites,
and that the channel is likely to be a homotetramer, it
seems reasonable to suppose that the channel has four
Ca2+ binding sites, one on each subunit. However, we
can not rule out more than four, particularly considering reports of Hill coefficients >4 for skeletal muscle
BK channels in the presence of millimolar internal
Mg2+ (Golowasch et al., 1986
; Oberhauser et al., 1988
).
Supporting this assumption, each mslo subunit
has a partially charged putative membrane spanning region (S4), whose counterparts in purely voltage-gated
K+ channels are thought to form, at least in part, each
of four voltage-sensing elements (Liman et al., 1991
;
Lopez et al., 1991
; Papazian et al., 1991
; Logothetis et
al., 1993
; Sigworth, 1994
; Yang and Horn, 1995
; Aggarwal and MacKinnon, 1996
; Larsson et al., 1996
; Mannuzzu et al., 1996
; Seoh et al., 1996
; Yang et al., 1996
).
Also, certain properties of the voltage dependence of
mslo macroscopic and gating currents suggest that the
channel multimer can undergo more than one voltage-dependent conformational change (see Horrigan et
al., 1996
; Ottolia et al., 1996
; Toro et al., 1996
; Cui et
al., 1997
, and discussion). It is possible, however, that
each subunit can undergo more than one voltage-
dependent conformational change. If this is the case,
the main conclusions of this paper are not affected.
Assumptions b and c allow us to consider each subunit of the mslo channel as being able to reside in four
physically distinct conformational states (Fig. 1 A): (a)
Ca2+ bound, (b) voltage activated, (c) both Ca2+ bound
and voltage activated, and (d) neither Ca2+ bound nor
voltage activated. A channel comprised of four subunits (assumption a) then could reside in 44 or 256 states.
However, without subunit expression, each of mslo's
subunits are identical. Some of these states will not be physically distinct. How many are redundant will depend upon the symmetry of the tetramer. If the channel were to have twofold rotational symmetry such as is
shown in Fig. 1 B, then it could occupy 73 physically distinct states. Several crystallized tetrameric proteins have
been found to display this sort of symmetry (Dittrich, 1992
). However, the mslo channel is thought to have a
single pore, and its homology with shaker K+ channels
suggests that the same portion of each subunit contributes to the pore's structure (MacKinnon, 1991
; Heginbotham and MacKinnon, 1992
; MacKinnon et al.,
1993
). It is more likely, therefore, that the channel has
fourfold rotational symmetry (Fig. 1 C). If this is the
case, then there are 55 distinct channel states. These states are represented schematically in Fig. 2 A. Those
states with a common number of bound Ca2+ are
grouped. Notice that to be true to the physical picture
(fourfold rotational symmetry), a distinction must be
made between a channel with two Ca2+ bound to adjacent subunits and two Ca2+ bound to subunits, one diagonally across from the other. Similarly, there is a natural distinction between a channel with two voltage sensors activated in adjacent as opposed to diagonally
opposite subunits. If the mslo channel exists as a homotetramer with a single Ca2+ binding site and a single
voltage sensor in each subunit, then the states displayed in Fig. 2 A represent the smallest number that
must be considered to describe the gating of the channel accurately.
Working with a 55-state kinetic model poses problems. First, due to the large number of equations necessary to represent such systems mathematically, testing them can be slow. Second, if there are energetic interactions between subunits (conclusion e, introduction), then the number of parameters necessary to define the system's behavior can be very large, far larger than can be constrained by our data. And third, large kinetic models tend to lose their explanatory power as the system grows more complex. Even if one is able to find parameters by which a highly complex model fits the data well, it may not be easy to discern from these parameters why the model behaves as it does. For these reasons, we have taken two further steps to reduce the number of kinetic states in the mlso model before trying to examine its performance. In doing so, however, we are aware that in terms of the idea represented in Figs. 1 A and 2 A we are over simplifying the system and may lose the ability to mimic certain important channel behaviors.
The first simplification is to make no distinction between two voltage- or Ca2+-activated subunits that are adjacent to one another and two voltage- or Ca2+-activated subunits diagonally opposed to one another. Examples of equivalent states under this assumption are shown in Fig. 2 B. The total number of conformations the channel may reside in then reduces to the 35 shown in Fig. 2 C (scheme I). Horizontal transitions represent Ca2+ binding and unbinding steps. Vertical transitions represent the movement of voltage-sensing elements. The states in each vertical grouping have a common number of Ca2+ ions bound. To avoid confusion, most of the allowable horizontal transitions are not represented; however, at all voltages the channel must bind four Ca2+ ions (undergo four horizontal transitions) to move from the leftmost column of states to the rightmost column.
To use scheme I to model channel gating, each state
must be designated as open or closed. The states in the
uppermost tier (horizontal row) may be designated closed,
as even at high Ca2+ concentrations strong hyperpolarization reduces the channel's open probability to very
low levels (Latorre et al., 1982; Moczydlowski and Latorre,
1983
; Adelman et al., 1992
; Wei et al., 1994
; DiChiara and Reinhart, 1995
; Cox et al., 1997
; Cui et al., 1997
).
The states in the lowermost tier may be designated
open, as strong depolarization can bring the mslo channel to very high open probabilities (>0.9) with or without bound Ca2+ (Cox et al., 1997
; Cui et al., 1997
).
Which intermediate states (those inside the box in Fig.
2 C) are open is not obvious. Bearing on this issue, the
mslo G-V relation can be approximated by a simple
Boltzmann function over a wide range of Ca2+ concentrations (Butler et al., 1993
; Wei et al., 1994
; Cui et al.,
1997
),2 and simulations indicate that for scheme I to
predict Boltzmann-like G-V relations, the voltage-
dependent transitions that link the uppermost closed
tier to the lowermost open tier must be highly concerted. By concerted we mean that the change in standard free energy associated with the last gating step before opening is much larger than the previous steps
such that the probability of the channel occupying an
intermediate state at equilibrium is low at most Ca2+
concentrations and membrane voltages (conclusion e in
introduction). If this is the case, we can approximate
scheme I by scheme II, shown in Fig. 3 A. Here the
four voltage-dependent steps in scheme I are represented by a single, and therefore completely concerted, voltage-dependent step. Which intermediate states are
open is no longer an issue. Further suggesting that
there is a single conformational change that is limiting
the rate of current relaxations under most circumstances (see below), macroscopic mslo kinetics are
nearly exponential over a wide range of Ca2+ concentrations and membrane voltages (see Fig. 9 and Cui
et al., 1997
). Condensation of intermediate voltage-
dependent steps to a single step, therefore, at least as a
first approximation, may be reasonable, although departures from strictly single exponential behavior are
known (Toro et al., 1996
; Cui et al., 1997
).
The hypothesis represented by scheme II may be described as follows. The channel can exist in two main
conformations, closed and open. The equilibrium between closed and open is voltage dependent, having associated with it a gating charge Q. The channel has
four identical Ca2+ binding sites, and the binding of
Ca2+ shifts the closed-to-open equilibrium towards open.
The simplest physical picture that corresponds to scheme
II is one in which, in either of the two main conformations of the protein, the Ca2+ binding sites act independently. The binding of one Ca2+ ion does not influence
the binding of subsequent Ca2+ to that same conformation; only undergoing the closed-to-open conformational change affects Ca2+ binding affinity, and then all
sites are affected equally. In this case, scheme II corresponds to a voltage-dependent version of the Monod-Wyman-Changeaux model originally proposed to describe the binding of oxygen to hemoglobin (Monod et
al., 1965). Here, the usual R and T nomenclature is replaced by O and C, and the central conformational
change is made voltage dependent. MWC models have
been used to describe the equilibrium behavior of many
of the best studied allosteric proteins (Dittrich, 1992
),
particularly tetrameric proteins: hemoglobin, phosphofructokinase, and fructose-1,6-bisphosphatase for example. However, unlike in the study of most allosteric
proteins, in studying channel gating we are better able
to measure the fraction of channels in the R, or open
state, than the fraction of binding sites occupied by
ligand. Nevertheless, the fact that an examination of
the gating behavior of mslo leads to models of the form
of scheme II suggests that perhaps this channel has
something mechanistically in common with these other
allosteric proteins. As described below, we examined
the ability of voltage-dependent MWC models to predict the macroscopic behavior of mslo channels. In addition, we have examined the general characteristics of
models that do not conform to the MWC constraint
that each binding site acts independently but are of the
form of scheme II.
mslo Steady State Behavior
Assuming that Ca2+ binding is not voltage dependent
(see Cui et al., 1997), the equilibrium open probability
(Popen) of models of the form of scheme II can be written as the following function of [Ca] and voltage:
![]() |
(2) |
where KCi is the dissociation constant for the subunit
binding the ith Ca2+ in the closed conformation, KOi is
the dissociation constant for the subunit binding the
ith Ca2+ in the open conformation, and L(V) is the
open-to-closed equilibrium constant ([C0]/[O0]) when
no Ca2+ ions are bound to the channel. If we assume
that the free energy of the channel varies linearly with
voltage, as is likely to be the case at moderate voltages
(Stevens, 1978), then L(V) can be defined as:
![]() |
(3) |
where, F is Faraday's constant, R is the universal gas constant, T is temperature, V is voltage, and Q is the equivalent gating charge associated with the closed-to-open conformational change, and L(0) is the open-to-closed equilibrium constant in the absence of bound Ca2+ at 0 mV. Combining Eqs. 2 and 3, Popen depends on voltage in the form of a Boltzmann function:
![]() |
![]() |
with
![]() |
![]() |
The steepness of this relation as a function of V is determined solely by the value of Q, and is therefore independent of [Ca]. Assuming no voltage dependence in Ca2+ binding, models of the form of scheme II will predict a G-V that does not change shape as [Ca] is varied. The position of the G-V curve on the voltage axis, however, will be determined by BL(0), and therefore will depend on L(0), [Ca], and all the Ca2+ dissociation constants for both the closed and open channel. For voltage-dependent MWC models, where the microscopic dissociation constants depend only on whether the channel is open or closed, Eq. 4 simplifies to
![]() |
(5) |
Shown in each panel of Fig. 4 are data from 22 macropatches expressing mslo channels. These patches were
excised from Xenopus oocytes and superfused with solutions of varying [Ca]. Each data point represents the
half maximal activation voltage (V1/2) of the ionic current in a particular patch at the indicated [Ca]. At each
[Ca], typically three to five current families were recorded and averaged before analysis. In three experiments, it was possible to expose the patch to eight or
more Ca2+ concentrations and determine complete
G-V relations. The V1/2 values from these patches are indicated as filled circles in Fig. 4. (patch 1, Fig. 4 A;
patch 2, Fig. 4 B; patch 3, Fig. 4 C), and the full G-V relations from these patches are shown in Fig. 5. Also
shown in Fig. 5 are voltage-dependent MWC model fits
to these data (solid lines) using Eq. 5.
The fits to each set of G-V curves are determined by four parameters: the closed Ca2+ dissociation constant (KC), the open Ca2+ dissociation constant (KO), and, as already defined, L(0) and Q. Voltage-dependent MWC models can fairly well describe the mslo G-V relation at very low [Ca] (Fig. 5, A and B, open circles), as well as the shifting behavior of the G-V relation as Ca2+ is increased to 124 µM. The calculated G-V curves displayed in bold correspond to the following [Ca]: ~2 nM, 0.84, 1.7, 4.5, 10.2, 65, and 124 µM. The best fitting parameters for each patch over this [Ca] range are listed in Table I. At higher [Ca] (dimmed data and fits), the mslo G-V relation is not well fitted by voltage-dependent MWC models. Possible reasons for this discrepancy will be considered later in some detail. Two G-V relations recorded with 1.7 µM [Ca] are included in Fig. 5 A, one recorded at the beginning of this ~40-min experiment and the other recorded at the end. The difference between these two data sets reflects the variability in G-V position over time, and therefore provides some assessment of how accurately we might expect the model to fit the steady state data if it were known a priori that such a model properly mimicked the underlying physical system. Given this variability, the model appears to mimic mslo steady state behavior over the [Ca] range ~2 nM-124 µM fairly well.
In some previous experiments, however, at very low
[Ca] (~0.5 nM), the maximum slope of the mslo G-V
relation was more shallow than that observed at ~2 nM
[Ca] for patches 1 and 2 in Fig. 5, (mean Boltzmann fit
parameters: z = 0.87, V1/2 = 195 mV, ~0.5 nM [Ca],
Cui et al., 1997), and it was more shallow than its value at 10.2 µM [Ca] by ~30% (Cui et al., 1997
). Also, some
variation in G-V curve steepness is observed throughout
the [Ca] range. On average, we have found this relation to be most steep at ~2 µM [Ca], and to become
somewhat more shallow at both higher and lower [Ca]
(Cui et al., 1997
). For voltage-depedendent MWC models, the value of Q determines the G-V relations steepness, and, supposing there is no voltage dependence in
horizontal transitions, dictates that the model G-V relation will have the same shape regardless of [Ca] (see
Fig. 6 for example). Variations in G-V curve shape as a
function of [Ca], therefore, cannot be accounted for
by these simplified models. Models of the form of
scheme I, however, could likely account for this behavior as scheme I allows for differences in the cooperative
interactions between voltage sensing elements when
different numbers of Ca2+ are bound to the channel.
Alternatively, adding a small amount of voltage dependence to the Ca2+ binding steps in scheme II, or assigning a nonexponential voltage dependence to the central closed-to-open conformational change can also give
rise to changes in G-V curve steepness as [Ca] is varied.
Properties of Scheme II Models
In Fig. 4 we used the voltage at which mslo currents are
half maximally activated (V1/2) as an index of the position of the channel's G-V relation on the voltage axis at
a given [Ca]. In the study of BK channel gating, this is
common practice (Wong et al., 1982; Wei et al., 1994
;
DiChiara and Reinhart, 1995
; McManus et al., 1995
;
Wallner et al., 1995
; Dworetzky et al., 1996
; Meera et
al., 1996
; Cui et al., 1997
). For models conforming to
scheme II
![]() |
(6) |
where B is given by Eq. 4b, and is thus a function of [Ca] and its dissociation constants. At constant Q and [Ca], a decrease in L(0) decreases the change in free energy required to open the channel, and the G-V relation moves leftward along the voltage axis. If L(0) and [Ca] are held constant and Q is increased, more energy is imparted to the central equilibrium by a given change in voltage, and V1/2 moves towards 0 mV. When [Ca] equals 0, B equals 1, and the position of the G-V relation is determined only by Q and L(0). The effects that changes in various voltage-dependent MWC model parameters have on model G-V relations at a series of [Ca] are illustrated in Fig. 6.
The parameters that best fit the mslo G-V data (Table
I) suggest an intrinsic standard free energy difference
between open and closed at 0 mV of 4.6-5.4 kcal mol1
at 23°C. However, the standard errors of estimates of
L(0) are large. The L(0) values could be further constrained by fits to kinetic measurements of gating. Smaller
values of Q, and therefore correspondingly smaller values of L(0), could give similar fits to the mslo G-V relations (dashed lines in Fig. 5), and, as will be discussed,
yielded better fits to the kinetic data. In fitting both
steady state and kinetic data, therefore, lower values of L(0) were favored. Typically, values of L(0) between
1,500 and 2,000 were used (see Table III) corresponding to an intrinsic free energy difference between closed
and open at 0 mV of 4.3-4.5 kcal mol
1 at 23°C.
Taking the derivative of Eq. 6 with respect to L(0) yields:
![]() |
(7) |
The Ca-dependent factor B in Eq. 6 does not appear in Eq. 7. Therefore, the effects of changing L(0) are the same at each [Ca], and as L(0) is varied, the G-V curves for a number of [Ca] move as a set along the voltage axis maintaining the spacing between them (compare Fig. 6, C to A). The shape of the V1/2 vs. log[Ca] relation does not change (Fig. 6 E).
Unlike L(0), the value of Q influences the spacing of the model G-V curves. For a given series of [Ca], increases in Q bring the model curves closer together (compare Fig. 6, B to A). This can be understood intuitively by considering that as the gating valence is increased the free energy contributed to the central conformational equilibrium by a given size electric field increases; therefore, smaller increases in the size of this field are required to compensate for the free energy contributed to this equilibrium by Ca2+ binding. It can also be understood mathematically by considering that Q appears in the denominator of Eq. 6. A large Q value will therefore mitigate the change in V1/2 brought about by increases in [Ca]. The slope of the V1/2 vs. log[Ca] relation becomes more shallow as Q is increased (Fig. 6 E). The relative spacing of the model G-V relations, however, which we might define as the difference in V1/2 between the G-V curves for any two [Ca] divided by the difference in V1/2 for any other two [Ca], is unaffected by changes in Q. This is a manifestation of the fact that a plot of V1/2 vs. log[Ca] scales linearly with (1/Q) (Eq. 6).
To this point, our analysis has been based on equations for the general form of scheme II. The effects described above are therefore not specific to voltage-dependent MWC models, but rather are general to all models of this form.
When Q and the number of binding sites are fixed, the spacing of the G-V curves along the voltage axis as [Ca] is varied is determined by all the Ca2+ dissociation constants in the system, KC and KO for voltage-dependent MWC models. These constants reflect the change in free energy that occurs as Ca2+ binds to either conformation of the protein, and their ratio determines the maximum shift in V1/2 as [Ca] is increased from 0 to a saturating value. Comparing Fig. 6 D to A for example, as KC is increased from 10 to 100 µM, the position of the model G-V relation at high [Ca] shifts considerably leftward due to an increase in the ratio KC/KO. Notice that this large leftward shift is accomplished by decreasing the affinity of the channel for Ca2+ in the closed state. Such shifts are therefore not necessarily indicative of an increase in Ca2+ binding affinity. The G-V relation at 0 [Ca], is unaffected. Within the range between 0 and saturating [Ca], the specific values of the Ca2+ dissociation constants determine the spacing of the G-V relations. In fitting mslo G-V relations to voltage-dependent MWC models, we consistently found that to create the observed spacing the values of KC and KO had to be close to 10 µM and 0.5-1 µM, respectively (see Table I for standard errors). In terms of these models, then, KC and KO are well determined.
Accounting for the mslo G-V Relation at High [Ca]
As is evident in Fig. 5, the position of the mslo G-V relation at very high [Ca] is not well accounted for by voltage-dependent MWC models. The dimmed data points
in this figure were recorded with 490 (triangles) and
1,000 (bowties) µM [Ca]. Model fits are also shown (gray
lines). Both real and model G-V relations are close together at 65 and 124 µM [Ca], suggesting a saturation of the effects of [Ca]. At higher [Ca], however, the mslo
G-V curve continues to move leftward, while the model
curve saturates. In fitting the data in Fig. 5, only the
[Ca] range, ~2 nM-124 µM, was included and, therefore, good fits at higher [Ca] might not be expected.
However, as shown in Fig. 7 A, when data recorded at
490 and 1,000 µM [Ca] are included in the fitting, voltage-dependent MWC models cannot account for the
position of the G-V curves at the highest [Ca]. Attempting to do so sacrifices fitting the data at lower [Ca]. Increasing the number of Ca2+ binding sites did not improve model fits at high [Ca]. For each patch, the residual sum of squares increased as the number of binding
sites in the model was increased.
We also tried to fit the data at high [Ca] by relaxing
the MWC constraint that there are no interactions between Ca2+ binding sites in either the closed or open
channel. The affinity constants KC1 KC4 and KO1
KO4 in scheme II were allowed to vary independently.
In physical terms, relaxation of this constraint allows
each Ca2+ binding to affect the affinities of the remaining unoccupied sites. The sequential nature of each
horizontal row in scheme II (Fig. 3 A), however, dictates that all unoccupied sites are affected equally by
any binding event. As shown in Fig. 7 B (solid lines), relaxing the MWC independent binding constraint significantly improved model fits to the mslo G-V relations at
high [Ca] (P < 0.01 for each patch, tests for the superiority of nested regression models where carried out as
described by Horn, 1987
, see methods). The general
10-state model parameters that best fit patch 2 in Fig. 7
B are listed in Table II, as are the parameters that best fit the G-V data from the two other patches in Fig. 5.
The dissociation constants from these fits suggest a
complex relationship between successive Ca2+ binding
events in both the open and closed conformation. The standard errors of these fit parameters, however, were
very large, sometimes several orders of magnitude larger
than the parameter values themselves. This result suggests that many different parameter combinations could
produce similar fits, and therefore no emphasis should
be placed on any particular parameter combination. It
also underscores the inadequecy of steady state data for
constraining complex models, and the need for both
steady state and kinetic data to obtain reliable parameter estimates.
A common property of each fit, however, is a large KC4 value relative to the other closed state Ca2+ dissociation constants. This decrease in the affinity of the last Ca2+ binding event in the closed conformation, coupled with no change or an increase in the affinity of the last Ca2+ binding event in the open conformation, increases the ratio (KC4/KO4) and thereby the power of the last binding event to push the G-V relation leftward along the voltage axis. This effect can account for most of the improvement over voltage-dependent MWC fits at high [Ca]. This is demonstrated by the dashed lines in Fig. 7 B, which represent a general 10-state model fit to these data with parameters that are similar to the voltage-dependent MWC model parameters for this patch (Tables I and III), except for a large KC4 value (for parameters, see legend to Fig. 7). It is possible, therefore, that negative cooperativity between the third and fourth Ca2+ ions binding to the closed conformation, which is not present in the open conformation, accounts for the position of the mslo G-V relation at [Ca] > ~100 µM.
Another means by which we might relax the MWC
constraint that the binding sites are acting independently is to suppose that in either the closed or open
conformations of the channel the affinity of a given
Ca2+ binding site changes depending on whether the
binding sites on one or two adjacent subunits are already occupied. Scheme III in Fig. 3 B represents this
idea. In this scheme, each tier is formally equivalent to
a square version of the well known KNF model of allosteric interactions between subunits (Koshland et al.,
1966). These types of interactions were also considered
by Pauling (1935)
. Because under this idea there is a
difference in free energy between a channel with two
adjacent sites occupied and one with two diagonally opposed sites occupied, an additional state in each tier is
necessary. Once the dissociation constants for a site
with zero (KXA) or one (KXB) neighbor occupied are defined, the dissociation constant for a site with two
neighbors occupied is determined (KXC = KXB2/KXA).
Two free parameters therefore determine the equilibrium binding of Ca2+ to each tier of the model. Shown
in Fig. 7 C is a fit of scheme III to the same data as
shown in Fig. 7, A and B. The dissociation constants for
this fit are KCA = 0.046 µM, KCB = 1.42 µM, KCC = 43.90 µM, KOA = 0.047 µM, KOB = 0.100 µM, KOC = 0.213 µM. In general, two-tiered KNF models provide
some improvement over voltage-dependent MWC models in fitting the mslo data at [Ca] above 124 µM (P < 0.01 for all patches). These models, however, do not fit
as well as general 10-state models (P < 0.01 for all
patches), and the small dissociation constants often associated with the best fit (KCA and KOA above) make it
difficult to fit the kinetic data (see below).
An alternative explanation for shifts in G-V curve position at very high [Ca] comes from the work of Wei et
al. (1994) (see also Solaro et al., 1995
). In their experiments they found that when 10 mM Mg2+ was present
in the internal solution no further shifting of the mslo
G-V relation was observed with [Ca] above 100 µM.
They suggested that there is a second divalent cation
binding site(s) on the mslo channel that is less specific
for Ca2+ over Mg2+ than what might be considered the
primary site(s), and that at high [Ca] it is Ca2+ binding
to this second site, perhaps screening a surface charge, that is responsible for the leftward G-V curve shifts
above ~100 µM [Ca]. Under this hypothesis, 10 mM
Mg2+ in the internal solution saturates this site and no
further effects of high [Ca] are expected. Clearly, more
investigation into this phenomenon will be important.
When the G-V data from [Ca] above 124 µM were excluded from the fitting, each of the models discussed in
relation to Fig. 7 performed similarly, with somewhat
better fits produced as the number of free parameters
increased.3
Popen as a Function of [Ca]
The steady state behavior of mslo and model channels
can also be compared by looking at Popen as a function
of [Ca]. In Fig. 8, the data of Fig. 5 A are shown converted to Ca2+ dose-response form (filled circles). Simulated voltage-dependent MWC model data are included
as well (open circles). Each curve represents a different
voltage. Both real data and simulated model points were
fitted (solid curves) with the Hill equation (Hill, 1910):
![]() |
(8) |
The parameters for these fits are plotted in Fig. 8, C-E.
As over this [Ca] range there is a fairly good agreement
between the model and the data in Fig. 5 A, it is not surprising that viewed in this way the real and simulated
data appear similar (Fig. 8, A and B). Fitting these data
with the Hill equation, however, serves to illustrate
some important characteristics shared by mslo and voltage-dependent MWC models. Both real data and simulated voltage-dependent MWC model behavior suggest
that the maximum extent of channel activation decreases at negative potentials (Fig. 8 C). Both systems
produce Hill coefficients between 1.5 and 2.0 over a
broad range of voltages (Fig. 8 D); however, mslo Hill
coefficients closer to 3 are often observed at voltages
greater than +50 mV.4 This is because the mslo dose-response curves are slightly more sigmoid than the
model curves at these potentials (Fig. 8, A and B). Both
systems also demonstrate an increase in the apparent affinity of the channel for Ca2+ as voltage is increased
(Fig. 8 E) (Barrett et al., 1982; Moczydlowski and
Latorre, 1983
; Markwardt and Isenberg, 1992
; Cui et
al., 1997
). This might appear to suggest that Ca2+ binding steps are voltage dependent (Moczydlowski and
Latorre, 1983
; Markwardt and Isenberg, 1992
). However, voltage-dependent MWC models predict this behavior without supposing voltage dependence in Ca2+
binding steps. This is because in the model both Ca2+
and voltage alter the free energy difference between
closed and open. As the voltage is increased, more energy is contributed to the central equilibrium by the
electric field, and, therefore, fewer Ca2+ must bind to
the channel to bring it to a given Popen. This is illustrated in Fig. 6 E where open probability is plotted as a function of the mean number of Ca2+ ions bound to
the model channel at several voltages.
Kinetic Behavior
A more demanding test of any gating model is to mimic
the nonstationary gating behavior of the protein being
studied. As described below, we attempted to fit mslo
macroscopic current kinetics with voltage-dependent
MWC models. To do so we had to reconcile the fact
that, but for a brief delay in the onset of activation,
both mslo current activation and deactivation can be described by an exponential function over a wide range of
[Ca] and membrane voltages (Cui et al., 1997), yet the
solution of a 10-state Markov system has nine exponential components. Some of these components may be
relatively small in amplitude, and some may have similar time constants. The challenge then is to find conditions under which scheme II behaves like an apparently
more simple gating system.
It seems reasonable to suppose that the time course
of current relaxation reflects, in part, the vertical rate
of flux between open and closed, and, in part, the time
it takes for the channels to redistribute horizontally as
the net motion between closed and open draws channels away from one tier and to the other. It might be,
therefore, that if one of these processes was considerably slower than the other, the kinetics of the system
would be dominated by the slower process. In fact, for
models of the form of scheme II it is possible to show
that in the limit that the equilibration of Ca2+ binding
steps becomes very fast relative to the vertical transition rates such that each horizontal elementary step can be
considered to be at equilibrium at all times, the kinetics
of scheme II become monoexponential (see Eigen,
1967; Wu and Hammes, 1973
). The time-dependent solution for scheme II in this limit becomes
![]() |
![]() |
![]() |
![]() |
where X and
X represent closed-to-open and open-to-closed vertical rates constants, respectively, and fCX and
fOX represent the fraction of closed (fCX) or open (fOX)
channels occupying state x at a given [Ca]. The macroscopic time constant given by (1/
+
) is determined
by an average of all the vertical rate constants in the
scheme, weighted by the fraction of closed (for
s) or
open (for
s) channels that precede each vertical transition. The simple kinetic behavior of mslo suggests something approximating this situation is occurring. If, on
the other hand, we suppose that voltage-dependent transitions are very rapid relative to Ca2+ binding transitions, the kinetics of scheme II do not converge to monoexponential behavior as is observed in the data (Cui et al., 1997
, and Fig. 9), but rather, each population of
channels with a given number of Ca2+ bound would appear to gate independently, producing five exponential components in the macroscopic current kinetics.
The kinetics of mslo macroscopic currents were fitted
to voltage-dependent MWC models using the following
procedure. The on rates for Ca2+ binding to each subunit were assumed to be at least as fast as is reasonable
(109 M1s
1) given the diffusion limit for Ca2+ binding
and the on rates reported for other Ca2+ binding proteins (Falke et al., 1994
; Cui et al., 1997
). Interactions between Ca2+ and surface charge could make the Ca2+
binding rates even faster. The Ca2+ off rates were then
constrained by the dissociation constants determined
from fitting the steady state data. Likewise, for each vertical step in the model one rate constant was free to
vary, while the other was then constrained by the steady
state fit. No other constraints were placed on the vertical rate constants except that closed-to-open rate constants were made to increase as the number of bound
Ca2+ increased. The total gating charge associated with
the central conformational change was determined by
the steady state fit while the proportion of charge moving in forward and backward transitions was allowed to
vary within this constraint. To obtain reasonable fits to
the kinetic data, we found that the total gating charge had to be less than that which produced the best fits to
the steady state data, this then required smaller L(0)
values as well. Good fits to the steady state data, however, were found with these parameters (Fig. 5, dashed
lines). The parameters used to model both mslo steady
state and kinetic behavior are listed in Table III.
Shown in Fig. 9 (left) are current families recorded
from the patch of Fig. 5 A at 0.84, 10.2, and 124 µM
[Ca]. On the right are traces generated from the voltage-dependent MWC model for this patch (Table III).
Each model trace was multiplied by a common factor
that made the amplitude of the model current at +90
mV equal to that of the data. Because under the conditions employed, and over this voltage range, the mslo
single channel i-V relation is linear (Butler et al., 1993;
Cox et al., 1997
), this normalization procedure is valid.
Clearly, there is a qualitative similarity between the data
and the model both in terms of kinetics and current amplitude. The parameters for this fit are listed in Table III (Patch 1). Similar fits were obtained with voltage-dependent MWC models for the two other patches
(Table III). In each case, model activation kinetics were
well described by a single exponential function. In Fig.
9, single exponential fits to the time courses of activation are superimposed on both model and mslo currents; however, the fits to the model currents are hard
to discern as they follow very nearly the activation time
course.
To compare better the kinetics of voltage-dependent
MWC models to those of mslo, data and model currents
were superimposed and scaled to have the same amplitudes (Figs. 10 and 11). At high final open probabilities, the data and model currents activate more rapidly
as the voltage is made more positive or [Ca] is increased (Figs. 9 and 10). As understood through the model, the
effect of voltage is due to an increase in the magnitude
of all the closed-to-open rate constants as the voltage is
made more positive. At high open probabilities, these
rate constants are largely responsible for the overall kinetic behavior. The acceleration of the activation time
course by [Ca] also can be understood through the
model. Considering Eq. 9, as more Ca2+ ions bind to
the channel, the fast closed-to-open rate constants on
the right hand side of the model become more heavily
weighted than the slower closed-to-open rate constants
on the left. Faster macroscopic activation kinetics result. Deactivation of model currents to low final open
probabilities is dominated by the open-to-closed rates
constants. Depolarization decreases the magnitude of
these rate constants and slows deactivation (Fig. 11).
Increases in [Ca] also slow deactivation because as [Ca]
is increased, more weight is given to the slower open-to-closed rate constants on the right hand side of the
model than to the faster open-to-closed rate constants
on the left.
While the time courses of mslo and voltage-dependent MWC model currents are often similar, there is a
clear tendency for the model currents to be faster than
the data when current relaxations are very rapid (at
highly positive voltages and high [Ca] for activation,
and highly negative voltages and low [Ca] for deactivation). Also, at negative potentials and low [Ca], the
time course of model deactivation deviates from exponential behavior. This is because under these conditions,
the open-to-closed transitions do not equilibrate slowly
enough relative to the Ca2+ binding equilibria so as to
allow the Ca2+ binding equilibria to always be very close
to steady state. In Fig. 11, for example, the model current in the lower left panel (0.84 µM [Ca]; 120 mV)
is better fit by a double rather than a single exponential. The fast component of this fit has a time constant
of 34 µs. No such fast component is evident in the data. However, in our experiments such a fast component, if
present, would have been difficult to distinguish from
the capacity transient, and would have been blunted by
the analogue filter employed (four pole low pass bessel,
10 kHz).
To compare mslo and model macroscopic kinetics
over a wider range of conditions, the relaxation time
courses of both model and mslo currents were fitted
with single exponential functions, and the time constants of these fits are plotted as a function of voltage
for six different [Ca] in Figs. 12 (activation) and 13 (deactivation). The major effect of an increase in [Ca] is to
shift both activation and deactivation -V curves leftward along the voltage axis (DiChiara and Reinhart,
1995
; Cui et al., 1997
) (notice the changes in the range
of the voltage axis in Figs. 12 and 13). As was evident in
Figs. 10 and 11, when relaxation is very rapid, the model
traces (time constants represented by open circles,
Figs. 12 and 13) often relax more quickly than the data
(time constants represented by filled circles, Figs. 12
and 13). There appears to be something limiting the kinetic behavior of the mslo channel that is not accounted
for by the model. Also, the slowest activation time constants are usually slower in the mslo data than predicted
by the model. Overall, however, voltage-dependent MWC
models do a fairly good job at mimicking the shifting
nature of the mslo
-V curves as [Ca] is varied.
To examine whether the discrepancies between the
shapes of the mslo and voltage-dependent MWC model
-V curves are due to the restrictions placed on the
Ca2+ dissociation constants by the MWC assumption
that the Ca2+ binding sites act independently, the kinetic data from patch 1 was fitted with a general form
of scheme II. The results of this fit are shown as solid
curves on the graphs in Figs. 12 and 13. The general 10-state model behaves very much like the voltage-dependent MWC model. This result suggests that it is not the
assumption of binding site independence that leads to
the discrepancies between the kinetic data and voltage-dependent MWC models, but, more likely, the assumption that the channel's voltage sensors move in a completely concerted manner may need to be relaxed.
We have approached the modeling of BK channel gating by first considering the most general system of
states that is required to conform to what is known, or
likely to be true, about the structure of these channels,
and then simplifying this system of states to a manageable number using adaptations of widely used models for allosteric conformational changes. From the simple
premise that each subunit can bind one Ca2+ molecule
and undergo one voltage-dependent conformational
change arises 55 physically distinct states that must exist
and therefore contribute to the kinetic behavior of the
channel. This large number of states is beyond what
can be dealt with analytically under non-steady state
conditions, and larger than Markov models typically
used to describe the gating of ion channels. The complexity arises here because BK channels respond to two
stimuli, Ca2+ and membrane voltage, and we have chosen to model the effects of changes in these stimuli simultaneously. In such circumstances, very simple physical pictures lead to many allowable protein conformations. The fact that slo subunits expressed alone form
homomultimeric channels helps to reduce the number
of distinct channel conformations. Even a homotetrameric slo channel, however, could have a conformational space comprised of many more than 55 states.
For example, if each subunit could undergo two sequential voltage-dependent steps instead of one, as has
been proposed for the shaker K+ channel, even without
making distinctions based on subunit position, 126 distinct conformations of the protein would result. This is without considering conformational changes occurring
at or below the time resolution of electrophysiological
measurements. In the case of mslo, however, two observations suggest that voltage-dependent gating steps are
highly concerted and therefore might be modeled as a
single conformational change without losing a great
deal of ability to mimic channel gating. The first is that
macroscopic current kinetics are often well approximated by a single exponential function over a wide range
of conditions (Cui et al., 1997
), suggesting a single rate
limiting step between closed and open. The second is
that the mslo steady state G-V relation can be approximated by a simple Boltzmann function at many [Ca]
(Butler et al., 1993
; Wei et al., 1994
; Cui et al., 1997
).
While the data do not exactly conform to these functions (as will be discussed below), the degree to which
they do conform suggests a concerted series of steps between closed and open. Combining the movement of
the channel's four hypothetical voltage sensors into a
single step leads to models of the form of scheme II.
The simplest form of scheme II occurs when each
Ca2+ binding site is independent in both the closed
and open conformations, and the opening of the channel changes the affinity of all Ca2+ binding sites equivalently. In this case, scheme II becomes a voltage-dependent version of the well known MWC model of allosteric proteins (Monod et al., 1965). MWC models have
been used to model the conformational equilibria of
many of the best studied allosteric proteins (see Perutz,
1976
; Neet, 1995
). This prompted us to examine their
ability to mimic mslo gating. As discussed above, several discrepancies between mslo gating and the behavior of
voltage-dependent MWC models were discovered: (a)
the models fail to predict the position of the mslo G-V
relation at very high [Ca], (b) under conditions in
which relaxations are very rapid, model traces often relaxed more rapidly than the data, (c) the slowest activation time constants at a given [Ca] are usually slower in
the data than predicted by the model, (d) any changes
in the maximum slope of the mslo G-V relation as a
function of [Ca] are not accounted for by the models,
and (e) at low [Ca] and negative potentials, the models
predict a fast component of deactivation that is not clearly evident in the data. These discrepancies are certainly important, and are very likely pointing towards a
better understanding of the gating of BK channels than
can be gained from models of this form. Relaxing the
constraint that the binding sites act independently did
to some extent ameliorate the difficulty in fitting the
mslo G-V relations at high [Ca]; however, it did not improve the kinetic fits.
On the other hand many aspects of mslo gating can
be understood through voltage-dependent MWC models: (a) monoexponential relaxation kinetics over a
wide range of conditions (Cui et al., 1997), (b) Boltzmann-like G-V relations, (c) macroscopic activation rate
constants that increase with voltage and increase with
[Ca] to saturation (DiChiara and Reinhart, 1995
; Cui
et al., 1997
), (d) macroscopic deactivation rate constants that decrease with voltage and decrease with
[Ca] to saturation (DiChiara and Reinhart, 1995
; Cui
et al., 1997
), (e) the apparent Ca2+ affinity of the channel, as judged by the channel's steady state open probability, is strongly voltage dependent, while the apparent affinity, as judged by saturation of the activation kinetics, is less so (Cui et al., 1997
), (f) at most potentials, it
takes less Ca2+ to saturate the steady state conductance
than it does to saturate the macroscopic rate constant
of activation (Cui et al., 1997
), (g) the channel can be
very nearly maximally activated by voltage without binding Ca2+ (Cui et al., 1997
), (h) plots of steady state conductance vs. [Ca] are often best fit with Hill coefficients
close to 2 (Barrett et al., 1982
; Methfessel and Boheim,
1982
; Golowasch et al., 1986
; Cornejo et al., 1987
;
Oberhauser et al., 1988
; Sheppard et al., 1988
; MacKinnon and Miller, 1989
; Reinhart et al., 1989
; Tabcharani and Misler, 1989
; Segal and Reuss, 1990
; Markwardt
and Isenberg, 1992
; Perez et al., 1994
; DiChiara and
Reinhart, 1995
), and (i) a nonlinear V1/2 vs. log[Ca]
plot (Wei and Salkoff, 1986
; DiChiara and Reinhart,
1995
; Cui et al., 1997
). In many respects, voltage- dependent MWC models appear to provide a reasonable general description of the channel's gating behavior and are useful in understanding the molecular
mechanisms of mslo gating.
The dose-response relationships of many allosteric
proteins have been modeled using the MWC formalism, and putative T and R states of several of these proteins have been crystallized. A list of these proteins
includes hemoglobin, aspartate transcarbamoylase, phosphofructokinase, glycogen phosphorylase, fructose bisphosphatase, pyruvate kinase, and others (for reviews
see Perutz, 1989; Dittrich, 1992
; Neet, 1995
). The allosteric behavior of some ion channels have also been
modeled in this way (Marks and Jones, 1992
; Goulding et al., 1994
; Tibbs et al., 1995
; Edelstein et al., 1996
;
Galzi et al., 1996
; Varnum and Zagotta, 1996
). Of these,
the cyclic nucleotide-gated (CNG) channels of retina
and olfactory epithelium are most similar to mslo.
These channels are thought to be tetramers with a single cyclic nucleotide binding site in each monomer
(Liu et al., 1996
). Like mslo, their amino acid sequences
suggest they are structurally similar to voltage-gated ion
channels with a specialized domain attached to facilitate ligand binding (Kaupp et al., 1989
). Also like mslo,
channel opening has been demonstrated in the absence of bound ligand (Tibbs et al., 1995
), subunits appear to change conformation in a concerted manner as
the channel opens (Varnum and Zagotta, 1996
), and
the cyclic nucleotide dose-response relations of CNG
channels can be fitted with MWC models (Goulding et
al., 1994
; Varnum and Zagotta, 1996
) (for a review of
cyclic nucleotide-gated channel properties see Zagotta
and Siegelbaum, 1996
). These similarities suggest a common mechanism by which ligand binding is coupled to
channel opening. Studies of either channel type may
therefore contribute to the understanding of the other.
Unlike CNG channels, however, BK channels are significantly voltage dependent. The equilibrium gating
of BK channels can therefore be examined under a
wider range of conditions than has been possible for
CNG channels. Determining the open probability of
mslo as a function of [Ca] at 10-mV increments spanning a 100-mV range (the experiment of Fig. 8) is in
many respects similar to measuring the substrate activation curve of a classic allosteric protein at many effector
concentrations. The main differences between these
experiments being that with channels we are in the unusual position of being better able to measure the fraction of channels in the open conformation than the
number of ligands bound to the channel, and voltage,
unlike an effector, can influence equilibria in either direction. Experiments involving both substrate and effector have contributed greatly to the functional understanding of many allosteric enzymes (for reviews see Perutz, 1989; Neet, 1995
). The ability to manipulate mslo
gating with both Ca2+ and voltage therefore holds
promise for a more detailed understanding of the gating of these proteins in the future. Also promising, with
voltage perturbations, the nonstationary kinetic behavior of slo channels can be examined in detail both with
macroscopic currents, as was done here, and at the single channel level. These types of data, however, put
greater demands on models of mslo gating than, for example, the models used to describe the gating of CNG
channels. The degree to which voltage-dependent MWC
models fail to mimic the gating of mslo therefore may
not indicate that these models are less valid for mslo
than other allosteric proteins, but rather in some instances the demands put on the models by the mslo data
are greater.
Often, careful examination of allosteric systems initially characterized as MWC systems reveals inconsistences with this simple model. For example, two main
conformations of hemoglobin (R and T) have been
crystallized (Perutz, 1976). However, experiments using the irreversible ligand Fe(III)-CN have revealed
that once a ligand is bound to one subunit of an
dimer, the protein is much more likely to convert to its
R conformation if the next ligand binds to a subunit of
the other
dimer rather than to the same dimer
(Holt and Ackers, 1995
). This indicates the presence of
intermediate states between the T and R conformations
and violates the MWC postulate that binding to any one
site affects all sites equivalently. Experimental evidence
also suggests that yeast pyruvate kinase can exist in an
intermediate state between T to R (Murcott et al., 1992
).
The discrepancies between our data and voltage-dependent MWC models are very likely indicating similar
complexities. As was shown, models based on the general form of scheme II do at least as well as voltage-
dependent MWC models in fitting mslo macroscopic
current behavior and, in some respects, better. General 10-state models, however, have six more free parameters than do voltage-dependent MWC models, and it
was often the case that for these models more than one
set of parameters could give reasonable fits to the
steady state data. As there are alternative explanations for the leftward shifting of the mslo G-V curves at [Ca]
over ~100 µM (Wei et al., 1994
; Solaro et al., 1995
), we
are not yet in a position to distinguish between models
containing interactions between binding sites in a
given conformation and those that do not. In fact, another model we have examined that conforms to scheme II and contains only three free parameters (at
equilibrium) can fit the mslo data with some success, although not as well as those described. This model is
based on the premise that the second Ca2+ ion that
binds to the channel provides all of the free energy that
stabilizes the open conformation relative to the closed. In the sense that a single binding event provides most
of the energy stabilizing the open state over the closed,
this model is similar to that proposed by McManus and
Magleby (1991)
to account for the Ca-dependent properties of single skeletal muscle BK channels under
steady state conditions. Whether the binding of Ca2+ to
some sites in the closed or open conformation affects
the affinities of the unoccupied sites will be important
to determine. However, even without supposing a deviation from strict MWC behavior, both the voltage-
dependent MWC models used here and the McManus and Magleby model predict that three Ca2+ ions must
bind to a channel at +30 mV before it is more likely to
be open than closed. Also, with a given number of Ca2+
ions bound, they both contain a single step between
closed and open.
Already, however, several lines of evidence indicate
that more than one voltage-dependent conformational
change is necessary for mslo channels to move from
closed to open, suggesting that better models of mslo
gating might be constructed if the intermediate states
in scheme I were not excluded. First, there is a brief delay in activation of mslo macroscopic currents, which
can be seen at most [Ca], including very low concentrations where the channels gate without bound Ca2+
(Toro et al., 1996; Cui et al., 1997
). To account for this
delay, multiple steps need to exist between closed and
open. Second, gating schemes with a single voltage-
dependent conformational change between closed and
open predict G-V relations described by a simple Boltzmann function, while better fits to higher powers of a
Boltzmann function are indicative of multiple voltage-dependent steps (Sigworth, 1994
; Zagotta et al., 1994a
,
1994b
). The mslo G-V relation often appears fairly well
fit by a Boltzmann function, but is usually best fit with a
Boltzmann function raised to a power between 1 and 3 (Cui et al., 1997
). And third, recent gating current
measurements from slo channels indicate that gating
charge movement precedes channel opening (Horrigan et al., 1996
; Ottolia et al., 1996
). Models of the
form of scheme II predict that ionic and gating current
relaxations follow the same time course. In the future,
an examination of the properties of models of the form
of scheme I may be fruitful as this scheme allows for
gating charge movement before channel opening.
Relation to Single Channel Studies
Single channel analysis of both native (McManus and
Magleby, 1988, 1991
; Art et al., 1995
; Giangiacomo et
al., 1995
; Wu et al., 1995
) and cloned (DiChiara and
Reinhart, 1995
; Lagrutta et al., 1996
) BK channels has
revealed that these channels exhibit stationary gating
kinetics with at least five exponential components to
their closed time distribution and at least three exponential components to their open time distribution.
How does a system display complex kinetic behavior at
the single channel level and relatively simple kinetic behavior when recorded from as a population? What sorts
of dwell time distributions do the voltage-dependent MWC models presented here predict? To answer the
second question, the Q-matrix methods described by
Colquhoun and Hawkes (1981
, 1982
, 1995b
) were used
to solve for the open and closed time distributions predicted by the voltage-dependent MWC model for patch
1. As might be expected, results of these calculations indicate that under most circumstances the distributions
are heavily dominated by a single component whose
time constant is well approximated by 1/
given by Eq.
9b for closed times and 1/
given by Eq. 9c for open
times. As discussed above, this is because, under most
conditions, model Ca2+ binding steps equilibrate rapidly relative to the closed-to-open equilibrium. However, as can be seen in Fig. 14, preliminary results suggest that when comparing model currents to the data,
voltage-dependent MWC models appear to mimic the
bursting of mslo, but fail to reproduce brief closures
during the bursts. Several studies indicate that these
brief closures contribute significantly to the complexity of the single channel dwell time distributions. When
brief (as well as very long) closures are ignored in the
analysis, BK channel burst time distributions are dominated by a single component (Methfessel and Boheim,
1982
; Magleby and Pallotta, 1983
; Moczydlowski and Latorre, 1983
, although a smaller component of bursts
comprised of brief single openings is also evident, Magleby and Pallotta, 1983
). The question then becomes,
how do these brief closures affect the time course of
macroscopic BK channel currents? This question has
been addressed by Colquhoun and Hawkes (1995a)
in the context of the following system, which bursts perceptibly when
is large relative to
and
.
[View Larger Version of this Image (21K GIF file)]
As pointed out, "when the gaps within bursts are brief,
noise and relaxation experiments will give a time constant that corresponds approximately to the mean
burst length rather than the mean open time" (Colquhoun and Hawkes, 1995a). Some of the complexity of channel gating is not evident in the macroscopic kinetic behavior. Thus, the observation that models designed to mimic macroscopic mslo currents do not predict brief closures within bursts is not unexpected.
However, this does not excuse the fact that voltage- dependent MWC models fail to predict this feature of
BK single channel recordings. What is the origin of
these brief closures? One possibility is suggested by
scheme I. If in this scheme Voltage sensor movement is
highly cooperative (as the data suggests), then under
most circumstances the final vertical equilibrium preceding each open state will be heavily biased towards
opening, creating a situation in which the channel is
likely to close briefly to states that precede open, and
then reopen producing brief flickers closed such as are
observed. The inability of the models of the form of
scheme II we have described to account for the complicated single channel kinetics of BK channels may be,
therefore, also a consequence of condensing several cooperative voltage-dependent steps into a single step.
Another possibility is that brief closures represent sojourns into closed states not involved in the activation pathway (see for example Hoshi et al., 1994
). Alternatively, McManus and Magelby (1991) have demonstrated that a subset of scheme II composed of five
closed and three open states can account for most of
the Ca-dependent stationary gating properties of native skeletal muscle BK channels at +30 mV, including the
flickering behavior. This demonstrates that models of
the form of scheme II are sufficient to generate such
complex single channel kinetics without supposing additional states. However, as they did not study the voltage-dependent properties of these channels, it is not
clear how well models like those they proposed could
mimic the voltage-dependent kinetics of skeletal muscle BK channel currents. Ultimately, models that can
account for both single-channel and macroscopic behavior over a wide range of [Ca] and membrane voltages are desirable. To build such models for mslo, a detailed kinetic analysis of single channel gating will be
necessary, and the existence of at least as many states as
shown in Fig. 2 will need to be taken into account.
Interpretation of Mutant Channel Behavior
Putting aside for the moment the problems with voltage-dependent MWC models, the success of these models in many respects suggests that they may be useful in
interpreting modifications or mutations of slo channels.
A common question, for example, might be: does a particular mutation that alters the apparent affinity of the
channel for Ca2+ affect the affinity of Ca2+ binding or
other conformational changes involved in gating? As has become common in the recent literature, changes
in channel behavior are often characterized in terms of
changes in the relationship between V1/2 and log[Ca].
Eq. 6 indicates that changes in L(0), and therefore the
intrinsic free energy difference between open and closed,
will affect the apparent Ca2+ affinity of the model channel, but will not affect the relative spacing of the G-V
curves determined at several [Ca]. A plot of V1/2 vs.
log[Ca], therefore, will not change shape from wild
type to mutant. It will simply translate along the voltage
axis. This is demonstrated in Fig. 15 A, where V1/2 vs.
log[Ca] is plotted for a hypothetical wild-type voltage-dependent MWC model channel (asterisks) and for the
case in which L(0) is reduced from 2,000 to 2 (triangles). Also shown are the effects of changing other model
parameters. For comparison, in Fig. 15 C, the curves
from Fig. 15 A have been shifted so that they have the
same V1/2 value at 0 [Ca]. Here, the curves corresponding to L(0) = 2,000 and L(0) = 2 superimpose, demonstrating that they have the same shape. This result
would suggest that a mutation has not affected the actual affinity of Ca2+ binding sites for Ca2+, but rather
L(0). On the other hand, according to scheme II, mutations that do affect the shape of the mslo V1/2 vs.
log[Ca] relation could arise in two ways: (a) a change
in the voltage sensitivity of the channel Q (Fig. 15,
squares), or (b) a change in one or more Ca2+ dissociation constants (Fig. 15, circles). As can be seen from Eq. 6, a change in Q scales a plot of V1/2 vs. log[Ca] by a
constant factor, while a change in the affinity of Ca2+
binding sites will affect the spacing of the G-V curves in
a less predictable manner. One way to distinguish between these possibilities then is to multiply the apparent gating charge of the channel determined from
Boltzmann fits by V1/2, and then to plot this value vs.
log[Ca]. This corrects for changes in the shape of the
V1/2 vs. log[Ca] relation due to alterations in Q. Plots of
QappV1/2 vs. log[Ca] are shown in Fig. 15 B. Notice that the wild type and increased Q plots superimpose, while
when KC is altered, the shape of the QappV1/2 vs. log [Ca]
plot differs from wild type. This type of plot would
therefore be preferable to a plot of V1/2 vs. log[Ca] for
differentiating changes in Ca2+ affinity from changes in
the intrinsic energy difference between closed and open.
Another approach to interpreting channel modifications would be to examine the wild type and mutant
channel's G-V relation at very low [Ca] where the channel is gating without binding Ca2+ (Meera et al., 1996;
Cui et al., 1997
). Here, according to scheme II, differences in the position of the G-V relation can only reflect changes in L(0) or Q. If it is only Q that has
changed, QappV1/2 will be the same for mutant and wild-type channels, while if L(0) has changed, this will not
be the case (see Fig. 15 B). Of course, if a mutation
were to make the shapes of the mslo G-V relations at different [Ca] grossly different, this would indicate that interpreting the effects of the mutation in terms of
models of the form of scheme II may not be valid.
Since scheme I may more generally represent mslo gating behavior than does scheme II, we might ask how modifications or mutations would affect the equilibrium properties of models of the form of scheme I? Restricting the analysis to conditions in which the G-V relations do not change shape as [Ca] is varied, simulations of the behavior of scheme I indicate that, as for scheme II, changes in the voltage-dependent equilibrium constant of each subunit (see Fig. 1 A) do not affect the shape of the V1/2 vs. log[Ca] curve, only its position along the voltage axis. Likewise, changes in the charge associated with each voltage-dependent step can to a very good approximation be compensated for by multiplying V1/2 by the apparent gating charge determined from a simple Boltzmann fit, and then plotting QappV1/2 vs. log[Ca]. For models of the form of scheme I, however, changes in the maximum slope of the G-V curves can also come about through alteration of the cooperative interactions between voltage-dependent steps. A decrease in cooperativity will make the G-V curves more shallow and less well approximated by a Boltzmann function. The spacing of the G-V curves, however, will not be affected by such a modification and, therefore, a plot of V1/2 vs. log[Ca] will maintain its shape. This is demonstrated in Fig. 15, A and C, where the asterisks can be thought of as representing scheme I with highly cooperative voltage-dependent conformational changes, and the diamonds represent scheme I with no cooperativity between voltage sensors. Although a plot of V1/2 vs. log[Ca] will maintain its shape after such a mutation, a plot of QappV1/2 vs. log[Ca] will not. It becomes more shallow as cooperativity decreases (Fig. 15, B and D). Also, unlike changes in Q or KC, a decrease in voltage sensor cooperativity causes a change in the value of QappV1/2 at 0 [Ca] (Fig. 15 B).
Considering the above discussion, according to scheme I, modifications that directly affect Ca2+ binding can be distinguished from those affecting gating steps intrinsic to the protein as follows. Given that the G-V curves have a similar shape at each [Ca] after the modification, if plots of both V1/2 vs. log[Ca] and QappV1/2 vs. log[Ca] change shape from wild type to mutant, the mutation is likely to be having a direct effect on Ca2+ binding constants. Otherwise, it is more likely that the effect is on conformational changes intrinsic to the protein. This is evident in Fig. 15, C and D where the only relation that differs in shape from wild type in both plots is that for changing KC.
In conclusion, the work described here demonstrates that many aspects of the gating of a BK channel can be understood through kinetic models that are based on the premise that the channel is a homomultimer with a Ca2+ binding site and voltage sensor in each subunit. Of these, voltage-dependent MWC models are the simplest, and yet they can mimic many aspects of mslo macroscopic kinetic and steady state behavior. This suggests that the allosteric mechanism of this channel may be similar to those of other allosteric proteins whose behaviors have been modeled with the MWC formalism. To gain a detailed understanding of mslo gating, however, more experimentation and more complex models along the lines of scheme I, or even more complex schemes, will be necessary. Very likely, such work will contribute to our understanding of allosteric proteins in general.
Address correspondence to Dr. Richard W. Aldrich, Department of Molecular and Cellular Physiology, Beckman Center B171, Stanford, CA 94305-5426. Fax: 415-725-4463; E-mail: raldrich{at}popserver.stanford.edu
Received for publication 4 April 1997 and accepted in revised form 11 June 1997.
1 Abbreviations used in this paper: CNG, cyclic nucleotide-gated; KNF, Koshland-Nemethy-Filmer; MWC, Monod-Wyman-Changeux.We gratefully acknowledge Larry Salkoff for providing the mslo clone, and Toshi Hoshi and Dorothy Perkins for developing software used in modeling channel behavior.
This work was supported by a National Institute of Mental Health Silvio Conte Center for Neuroscience Research grant (MH 48108). J. Cui was supported by a postdoctoral fellowship from the Muscular Dystrophy Association. R.W. Aldrich is an investigator with the Howard Hughes Medical Institute.
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