INTRODUCTION

Large 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.

METHODS

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 = G_max/(1 + e^zF(^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:

(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 M¹s¹ 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 M¹s¹ 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.

n 

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 k_f and n  k_g degrees of freedom at the desired level of significance. Here, SSE_f represents the residual sum-of-squares for the fit with model f, which is a subhypothesis of model g; SSE_g represents the residual sum-of-squares for the parent model g, n represents the number of data points, and k_f and k_g 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 [View Table]

Table II. General 10-State Model Parameters [View Table]

Table III. Voltage-dependent MWC Model Parameters [View Table]

RESULTS

For simplicity, in modeling mslo macroscopic currents we have made the following assumptions about the structure of the channel.

(a) When expressed alone, mslo  subunits form homotetrameric channels.

The 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).

(b) Each mslo  subunit contains a single Ca²⁺ binding site.

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).

(c) Each mslo  subunit contains a single voltage-sensing element.

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 (P_open) of models of the form of scheme II can be written as the following function of [Ca] and voltage:

P_open = 

(2)

where K_Ci is the dissociation constant for the subunit binding the ith Ca²⁺ in the closed conformation, K_Oi 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, P_open depends on voltage in the form of a Boltzmann function:

with

B = 

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 (V_1/2) 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 V_1/2 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 (K_C), the open Ca²⁺ dissociation constant (K_O), 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, V_1/2 = 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 (V_1/2) 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 V_1/2 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 mol¹ 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 V_1/2 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 V_1/2 brought about by increases in [Ca]. The slope of the V_1/2 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 V_1/2 between the G-V curves for any two [Ca] divided by the difference in V_1/2 for any other two [Ca], is unaffected by changes in Q. This is a manifestation of the fact that a plot of V_1/2 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, K_C and K_O 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 V_1/2 as [Ca] is increased from 0 to a saturating value. Comparing Fig. 6 D to A for example, as K_C 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 K_C/K_O. 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 K_C and K_O 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, K_C and K_O 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 K_C1  K_C4 and K_O1  K_O4 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 K_C4 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 (K_C4/K_O4) 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 K_C4 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 (K_XA) or one (K_XB) neighbor occupied are defined, the dissociation constant for a site with two neighbors occupied is determined (K_XC = K_XB²/K_XA). 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 K_CA = 0.046 µM, K_CB = 1.42 µM, K_CC = 43.90 µM, K_OA = 0.047 µM, K_OB = 0.100 µM, K_OC = 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 (K_CA and K_OA 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

P_open as a Function of [Ca]

The steady state behavior of mslo and model channels can also be compared by looking at P_open 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)

1

1

1

1

1

1

1

1

1

1

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 P_open. 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 f_CX and f_OX represent the fraction of closed (f_CX) or open (f_OX) 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 M¹s¹) 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 Ta