Macroscopic current at any condition reflects the number of channels (*N*) in a patch or cell, the elementary current (*i*) through an individual channel, and the probability that a channel is open (*P*_{o}) at a given condition. Since for most situations, *N* is invariant and, at least over some range of voltages, the elementary current varies ohmically with voltage, the macroscopic conductance (*G*) reflects solely the distribution of channels between closed and open states. Thus, *G*(*V*) = *N***g** *P*_{o}(*V*).

For Slo3 channels, increases in pH cause some shift in the

*G*-

*V* relationship to more negative potentials qualitatively similar to regulatory effects of Ca

^{2+} on the homologous Slo1 K

^{+} channel (

Zhang et al., 2006). However, the most dramatic effect of pH appears to be an increase in the apparent maximal conductance (

*g*_{max}) of Slo3 macroscopic currents as pH is increased. This contrasts with Slo1 in which sufficiently positive activation potentials are able to drive Slo1 channels into similar saturating open probabilities (

Cox et al., 1997), even at low [Ca

^{2+}]. For Slo1, the similarity of the saturating

*P*_{o} at different [Ca

^{2+}] is expected to arise from the intrinsic voltage dependence of the closed–open equilibrium and the strength of coupling of voltage sensor movement to channel opening (

Horrigan and Aldrich, 2002). For Slo3 channels, pH may exert effects on channel properties that are independent of effects on

*P*_{o}, for example, by direct effects on ion permeation. Therefore, it is imperative that the effects of pH on Slo3 channel behavior be determined to establish the validity of the macroscopic measurements for mechanistic analysis.

Three particular properties of Slo3 single channels are of interest: first, the effect of pH and voltage on apparent single channel current amplitude, second, the single channel open probability (*P*_{o}) over the range of pH and voltages of the macroscopic current measurements, and, third, the maximal single channel *P*_{o}. To answer all of these questions would require the measurement of single channel activity at potentials positive to +200 mV. Because of the technical challenges to recording channel activity of sufficient duration at positive potentials, we have approached this issue in two ways. First, we have made single channel measurements at potentials from +40 to +200 mV, a range over which substantial changes in macroscopic conductance are observed. Second, to obtain estimates of single channel behavior at potentials up to +300 mV, we have employed σ^{2}*/I* analysis.

Slo3 Channel Openings Exhibit Complex Open State Behaviors

For all results in this paper, Slo3 single channel activity or ensemble current activity at a given command potential was activated by repeated voltage steps usually of 200 ms duration. Examples of current openings at +100, +140, and +180 are shown for one patch in Activity was characterized by very brief openings and closings that preclude clear resolution of a well-defined open current level. Total amplitude histograms generated for a set of sweeps at the three command potentials reveal a complex distribution of current values (), suggestive of multiple current levels, each with rather substantial variance. The shape of the open current distributions was clearly not consistent with a simple filtering of a rapid two-state open to closed transition. The complexity in the open current behavior is also evident in the fastest time-base examples during which there is considerable variance in open current levels occurring on a time scale of tens to hundreds of milliseconds. Such variance in current levels is not found during complete closings of the Slo3 channel.

Total amplitude histograms were fit by a sum of Gaussian components. Visually, histograms from activity at the most positive voltages (positive to +100 mV) revealed two broad peaks in addition to baseline current values (). However, fits of three Gaussian components (; one baseline and two open levels), although adequate at the less positive potentials (e.g., +100 mV) did not adequately fit the histogram at the most positive voltages (e.g., +180 mV). Hence, over potentials of +120 to +180 mV, total amplitude histograms were best described by four Gaussian components (; three open levels and one baseline). Although a mean value for each of the open levels could be defined, each open level exhibited large variance that was relatively symmetric around the mean values. We have treated the smallest open level as a skewed shoulder in the baseline peak and have limited our analysis to the two primary open level components in the total amplitude histograms.

At Positive Potentials, Average Single Channel Current through Slo3 Channels Varies Ohmically with Voltage

For a set of four patches, the two open level components were measured for voltages of +120 through +180 mV. The mean amplitude of each component varied approximately ohmically with voltage (), yielding estimates of apparent mean conductance of 108.5 ± 5.1 pS (component O1) and 56.6 ± 1.7 pS (component O2). The relative contribution of each component to the distributions did not vary significantly with command voltage, and the smaller amplitude component (O2) contributed ~60% of all open current values (). The lack of change in the ratio of the two components indicates that voltage-dependent changes in macroscopic Slo3 conductance do not arise from a voltage-dependent change in the fractional occupancy of the different conductance levels observed in the single channel behavior. Thus, at least over this limited voltage range, despite the complexity in the Slo3 behavior, the average mean current for a channel varies with voltage in an approximately ohmic fashion (). A weighted mean single channel current amplitude was defined based on both open level components, yielding an average single channel conductance of 77 ± 2 pS.

We next examined the effect of voltage on the fraction of time a Slo3 channel was in open states. From the total amplitude distributions, the fraction of time channels were in components O1 and O2 provides an estimate of the probability that a channel is in open states. *P*_{o} estimates made from the sum of the fractional time in O1 and O2 show that depolarization increases *P*_{o}, but that at +180 mV, channels are open only ~20–25% of the time ().

The generally ohmic behavior of average Slo3 single channel current and absence of effects of voltage on the fraction of O1 and O2 provide some confidence that Slo3 macroscopic currents can be used to provide a description of the overall closed–open equilibrium behavior. However, this conclusion must be tempered because of the limited range over which such single channel properties were determined.

Average Unitary Currents of Slo3 Channels Vary Linearly with Voltage

To evaluate properties of Slo3 unitary currents over a wider range of activation conditions, we have employed analysis of Slo3 current variance. In

Appendix, we derive and evaluate the predictions for σ

^{2}/

*I* relationships resulting from a population of channels in which individual channels can open into either of two conductance levels. In brief, the expected σ

^{2}/

*I* relationship remains parabolic. Furthermore, it is shown that for the case under consideration here, i.e., the single channel conductance of O2 is about half of O1, and the fractional occupancy within either open level is about the same, application of

Eq. 1 will result in an estimate of apparent

*i* that will exceed the average open channel current level estimated from single channel measurements by ~10%. In addition, the estimate of

*P*_{o} derived from application of

Eq. 1 will systematically underestimate the true values by ~10%. We therefore consider the estimates of

*i* and

*P*_{o} derived from application of

Eq. 1 to Slo3 σ

^{2}/

*I* relationships as acceptable estimates of the underlying unitary behavior and that such estimates can be meaningfully compared with the direct single channel estimates described above.

The variance and mean of currents resulting from sets of 100 activation steps at a given voltage and pH were generated. For a set of 10 patches, σ

^{2}/

*I* relationships at any specified pH and voltage did not reveal any well-defined parabolic behavior. This suggests that the average

*P*_{o} even at +280 mV and pH 8.5 was rather low, and well below 0.5. Based on the fact that increasing the number of sweeps used in the analysis helps improve reliability in the fitted estimates in σ

^{2}/

*I* analysis (

Lingle, 2006), we used a strategy in which up to nine separate sets of σ

^{2}/

*I* relationships were simultaneously fit (

Eq. 2) over voltages from +80 through +280 mV and including data at pH 8.5 () and pH 7.6 (). From this procedure, estimates of unitary single channel conductance (

*g*) and the number of Slo3 channels in the patch (

*N*), were obtained for each patch. Over the range of voltages from +80 to +280 mV (; ), the initial slope of the σ

^{2}/

*I* relationships was well described, indicating that the apparent single channel current does vary ohmically with voltage. Furthermore, this estimate of unitary conductance was similar at both pH 7.6 and pH 8.5 (), indicating that the effects of pH on macroscopic currents do not arise from an effect of pH on single channel current. For some patches, the σ

^{2}/

*I* relationship of binned mean current values (

Steffan and Heinemann, 1997) was also determined to minimize excess weighting by variance estimates where mean current is relatively unchanging (). Estimates using the binned σ

^{2}/

*I* approach were similar to those obtained using all mean current values.

After

*N* was defined from the simultaneous fit of all σ

^{2}/

*I* relationships in a given patch, each individual σ

^{2}/

*I* relationship was then refit with

*N* constrained to the value defined from the simultaneous fit. This provided individual estimates of

*i* at different voltages and pH (), yielding single channel estimates of 81 ± 2 pS for all patches at pH 8.5 and 76 ± 16 pS for all patches at pH 7.6. Consistent with the direct unitary current measurements from single channel patches (, red circles), the average unitary current estimate from σ

^{2}/

*I* analysis varied ohmically over the range of +80 to +280 mV. The conductance estimates from σ

^{2}/

*I* analysis are therefore quite similar to the average single channel current estimates from single channel experiments. Based on the considerations given in the

Appendix, we would have expected the apparent estimate of

*i* from the σ

^{2}/

*I* to exceed the weighted mean of the single channel current amplitudes by ~10%. Given the nature of each type of measurement and the procedures used to extract the average unitary current values, we simply view the results as support for the idea that both methods are yielding similar information about the unitary current properties. The key point of significance for our understanding of Slo3 gating behavior is that the estimates of unitary current from the σ

^{2}/

*I* method vary ohmically with voltage and are not influenced by pH. At pH 8.5, our results strongly argue that the relative occupancy of channels in O1 and O2 open levels is not influenced by either voltage or pH. At pH 7.6, we do not have direct measurements of fractional occupancy in O1 and O2. However, based on the considerations in the

Appendix and the fact that the unitary current estimates from the σ

^{2}/

*I* method are identical at both pH, a pH-dependent change in fractional occupancy of O1 and O2 would require that the ratio of the amplitudes I1 and I2 was also changing. A simpler explanation for the equivalence of the o

^{2}/

*I* estimates of single channel current at both pH 7.6 and 8.5 is that fractional occupancy in O1 and O2 is not influenced by pH.

Slo3 Open Probability Is Increased by Both Depolarization and Increases in pH

The estimated values for

*i* and

*N* for a given patch were used to calculate

*P*_{o} (

*P*_{o} =

*I*_{max}/

*i**

*N*) at each voltage and pH. At pH 8.5 and +280 mV, the maximal

*P*_{o} was ~0.3 (), suggesting that the limiting

*P*_{o} for Slo3 channels is quite low compared with Slo1 channels. As developed in

Appendix, these estimates are expected to be on the order of ~10% less than the true

*P*_{o}, a deviation certainly within the variability among patches. Is it possible that the estimates of

*P*_{o} by this method are severe underestimates? This seems highly unlikely since if the true

*P*_{o} was substantially higher, it would require that the σ

^{2}/

*I* relationships should show substantially more curvature. As illustrated ( and ), a higher

*P*_{o} (than estimated) requires that

*N* be smaller than what we have observed. A decrease in the estimate of

*N* on the order of 25% results in σ

^{2}/

*I* relationships that are clearly inconsistent with the observed data. On the other hand, it is more difficult to exclude the possibility that we are overestimating the true

*P*_{o}, since the observed σ

^{2}/

*I* relationships reveal only a slight curvature. Irrespective of this uncertainty, the results clearly argue that the limiting Slo3 open probability even at pH 8.5 and +280 mV probably does not exceed 0.3. Furthermore, irrespective of the actual limiting values at +280 mV, the limiting values do differ between pH 8.5 and pH 7.6, since these estimates are defined within the same patch.

*P*_{o} estimates from the σ

^{2}/

*I* analysis are summarized in with

*P*_{o} estimates from the unitary single channel current analysis also included for comparison. Both procedures provide essentially identical estimates for single channel

*P*_{o}.

To compare the voltage dependence of Slo3 *P*_{o} to the macroscopic Slo3 conductance, the value of the macroscopic conductance at pH 8.5 and +280 mV was normalized to the σ^{2}/*I* estimate of *P*_{o} at pH 8.5 and +280 mV (). This normalization procedure shows that the dependence of Slo3 single channel *P*_{o} at pH 7.6 and 8.5 follows a similar dependence on voltage and pH as the macroscopic conductance estimates. Similarly, using all macroscopic conductance values, the complete dependence of Slo3 Po on voltage and pH is shown in . Overall, this analysis supports the view that the limiting *P*_{o} at positive potentials differs at different pH.

Reduced Apparent Single Channel Amplitudes at Negative Potentials Probably Arises from Effects of Filtering on a Fast Channel Closing Process

Macroscopic recordings show that Slo3 currents deactivate very rapidly at negative potentials such that the peak of the Slo3 tail current exhibits a markedly nonohmic behavior (

Zhang et al., 2006). In fact, peak tail current amplitude measured at 10 kHz bandwidth is essentially flat from −80 through −200 mV, despite the over twofold increase in driving force on the currents.

We examined the properties of unitary Slo3 currents under similar conditions by undertaking an analysis of Slo3 current variance during tail currents. Currents were initially activated by repeated steps to +240 mV and then tail currents monitored at potentials of −60, −100, −140, or −180 mV (). From the averages of four sets of currents, an estimate of *N* was obtained from the fit of the binned σ^{2}/*I* relationship at +240 mV (). The σ^{2}/*I* relationships at each negative voltage were then fit () with *N* constrained to the value estimated at +240 mV. Although the estimate of *i* at −60 mV was essentially ohmic with the estimate at positive potentials, at more negative potentials, the estimate of *i* showed marked rectification (). If this nonlinearity in Slo3 single channel conductance reflects a true change in conductance rather than a consequence of filtering of rapid kinetic transitions underlying Slo3 gating equilibria, this would cause errors in attempts to estimate Slo3 conductance at negative potentials.

We have therefore evaluated the extent to which the reductions in tail current amplitude and estimates of single channel current amplitudes are predicted by the filtering levels used in our experiments. examines the reduction in idealized tail current amplitude at a given voltage that would be expected based on different deactivation time constants (τ_{d}: 10, 20, and 50 μs) and filtering settings (10, 20, and 50 kHz 8-pole Bessel). With τ_{d} = 50 μs, marked reduction of tail current amplitude is observed at both 10 and 20 kHz filtering, while at τ_{d} = 10 μs, even the 50 kHz filtering reduces peak tail current by almost 40%. For comparison, the major component of Slo3 deactivation is ~15–30 μs, becoming somewhat faster at more negative potentials. The typical filtering used in our experiments was 10 kHz. Based on the measured τ_{d} for Slo3 over voltages of −60 to −200 mV, the relationships developed in allow a calculation of the fractional reduction in tail current amplitude expected for the measured Slo3 τ_{d} estimates. This predicted fractional reduction both for 10 and 50 kHz is compared with the measured estimates of single channel current (from σ^{2}/*I* analysis; ) at both 10 and 50 kHz (). The nonlinearity in the single channel current estimates generally approximates the expected reduction in tail current amplitude. Similarly, 10 kHz filtering of idealized single channel closures following a voltage step shows that for openings lasting <30 μs, the observed single channel amplitude will be markedly reduced (). This suggests that the σ^{2}/*I* estimate of single channel current at negative potentials represents the average open channel current level from largely unresolved openings, and not a true reduction in single channel current amplitude.

These considerations argue that the flattening of the single channel current estimates at negative potentials represents the rapidity of the underlying Slo3 gating mechanisms and not a true effect on conductance. Since the intrinsic value of single channel conductance does not vary with voltage, the net charge through Slo3 channels at negative potentials should therefore provide a direct measure of the gating equilibrium–dependent open probability at those potentials, irrespective of the effects of filtering. This is summarized in , in which it is shown that the net charge through any open channel is unaffected by filtering, no matter how brief the opening.

Slo3 Channel *P*_{o} at Negative Potentials Is Appreciable and Relatively Voltage Independent

Macroscopic measurements suggest that Slo3 conductance at negative potentials and pH 8.5 may be ~0.1–1% of the Slo3 conductance at positive potentials and be relatively voltage independent (

Zhang et al., 2006). Here in patches with very few channels, we examine Slo3 channel activity at negative potentials. A primary difficulty in these experiments is that the brief and flickery nature of the Slo3 channel openings make it difficult to assert unambiguously whether particular events are, in fact, Slo3 openings. However, such pH-dependent activity is not observed in patches from DEPC-injected (

*n* = 3) or uninjected (

*n* = 2) oocytes. As a consequence, here for the moment we simply assume that the events we see observed are Slo3 openings. We then ask whether the observed events have properties we might expect for Slo3 openings.

Examples of Slo3 channel activity in a two-channel patch are shown for each of four potentials (+100 mV, ; −100 mV, ; +200 mV, ; −200 mV, ). A clear increase in channel *P*_{o} is seen in moving from −100 to +100 mV and then to +200 mV. Whereas the openings to the largest conductance level expected for a single channel (O1: 110 pS) exhibit a generally ohmic behavior at +100 mV and +200 mV, the brief open events at −100 and −200 mV fail to approach the 110 pS level. Based on rapid deactivation at negative potentials and the effect of filtering, the brevity and amplitude of these events are not unexpected. Faster time-base examples of activity at each of these four potentials is shown in . The less than ohmic peak amplitudes of the brief events at negative potentials are generally consistent with the reductions seen in unitary current amplitudes from the σ^{2}/*I* analysis of the tail currents ( and ).

To make estimates of relative conductance for such patches, total amplitude histograms for all traces at a given voltage were generated () with care taken to ensure that the bins corresponding to the fully closed channels were symmetrically distributed around 0 current. For all current values, the average current in excess of the baseline was then determined for each voltage. From this, the average conductance at each voltage was calculated (). The results from this analysis indicate that the brief Slo3 openings observed at −100 and −200 mV at pH 8.5 contribute a total conductance that is approximately two to three orders of magnitude less than that observed at the most positive activation potentials. Although stochastic variability is most certainly an issue in such patches, any change in the measured conductance between −100 and −200 mV appears minimal compared with changes observed between −100 and +200 mV. A fit to the points at −100 and −200 yields an estimate of the voltage dependence of the closed–open equilibrium of 0.075 *e*. For the points at −100, +100, and +200 mV, the fitted slope yields an effective valence of 0.354 *e*, presumably reflecting primarily the contributions of voltage sensor equilibrium to current activation.

In the absence of established pharmacological tools to block Slo3 channels, it is difficult to prove that the brief events seen at negative potentials are unquestionably Slo3 openings. Yet, the observed openings have the reduced amplitude and rapid kinetic behavior expected for Slo3 openings, and also contribute to the overall conductance in a way that is consistent with our expectations from other measurements. Thus, it seems likely that the events we are observing at −100 and −200 mV are Slo3 openings.

As an additional test of the properties of unitary Slo3 openings at negative potentials, we have examined single channel tail currents conditional on an opening occurring at the end of a preceding depolarizing step. Leak and uncompensated capacitance were removed using an idealization of an adjacent null sweep. For two patches, tail current openings were monitored at either −80 or −100 mV. A set of 60 traces was identified in which both an opening occurred at the end of the depolarization and an adjacent sweep contained no openings. After subtraction, 22 of the 60 openings revealed clear tail current openings that closed in much less than 1 ms. This frequency of detected closures fits closely with the expectation for channels closing at a rate approaching 50 kHz with 10 kHz filtering (). Most closures would simply not be detected, since closure would occur during the transition from outward to inward current. Selected examples of cases in which such openings were observed are shown in . These tail current closures of Slo3 channels are qualitatively similar in terms of brevity and amplitude to the openings seen at −100 mV in . In rare cases, similar reopenings are also observed (e.g., , third trace from top) at later times in the trace. These observations lend support to the idea that the measured steady-state channel activity examined in does, in fact, reflect Slo3 channels.