Novel description of the large conductance Ca2+‐modulated K+ channel current, BK, during an action potential from suprachiasmatic nucleus neurons

Abstract The contribution of the large conductance, Ca2+‐modulated, voltage‐gated K+ channel current, IBK, to the total current during an action potential (AP) from suprachiasmatic nucleus (SCN) neurons is described using a novel computational approach. An experimental recording of an SCN AP and the corresponding AP‐clamp recording of IBK from the literature were both digitized. The AP data set was applied computationally to a kinetic model of IBK that was based on results from a clone of the BK channel α subunit heterologolously expressed in Xenopus oocytes. The IBK model result during an AP was compared with the AP‐clamp recording of IBK. The comparison suggests that a change in the intracellular Ca2+ concentration does not have an immediate effect on BK channel kinetics. Rather, a delay of a few milliseconds may occur prior to the full effect of a change in Cai 2+. As shown elsewhere, the β2 subunit of the BK channel in the SCN, which is present in the daytime along with the α subunit, shifts the BK channel activation curve leftward on the voltage axis relative to the activation curve of BK channels comprised of the α subunit alone. That shift may underlie the diurnal changes in electrical activity that occur in the SCN and it may also enhance the delay in the effect of a change in Cai 2+ on BK kinetics reported here. The implication of these results for models of the AP for neurons in which BK channels are present is that an additional time dependent process may be required in the models, a process that describes the time dependence of the development of a change in the intracellular Ca2+ concentration on BK channel gating.


Introduction
BK channels are a significant factor underlying membrane excitability of suprachiasmatic nucleus (SCN) neurons (Cloues and Sather 2003;Jackson et al. 2004;Colwell 2006;Meredith et al. 2006;Pitts et al. 2006;Kent and Meredith 2008;Belle et al. 2009;Montgomery and Meredith 2012;Montgomery et al. 2013;Whitt et al. 2016). Specifically, they have been implicated in the diurnal changes in spontaneous firing that occur in the SCN (Colwell 2006;Meredith et al. 2006;Whitt et al. 2016). SCN neurons fire spontaneously at rates of 8-10 Hz during the day (Jackson et al. 2004). At night, the activity is suppressed typically to <2 Hz with many cells in the silent state (Inouye and Kawamura 1979;Green and Gillette 1982;Groos and Hendriks 1982;Shibata et al. 1982;Yamazaki et al. 1998). These neurons express a subunit, b2, having the potential to modify BK channel properties (Montgomery and Meredith 2012;Whitt et al. 2016). The b2 subunit produces inactivation of BK (Wallner et al. 1999;Xia et al. 1999). Inactivating BK currents are referred to as BK i . BK currents that exhibit relatively little inactivation are referred to as BK s . The primary BK component in the SCN during the day is BK i , whereas BK s is the primary BK current in the SCN at night (Whitt et al. 2016). The latter group has suggested that the difference in the inactivation properties of these two BK channel types underlies the differences in excitability in SCN neurons between night and day; repetitive firing during the day (BK i ), versus relative quiescence at night (BK s ).
The focus of this report is on the contribution of BK i channel current, I BK , to the total current during an action potential (AP) in the daytime. Those results have been measured in SCN neurons using the AP-clamp technique (Jackson et al. 2004;Whitt et al. 2016), a method in which a previously recorded AP is applied to a neuron before and after addition to the external medium of a specific ion channel blocker (Llinas et al. 1982;Bean 2007). For example, the rapidly activating and inactivating sodium ion current, I Na , has been measured in SCN neurons using the AP clamp together with tetrodotoxin, a specific blocker of I Na (Jackson et al. 2004). The amplitude and time course of I Na during an SCN AP was determined from the difference between the test and control results of the experiment. Jackson et al. (2004) also reported voltage clamp step recordings of I Na , the traditional method for analyzing an ionic current (Hodgkin and Huxley 1952). The two approaches together provide a more complete description of a particular ion current component than is provided by voltage clamp step results alone. Jackson et al. (2004) reported a similar analysis for the calcium ion current, I Ca . The I Ca component is required for BK channel activation during an AP. In a recent study the I Na and I Ca results from Jackson et al. (2004) were analyzed computationally to obtain models of these components suitable for SCN neurons (Clay 2015). In this report that analysis has been extended to I BK . The long-term goal of this work is a mathematical model of the AP for SCN neurons.

Methods
The results described below require an AP from an SCN neuron, the corresponding AP-clamp measurement of I BK , both of which are provided by Jackson et al. (2004;their Fig 12 with I BK corresponding to I KCa ), and a kinetic model of I BK gating. The AP and I BK recordings were digitized (Clay 2015). Those records are shown in Figure 1 with lines connecting the points. Figure 1 also contains simulations of I BK at various points during the AP. The data sets V i vs. t i for the AP and I BK,i vs. t i with i = 0,1,2,. . .., are the basis for the analysis that follows with V membrane potential in millivolts, time t in milliseconds, and I BK in picoamperes. The V i vs. t i data set for the AP was applied computationally to the model of I BK given below using Mathematica (Wolfram Research, Champaign, IL;Clay 2015). That model is based on voltage clamp step recordings of Cui et al. (1997) from mslo, a clone of the BK channel a subunit heterologously expressed in Xenopus oocytes. Those results were obtained using the inside-out voltage clamp mode so that Ca i 2+ could be controlled during the experiments. A novel feature of the I BK model concerns a putative Ca i 2+ dependence of some of its parameters. The model is given by: where g BK is BK channel conductance, g BK = 38.5 nS, n (V,t) is the voltage-and time-dependent gating variable of the channel, and E K is the K + reversal potential, E K = À96 mV (Jackson et al. 2004). The gating variable is determined by: The various parameters of the model were determined by comparing its predictions with the recordings of Cui et al. (1997). The model in Equations 1-3 is similar to the one used by Hodgkin and Huxley (1952) for their analysis of I K in squid giant axons with a Ca i 2+ dependence assigned to the model via V Ca , a Ca , and b Ca (Equation 3; Results). The V i vs. t i data set of the AP in Figure 1 was applied to Equation 2 using a procedure described in Clay (2015) to determine BK channel activation throughout the AP. The start point of the AP in Figure 1 is t 0 = 0, V 0 = À58 mV. This level of V is below activation of I BK during the interspike interval since Ca i 2+ during that time is~50 nmol/L, the level of Ca i 2+ in a resting neuron (McCormick and Huguenard 1992). At this level of Ca i 2+ , V = À58 mV is below the activation range of I BK (Cui et al. 1997;Xia et al. 1999), and so the start value for n, n 0 , is assumed to be 0. The next iterative value of n, n 1 , was determined from Equation 2 with NDSolve (Mathematica), using The iterative values of Ca i 2+ were obtained as described in Clay (2015) and shown below (Results). This procedure was continued throughout the V i versus t i data set of the AP waveform. The resulting digitized values of I BK corresponding to Equation 1 are Those results are shown in Figure 1 along with the experimental recording of I BK . The model for I BK does not contain an inactivation parameter since the results of Cui et al. (1997) do not clearly show inactivation over the duration of the voltage clamp steps -20 msec -used for those results (Discussion).

Results
Cai 2+ during an AP As noted above, the intracellular calcium ion concentration, Ca i 2+ , during an AP from SCN neurons is required to determine I BK . A model of I Ca is, in turn, required for this result. This component cannot be described by I Ca = g Ca (V-E Ca ), where g Ca is Ca 2+ channel conductance. Rather, I Ca has a nonlinear dependence on the Ca 2+ driving force (McCormick and Huguenard 1992). That relationship is well described by the Goldman-Hodgkin-Katz (GHK) equation, I Ca = a(exp(zq(V-E Ca )/kT)-1)/(exp(zqV/ kT)-1), where a is a constant related to Ca 2+ membrane permeability, z is the ionic valence of a Ca ion, z = 2, q is the unit electronic charge, k is the Boltzmann constant and T is the absolute temperature. At room temperature, which was used by Jackson et al. (2004), kT/ 2q = 12.5 mV. The extracellular concentration of Ca 2+ for the results in Jackson et al. (2004) (Fakler and Adelman 2008), which is also considerably less than 1.2 mmol/L. For the purposes of the GHK equation Ca i 2+ may assumed to be zero without significantly altering the results in this report over the range of potentials spanned by an AP. For Ca i For the purposes of BK channel activity as well as other features of SCN neuron behavior, Ca i 2+ is, of course, not zero. In particular, Ca i 2+ undoubtedly has "hot spots" adjacent to the Ca/BK complexes within the membrane during an AP (Fakler and Adelman 2008). This distribution has been simplified for modeling purposes by the assumption of two intracellular compartments for Ca i 2+ (Yamada et al. 1998;Diekman et al. 2013). One compartment corresponds to a thin spherical shell 0.1 lmol/L in thickness near the membrane surface (McCormick and Huguenard 1992). The Ca 2+ concentration in this compartment is denoted by Ca s . The other compartment corresponds to the cytosol. The Ca s parameter is given by: with K 1 = 3 9 10 À5 M/nC, K 2 = 0.04 ms À1 (Purvis and Butera 2005), c s = 2 nmol/L/ms, and I Ca ðV; tÞ ¼ ÀGHKðVÞf ðtÞ½305r 2 1 ðV; tÞ þ 31r 2 ðV; tÞ (5) where f(t) is I Ca inactivation, and r 1 and r 2 are the gating activation parameters for the two types of I Ca channels present in SCN neurons, one of which appears to be the L-type channel based on its sensitivity to nimodipine. The other component is nimodipine-insensitive (Jackson et al. 2004). A full description of the I Ca model is given in Clay (2015). The model was previously applied to the AP in the inset of Figure   noted above, corresponds to the AP used by Jackson et al. (2004) for their AP-clamp recording of I BK . The I Ca analysis must necessarily be carried out for the same AP used for the AP-clamp recording of I BK .
I BK model The model for I BK , Equations 1-3, predicts a response to a rectangular voltage clamp step that is consistent with a single exponential function of time, as in the results of Cui et al. (1997). Similar results have been reported for BK splice variants heterologously expressed in HEK cells (Shelley et al. 2013), as well as results from mslo/CaV channel complexes also heterogously expressed in HEK cells (Cox 2014 Cui et al. (1997). The deactivation kinetics in the right-hand panel of Figure 3 are to be compared with the corresponding results in Figure 1B of Cui et al. (1997).

Ca i 2+ dependence of I BK results
Cui et al. (1997) reported results for Ca i 2+ = 0.84, 1.7, 4.6, and 10.2 lmol/L, as well as higher levels of Ca i 2+ . The channel activation curves for 0.84 ≤ Ca i 2+ ≤ 10.2 lmol/L, given in their Figure 5B, are reproduced here in Figure 4. These results represent currents at the end of 20 msec duration voltage steps, sufficiently long so that the channel kinetics at any given voltage were at their steady-state level. In the model these results correspond to n ∞ = a BK (V,V Ca )/(a BK (V,V Ca )+ b BK (V,V Ca )) with a BK and b BK as given by Equation 3. The predictions of this equation for n ∞ are represented by the curves in Figure 4. The time constants of the model, s BK , are given by s BK = 1/(a BK +b BK ). This equation predicts the bell-shaped curves in Figure 5 shown along with experimental results for s BK from Cui et al. (1997) corresponding to Ca i 2+ = 0.84, 1.7, and 10.2 lmol/L. The curve for Ca s = 0.84 lmol/L is shifted leftward on the voltage axis by an increase in Ca s similar to the results for the activation curves (Figure 4). The maximal time constant at any given level of Ca s is itself dependent upon Ca s , a bell-shaped dependence as indicated in Figure 1D of Cox (2014). Results for s BK from Cui et al. (1997) and Cox (2014) Figure 6). The former is below the resting level of Ca i 2+ , which is 0.05 lmol/L (McCormick and Huguenard 1992). The latter, Ca i 2+ = 118 lmol/L, would appear to be well above the maximum level of Ca i 2+ reached during an AP (10-20 lmol/L, Berkenfeld et al. 2006;Fakler and Adelman 2008). Cox (2014) described a detailed model of BK channel activation including the effects of Ca i 2+ on activation gating. The model in this report is relatively simple, which is useful for the analysis of I BK during an AP. Those results are qualitatively similar to the experimental recording of I BK , although their timing does not completely match experiment. The simulations assume that the effect of a change in Ca s on BK Figure 3. Predictions of the model in Equations 1-3 for I BK . Left panel: Currents elicited from the model with 6 msec duration voltage clamp steps from +20 to +110 mV, with 10 mV increments between each step. Initial value of n(t) in Equation 2 at the beginning of each step was n = 0. Right panel: Deactivation currents for an initial value of n = 1 with V = À10, À30, À50, and À70 mV. channel kinetics occurs instantaneously. Alternatively, a delay of a few milliseconds may be involved (Hille 2001;Martinez-Espinosa et al. 2014), which could account for the difference between experiment and theory in Figure 1.

Discussion
As noted above (Introduction), Whitt et al. (2016) have suggested that BK channel inactivation is a significant factor in SCN excitability. Specifically, BK i , the primary BK channel component during the day, exhibits pronounced inactivation during voltage clamp step recordings, whereas BK s , the primary BK component at night, does not. However, significant inactivation of BK occurs only for V > +30 mV (Whitt et al. 2016; Fig. 2a), which is above the maximum overshoot potential of the SCN AP (Jackson et al. 2004; Fig. 2). Moreover, the time constant of inactivation at potentials for which it does occurs is 45 msec, and that result is relatively insensitive to changes in V (Whitt et al. 2016). The duration of the SCN AP at its midpoint is~4 msec (Jackson et al. 2004), significantly less than the time constant of inactivation. The membrane potential spends considerably less time than 4 msec at potentials greater than 0 mV. These observations taken together suggest that BK inactivation is not an important factor for excitability in the SCN. Another aspect of BK channel gating, its activation curve, could be significant. The b2 subunit not only produces inactivation of BK at strongly depolarized potentials, it also shifts the BK channel activation curve leftward on the voltage axis for a given level of Ca i 2+ (Wallner et al. 1999;Xia et al. 1999). Results from BK i and BK s channels in rat adrenal chromaffin cells (RCC) may be relevant to the roles of these channels in the SCN. The activation curve for BK i in those cells is also shifted leftward relative to the activation curve for BK s (Sun et al. 2009). Constant current injection in a model of an RCC with BK i produces repetitive firing, whereas constant current injection in model cell having BK s produces only one or a very few APs (Sun et al. 2009). As noted by Sun et al. (2009), ". . .These differences arise, not because of the inactivation behavior of BK i current, but from the more negatively shifted range of activation of BK i channels at a given Ca i 2+ in comparison to BK s current." A similar conclusion may apply to the SCN. Specifically, the diurnal changes that occur in the SCN may be attributable to a voltage shift of the BK channel activation curve produced by the b2 subunit rather than inactivation of the BK channel which, additionally, appears to occur outside of the range of membrane potentials spanned by an AP. The results in this report are based on a clone of the a subunit, a BK channel which does not possess the b2 subunit. A leftward shift of the activation curve attributable to that subunit may produce an increase in the difference between experiment and theory in Figure 1, a delay in the onset of the  Cui et al. (1997). Specifically, the results for Ca s = 0.84, 1.7, and 10.2 lmol/L were taken from Figure 2A and 3A, 2B and 3B, and 2C and 3C, respectively (Cui et al. 1997 Cui et al. (1997) and Cox (2014). The results for Cui et al. (1997) correspond to the maximum values of the curves in Figure 5: Ca s = 0.84 lmol/L, 5.3 msec; Ca s = 1.7 lmol/L, 5.0 msec; Ca s = 10.2 lmol/L, 3.1 msec. The results from Cox (2014) were taken from Figure 1D of that report: Ca s = 0.9 lmol/L, 5.8 msec; Ca s = 2.4 lmol/L, 4.2 msec; Ca s = 7.8 lmol/L, 1.9 msec, and Ca s = 22 lmol/L, 1.5 msec. These results were normalized relative to the Ca s = 0.9 lmol/L result from Cox (2014). The curve corresponds to 1/[1 + (Log{Ca s /Ca o }) 2 ] with Ca o = 1 lmol/L. effects of a change in Ca s on BK channel gating that is greater than the delay indicated in Figure 1. Additional experiments and simulations of I BK during an AP may be needed to clarify this issue.
The effects of Ca i 2+ on BK channels are similar, in some respects, to the effects of Ca i 2+ on synaptic vesicle release in nerve terminals following an AP (Neher 1998;Augustine et al. 2003). An AP triggers entry of Ca 2+ into the terminal via voltage-gated Ca 2+ channels. These channels almost certainly lie in close proximity to vesicle release sites. As a result, the local Ca 2+ concentration may briefly rise to levels of 100 lmol/L, or higher (Neher 1998). BK channel activation in presynaptic nerve terminals has been used to report these levels of Ca i 2+ (Yazejian et al. 2000). Following an AP, vesicle release drops precipitously, which is consistent with a similarly rapid drop in Ca i 2+ , probably due to various Ca 2+ buffering mechanisms (Parnas and Parnas 1994;Neher 1998). In SCN neurons Ca 2+ channels are likely to be in close proximity to BK channels similar to the relationship between Ca 2+ channels and vesicle release sites in presynaptic nerve terminals. During an AP, the Ca 2+ concentration in the vicinity of BK channels may rise to somewhere in the 10-20 lmol/L range, or perhaps higher (Berkenfeld et al. 2006;Fakler and Adelman 2008). The simulation in Figure 2 is consistent with this result. Specifically, Ca s equals 15 lmol/L midway through repolarization of the AP. Calcium ion buffers may not be required to reduce this level of Ca i 2+ thereby terminating BK channel activation because repolarization of the AP, a process to which I BK contributes, brings the membrane potential below BK channel threshold even with Ca s in the 1-10 lmol/L range. In this sense I BK is self-limiting. The time between APs in a spontaneously firing SCN neuron during the day is approximately 100 msec, considerably longer at night, either of which is sufficient for Ca 2+ to diffuse passively away from the membrane bringing Ca s to its baseline level prior to the subsequent AP. Indeed, Ca s in the simulation in Figure 2 is reduced below 15 lmol/L by passive diffusion even before the AP has ended.
Voltage-dependent activation of BK channels may not require the presence of Ca i 2+ , although strong depolarizations are needed for channel activation with very low Ca i 2+ . For example, Cui et al. (1997) reported mslo channel currents with Ca i 2+ = 0.5 nmol/L. Cox (2014) reported similar results with Ca i 2+ = 3 nmol/L. The midpoint of the channel activation curve for these conditions is V = +200 mV for Ca i 2+ = 0.5 nmol/L and V = +150 mV for Ca i 2+ = 3 nmol/L, both of which are well positive to the range of potentials spanned by an AP even with a 50-60 mV leftward shift of the activation curve on the voltage axis caused by the b2 subunit (Xia et al. 1999).
As noted above, this work may have significant implications for computational models of the AP for neurons in which BK channels are present. In traditional models, such as the Hodgkin and Huxley (1952) model of the AP in squid giant axons, the intracellular and extracellular concentrations of permeant ions, Na + and K + in the case of squid axons, are fixed throughout the AP although the effective extracellular K + concentration can change due to K + accumulation in the extracellular space between the axolemma and the surrounding Schwann cell (Frankenhauser and Hodgkin 1956). This effect can be accounted for in the AP model by assigning a time dependence to E K (Clay 1998). The intracellular Ca 2+ concentration provides another example of an ion concentration not remaining fixed during an AP, especially in the vicinity of BK channels. This result does not significantly change the driving force, V-E Ca , over the range of potentials spanned by the AP (Clay 2015). The intracellular Ca 2+ concentration, Ca i 2+ , does not appear in the expression given above for GHK(V) -RESULTS. A change in Ca i 2+ does modify BK kinetics, perhaps with a delay and that delay would have to be accounted for in a model of the AP.