INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

Some vertebrates use electrical tuning of sensory hair cells as a filtering mechanism to distinguish different frequencies of sound. The mechanism was first described in the turtle (Crawford and Fettiplace, 1980), and subsequently in amphibia, other reptiles, birds, and fish (Ashmore, 1983; Lewis and Hudspeth, 1983; Fuchs et al., 1988; Fuchs and Evans, 1988; Sugihara and Furukawa, 1989). Electrical tuning occurs by an interplay between voltage-gated calcium current and potassium current through neighboring large conductance, calcium and voltage-activated potassium (BK) channels (Lewis and Hudspeth, 1983; Art and Fettiplace, 1987; Hudspeth and Lewis, 1988a). Different electrical tuning frequencies result from the varied, cell-specific kinetics of the BK channels (Art and Fettiplace, 1987, Art et al., 1995; Hudspeth and Lewis, 1988b). Kinetically distinct BK channels may be formed by a combination of two mechanisms: alternative splicing of the slo- gene that codes for the pore-forming BK channel subunit as well as modulation by accessory  subunits (Ramanathan et al., 1999, 2000; Jones et al., 1999). The present study examines the sufficiency of these mechanisms by incorporating the gating constants of cloned BK channels into a computational hair cell model.

The steady-state and kinetic properties of BK channels cloned from hair cells have been described previously by expression of the  (encoded by the slo gene) and  subunits in human embryonic kidney (HEK)293 cells (Ramanathan et al., 1999, 2000), as well as in oocytes (Jones et al., 1999). Certain alternatively spliced slo- channels from the chick's cochlea (₀ and ₆₁) have two- to threefold differences in deactivation kinetics as homomers, whereas combination with  subunits slows the kinetics of any  subtype substantially. Consistent with slowing of channel gating by , in situ hybridization showed that the  mRNA is expressed at highest levels in hair cells at the apical (low frequency) end of the tonotopically organized sensory epithelium (Ramanathan et al., 1999).

These findings raised the suggestion that a smooth progression in hair cell-tuning frequencies might be achieved by "titration" with the modulatory  subunit. At least two possibilities can be imagined. Because BK channels are tetrameric structures,  subunits might combine progressively to generate as many as five classes of channel with intermediate kinetics from any one  subunit (subunit mixing). Alternatively, intermediate tuning frequencies may be achieved by mixing different ratios of  only and -modified channels within each hair cell (channel mixing). This mechanism would be required if, for example,  subunits modulated BK gating in an all-or-none manner.

Earlier modeling efforts showed that the gating properties of native BK channels recorded from turtle hair cells could be used to reconstruct electrical tuning (Wu et al., 1995; Wu and Fettiplace, 1996). This model used subunit mixing to generate a population of five heteromeric channels whose graded expression generated the tonotopic array of tuned hair cells. The gating constants were taken from those of native BK channels of turtle hair cells, and did not explicitly test  subunit effects, which were unknown at the time. The present effort is also based on the turtle hair cell model. In contrast, we used gating constants from cloned and heterologously expressed BK channel subunits to test the possibility that a continuum of tuning also can be produced simply by mixing two kinetically distinct channels in each cell, -only and -plus- channels. That is, the gradient in  subunit expression simply alters the fraction of BK channels that are -modified in each cell, a mechanism consistent with the all-or-none modulation of  subunits suggested by expression studies in Xenopus oocytes (Jones et al., 1999). This modeling effort demonstrates the ability of specific gene products to recreate electrical tuning equivalent to that based on the properties of native BK channels from hair cells, confirming that the  and  genes can be regulated to generate the tonotopic gradient. Further, the model shows that a gradient in tuning frequencies can result from even the simplest outcome of a  expression gradient (the varying mixture of only two channel types) despite substantial differences in kinetics of the individual channels. Finally, we have incorporated temperature effects on channel transition rates and conductance as well as on calcium diffusion parameters to account for tuning frequencies seen at the higher body temperature of the bird (Schermuly and Klinke, 1985; Fuchs and Evans, 1990).

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

Expression of BK channel genes in HEK293 cells

Two splice variants of the slo gene cloned from a chick's cochlear cDNA library were used in the present study. The pore-forming  subunits, ₀ and ₆₁, correspond to the original cslo1 (Accession #U23821; Jiang et al., 1997), here called ₀, and to this cDNA with a 183 basepair insert at splice site 4 (₆₁) as described previously (AF076268; Ramanathan et al., 2000). The variant exon ₆₁ is closely related to the Strex exon described in adrenal chromaffin cells (Xie and McCobb, 1998; Xia et al., 1999). The slo  gene was cloned from quail papilla and shown to be identical to that first found in transformed fibroblasts (U67865; Oberst et al., 1997). A combination of  and  cDNAs, or single  cDNAs were transfected into HEK293 cells by calcium phosphate precipitation, as previously described (Ramanathan et al., 2000). Excised inside-out patches were formed from the transfected cells (detected by fluorescence of cotransfected GFP: "pGreen Lantern," Life Technologies, Carlsbad, CA). Voltage-clamp protocols to activate currents through BK channels in the presence of different concentrations of calcium were detailed previously (Ramanathan et al., 2000). For this study, it is sufficient to note that excised patches contained tens to hundreds of channels so that "macroscopic" currents were used for steady-state and kinetic fits. Also,  cDNA was added in a molar ratio of ~10:1 to that of the  subunit cDNA, in an effort to ensure a saturated effect on BK gating. Consistent with that expectation,  and  cotransfections produced homogeneous "-altered" currents, with no indication that -only channels coexisted with channels in cotransfected HEK cells.

Modeling BK channel gating

Steady-state parameters

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View larger version (14K): [in this window] [in a new window]   FIGURE 1   BK channel and hair cell models. (A) The allosteric (voltage-dependent MWC) scheme for BK channels has horizontal transitions that are calcium dependent and vertical transitions that are voltage dependent. The voltage-dependent MWC version assumes that each calcium-binding step has the same affinity. However, binding of calcium to the closed states may differ from the binding to the open states. The subscripts to the closed and open states indicate the number of calcium ions bound. Voltage-dependent gate movements between closed and open states are thought to occur allosterically with a single rate constant. ax and bx are the vertical transition rates that are dependent upon voltage. KCX and KOX are the dissociation constants for the binding of xth calcium ion to the closed and open states respectively. (B) The hair cell model incorporates BK channels into functional units with two adjacent voltage-gated calcium channels. These are shown as clusters, as might occur at transmitter release active zones, but each BK channel is gated independently by its associated calcium channels. The number and kinetic properties of BK channels varies between model cells, but the ratio of two calcium channels to one BK channel remains constant, as do the gating properties of the voltage-gated calcium channels.

Deactivation kinetics

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The hair cell model

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	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

Fitting cloned BK channel currents to the allosteric (voltage-dependent MWC) gating model

The effect of  modulation on BK channel gating was determined by comparing the properties of  subunits expressed alone, with those coexpressed with  subunits. Fig. 2 shows fits to average steady-state open probabilities of ₀ (A) and ₀ (B) channels expressed in HEK293 cells as a function of voltage at different concentrations of calcium, calculated according to Eqs. 1 and 2. As described in Methods, these fits were achieved by simultaneous optimization for K_C, K_O, and L(0) at all voltages and calcium concentrations. These constants were then used in a subsequent test of the relationship between the half activation voltage and calcium concentration.

View larger version (21K): [in this window] [in a new window]   FIGURE 2   Voltage gated-MWC model fits to steady-state open probability of 0 and 0. (A) Channel open probability for 0 expressed in HEK293 cells, determined from tail current amplitudes, is plotted as a function of membrane voltage for different cytoplasmic calcium concentrations (1.0, 2.0, 5.0, and 25.0 µM from right to left). The datapoints are fit with open probability functions defined for a voltage-dependent MWC scheme (solid lines) as defined by Eq. 1. (B) Fits for 0 are performed as in A, for calcium concentrations of 0.2, 1.0, 5.0, 25.0, and 50.0 µM (from right to left). The gating charge Q was held constant at 1.43 e for all fits. Fits were performed simultaneously for all calcium concentrations and voltages for each channel type.

By rearranging Eq. 1, the half activation voltage V_1/2 (where P_open is 0.5), may be written as a function of calcium concentration.

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L(0) and B are defined above where B depends upon the calcium concentration (Eq. 2). The ratio between K_O and K_C determined the slope of the curve that fits the V_1/2 versus [Ca²⁺] relationship and the value of L(0) determined the vertical position of the curve. This relationship of V_1/2 versus calcium concentration was used to fit voltage-clamp data for ₀ and ₀ in Fig. 3 (solid line). The allosteric gating model (or voltage-dependent MWC model) fits the V_1/2 relationship better than did a simple two-state voltage dependent calcium binding process (dotted line) used earlier (Ramanathan et al., 2000).

View larger version (12K): [in this window] [in a new window]   FIGURE 3   Voltage-dependent MWC model fits for the dependence of V1/2 on calcium concentration. The V1/2 vs. [Ca2+] relationship for 0 and 0 is fit with the voltage-dependent MWC scheme (solid line) using Eq. 10. The values of the parameters are listed in Table 1. The dotted line shows the fit provided by a simple two-state model (Ramanathan et al., 2000).

Steady-state open probability functions for the splice variant ₆₁ (Genbank #AF076268) and its combination with , (₆₁) were also fit with the allosteric gating model (parameters in Table 1). The values show that ₆₁ had slightly higher calcium affinities (corresponding to lower concentrations) than did ₀. There was a larger reduction in both K_C and K_O upon addition of  subunits to either channel, indicating a tighter binding of calcium to both closed and open states. The open-state affinity was more affected by  addition. The L(0) term (reflecting the intrinsic voltage-sensitivity of each channel) also varied among the four channel models, but here the differences were greatest when comparing splice variants, and were affected less by addition of  subunits to any one splice variant. The gating charge (Q = 1.43) was assigned to each model.

The macroscopic time constant given by (1/a+b) is determined by an average of all the vertical rate constants in the scheme weighted by the fraction of channels in each closed (a) or open (b) state. Equation 3 was used to fit simultaneously deactivation time constants measured at calcium concentrations of 1 and 5 µM for ₀ and ₀ expressed in HEK293 cells (Fig. 4., A and B). Values of the kinetic parameters for those channels and from similar fits of the ₆₁, ₆₁ channels, are listed in Table 2 along with the gating charge. Transition rates were reduced, both by addition of  subunits and by insertion of the 61 aa exon, although the  subunit produced larger effects than did alternative splicing.

View larger version (14K): [in this window] [in a new window]   FIGURE 4   Channel deactivation time constant fit with a voltage-dependent MWC (allosteric) model. (A) Tail current deactivation time constants are plotted as a function of voltage for 0 expressed in HEK293 cells and fit with model time constants by fitting the data points to the voltage-dependent MWC scheme (Eq. 3). The values of KC, KO, and L(0) were the same as those used in fitting steady-state relationships (Table 1). The free parameters are listed in Table 2. (B) Voltage-dependent MWC scheme fit to the deactivation kinetics of 0 as in (A).

View this table: [in this window] [in a new window]   TABLE 2   Kinetic rates and gating charge for 0, 0, 61, and 61

Constructing the model hair cells

With parameters for the channel models in hand we next incorporated these into an elementary model of chicken hair cells (Fig. 1 B) based on previous quantitative models of turtle hair cells (Wu et al., 1995). The basolateral membrane of the hair cell is thought to contain clusters of BK channels and calcium channels at distinct regions where transmitter release occurs (Roberts et al., 1990; Issa and Hudspeth, 1994; Martinez-Dunst et al., 1997). The total number of voltage-gated calcium channels per hair cell was set to be twice the total number of BK channels, and the open probability and dynamics of current through the calcium channels were based on experimental measurement from turtle (Art and Fettiplace, 1987), whose voltage-gated calcium channels are essentially identical to those of chickens (Zidanic and Fuchs, 1995). Although we mention that calcium channels and BK channels cluster at release sites, for the purpose of this model it is sufficient to assume that the calcium level near each BK channel is influenced by two nearby calcium channels. Indeed, each such BK channel is treated as an independent entity, gated solely by its attendant calcium channels.

A first order linear differential equation was used to describe calcium diffusion from an open calcium channel to a neighboring BK channel (Wu et al., 1995).

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S_F determines the calcium concentration achievable near a BK channel and reflects the buffering conditions and the distance between the BK channels and their closest calcium channels. k_R is the rate of calcium absorption by the ambient buffer. The ratio of S_F to k_R determines the available calcium at the pore of the BK channel. Resonance could be obtained for different values of S_F and k_R, but was sensitive to their ratio.

Fig. 5 shows responses of model hair cells operating at 22°C, each constructed with a single type of BK channel based on the properties of cloned, expressed  and  subunits. The panels are arranged in decreasing order of frequency. ₀, the channel with the most rapid kinetics generated a ringing frequency of 282 Hz and ₆₁, the slowest channel produced a hair cell that oscillated at 88 Hz. ₆₁ and ₀ formed intermediate frequencies of 195 and 121 Hz, respectively. Thus, BK channel subunits cloned from the avian basilar papilla were sufficient to generate electrical tuning in a manner similar to that provided by native hair cell BK channels. Further, differences in the kinetics of those channels gave rise to different tuning frequencies.

View larger version (14K): [in this window] [in a new window]   FIGURE 5   Electrical tuning of model hair cells expressing four types of BK channels. Model hair cells expressing 1of the 4 channel types were subject to current injection of 100 pA to excite electrical resonance. Channel numbers and model parameters were varied for each trial. Channels with faster kinetics (0 and 61) were present in larger numbers whereas those with  subunits were present in smaller numbers (Wu et al., 1995). The four channels span a electrical tuning frequency range from 88 Hz to 282 Hz. The resonant frequency and the quality of tuning for positive current injection are shown below each trace. 0 and 61 produced electrical resonance at resting hair cell voltages of ~55 mV. Hair cells containing 0 or 61 resonated at resting potentials that were more depolarized (~25 mV).

One feature of this simulation departed from expectation; that was the resting potential of hair cells tuned to the lower frequencies, namely those generated by ₀ and ₆₁ channels. Both these model cells had resting potentials near 25 mV, compared to resting potentials between 40 and 60 mV in live hair cells. This difference may arise from the fact that other voltage-gated potassium channels, such as the inward rectifier, increase the total resting potassium conductance of living low frequency hair cells (Fuchs and Evans, 1990; Murrow, 1994; Holt and Eatock, 1995; Goodman and Art, 1996), but are absent from the model cells.

As for any resonance system, a hair cell's electrical resonance can be described by its fundamental resonant frequency F₀ and the quality of the resonance Q_3dB. The quality of resonance is equivalent to the sharpness of tuning, with a larger Q indicating sharper tuning. Higher quality resonances also have prolonged "ringing" responses to a transient stimulus, and so the Q can be derived from the time constant of decay of oscillations () to such a transient stimulus (Crawford and Fettiplace, 1981; Wu et al., 1995).

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As observed in experimental recordings from living hair cells, the Q_3dB of the model cells varied between 2.5 and 6.0. The parameters used in each model hair cell model and the resulting tuning are listed in Table 3.

Two parameters were preset for each hair cell model. The calcium extrusion rate K_R was 40 in all cases. The number of BK channels per hair cell (N) was set to replicate the relationship between tuning frequency and channel number found in electrically tuned turtle hair cells (Wu et al., 1995). We estimated the expected tuning frequency for each model from the time constant of decay of macroscopic currents through the cloned channel type that constituted that model (Ramanathan et al., 2000). An initial value for S_F was based on previous modeling studies of turtle hair cells (Wu et al., 1995) and then adjusted empirically in order to provide a quality of resonance (Q) between 2 and 11. Much lower or higher values of S_F did not result in voltage oscillations for each channel type.

Temperature effects on resonance

The preceding results were obtained at a temperature of 22°C, consistent with experimental studies of cloned channels in HEK293 cells and work on turtle hair cells. However, body temperature in birds such as the chicken is near 40°C, and both hair cell oscillation frequency (Fuchs and Evans, 1990), as well as the tuning frequency of afferent fibers (Schermuly and Klinke, 1985; Wu et al., 1995), have been shown to vary approximately twofold per 10°C temperature change. Thus, we applied temperature corrections to channel kinetics as well as other features of the model to determine tuning behavior at 40°C. These were:

1. Rate constants for BK and calcium channels were scaled by a factor of 7. BK channels in guinea pig urinary bladder myocytes had Q₁₀ values near 3.0 for activation and deactivation rates (Markwardt and Isenberg, 1992). Q₁₀ is defined as the increase in the rate for a 10-fold rise in temperature. Hence, for an 18°C rise in temperature, the new rates R_40°C = R_22°C · (Q10)^1.8.

2. Single channel conductances of BK channels and calcium channels were doubled. Typical Q₁₀ values for conductance are between 1.3 and 1.6 for most channels. Potassium channels in the axons of Xenopus laevis had Q₁₀ values ~1.5 (Frankenhaeuser and Moore, 1963). BK channels in cultured rat muscle had a similar temperature dependence (Barrett et al., 1982).

3. Channel numbers were increased two- to threefold. This is in formal agreement with voltage-clamp measurements which show that hair cells tuned to higher frequency have larger numbers of calcium and potassium channels (Martinez-Dunst et al., 1997; Art et al., 1987). In the models, higher channel numbers produced higher quality of resonance (Q_3dB). Increasing channel numbers by threefold produced quality factors between 3.0 and 8.0. Although using the smaller number of channels for the simulation of higher temperature did not change the resonant frequency significantly, the numbers were increased to preserve the quality factors, which were lower when the channel numbers were not increased. The parameters used for the increased body temperature of 40°C for electrical tuning simulations are listed in Table 4.

Using the new model parameters, simulations were performed to determine what oscillation frequencies these model hair cells would produce. Fig. 6 shows the results from the simulation of the same four channel types as in Fig. 5. The frequencies have increased almost fourfold to span a range from 307 Hz to 1133 Hz. This increase in frequencies with temperature provides a Q₁₀ value of 1.84, which is essentially that observed by Fuchs and Evans (1990) for chick hair cell electrical resonance frequencies and by Schermuly and Klinke (1985) for auditory nerve fiber tuning (Q₁₀ = 2.0 in both cases). Hence, the temperature dependence of biophysical properties could be entirely responsible for the temperature dependence of tuning obtained by direct measurements from hair cells and auditory nerve fibers. Moreover, the resting hair cell potentials (51 to 59 mV) were very near those observed in isolated chick hair cells (40 to 60 mV), presumably as a result of the increase in potassium conductance.

View larger version (14K): [in this window] [in a new window]   FIGURE 6   Model hair cell electrical tuning at chicken body temperature. When temperature corrections were made to simulate resonance at the body temperature of the chick (40°C), the frequencies increased almost fourfold spanning a range from 307 Hz to 1133 Hz. The resting membrane voltages were between 50 and 59 mV. The increase in temperature caused models containing  subunits to oscillate near hair cell resting potentials (cf. Fig. 5).

subunit titration

We have modeled four types of cloned channels that encode frequencies extending from 307 to 1133 Hz at 40°C. The next question was to test a method of channel expression that could result in a gradient of tuning along the tonotopically organized auditory epithelium. Both alternative splicing of  subunits and combination with  subunits alter the gating kinetics of the resulting channels. Only limited information is available on the tonotopic distribution of  splice variants in the chick (Navaratnam et al., 1997; Rosenblatt et al., 1997). However, we know from in situ hybridization experiments that there is a smooth gradient in  mRNA expression along the length of the basilar papilla (Ramanathan et al., 1999). Thus, we have used the model hair cells to ask whether  subunits might be titrated to alter BK channels kinetics and hair cell tuning. In particular, we have asked whether it is possible simply to mix together -only and -plus- channels to produce hair cells with intermediate tuning. Intuitively, one would question whether the relatively large differences in kinetics between these channels could support the "homogeneous" behavior necessary for voltage oscillations.

We tested this idea using model cells with different ratios of ₀ and ₀ channels, operating at 40°C. Each BK channel was again assumed to obtain its calcium ions from two neighboring calcium channels. Hence, all BK channels would "see" the same amount of calcium for a given membrane potential. The total number of BK channels was varied so that lower frequency cells had fewer BK channels than their high frequency counterparts, corresponding to previous observation (Art et al., 1987; Fuchs et al., 1988). The total potassium current was obtained by multiplying the current through each BK channel by their respective numbers in the model cell. The responses of these model hair cells to current injections are shown in Fig. 7. Two conclusions can be drawn: 1) a hair cell expressing a mixture of channel types can oscillate with a single frequency and 2) its tuning frequency is intermediate to those produced by each channel type expressed on its own. Two intermediate combinations of ₀ and ₀ were tested in Fig. 7. The model hair cells oscillated at 968 Hz when one-third of the BK channels were of the ₀ form, and at 717 Hz when two-thirds were of the ₀ form. The resonance was similar to that of model hair cells containing a homogenous mixture of BK channels and produced exponentially decaying oscillations in membrane voltage that could be described by Eq. 13.

(13)

The resonant frequency F₀ and  are as defined before (Eq. 12), V₀ is the maximum amplitude, and  is the phase delay of the oscillations. This voltage response with a single characteristic frequency of a model hair cell containing a heterogeneous mixture of BK channels suggests three things. One, the population response of channels with widely different kinetics within a hair cell can be represented by a single kinetic parameter. Second, individual channels with intermediate kinetics are not mandatory for hair cells tuned to intermediate frequencies. A hair cell may express a complex mixture of BK channel subtypes with varying kinetic properties and yet have a unique resonance frequency that reflects an ensemble average of the properties of the constituent channels. Third and finally, hair cell-tuning can be produced by a functional gradient in  subunit expression.

View larger version (16K): [in this window] [in a new window]   FIGURE 7   Channel titration to obtain intermediate frequencies in model hair cells. The effect of mixing two channel types in a model hair cell is examined. Different ratios of 0 and 0 are inserted in model hair cells, which are tested with a 100-pA current injection. Cells that express either 0 or 0 were the same as in Fig. 6. When the ratio of 0:0 is 2:1, the tuning frequency was 968 Hz, and when the ratio was 1:2, the frequency was 717 Hz.

Voltage-clamp responses

In the previous section it was shown that a model hair cell may contain channels with different kinetic properties, yet still produce a unimodal voltage resonance. This might arise by voltage feedback through membrane resistance and capacitance so that the membrane time constant serves to integrate or smooth multiple current kinetics. Thus, a further examination of model behavior was made under voltage clamp to examine current kinetics. Can multiple exponential components in current activation and deactivation be distinguished? We simulated the same models under voltage clamp mode and studied the kinetics of the "tail" currents, which in native hair cells have monoexponential kinetics whose time constants vary tonotopically. The calcium current was calculated at each time point from calcium channel open probability and fed back to the BK channel model to determine the potassium current. The resulting whole-cell current response is shown in the middle panel of Fig. 8 for a hair cell containing an equal number of ₀ and ₀. Activation and deactivation for this cell is intermediate between those of the -only and all model cells (the upper and lower responses in Fig. 8).

View larger version (8K): [in this window] [in a new window]   FIGURE 8   Voltage clamp responses of model hair cells. Model hair cells expressing different ratios of 0 and 0 were tested under voltage clamp. Their activation and deactivation kinetics were examined for multiple exponential components. The total number of BK channels and other model parameters were kept constant. Tail currents of hair cells containing equal amounts of 0 and 0 decayed with a time constant of 3.16 ms which was intermediate to the decay times of cells containing either 0 (0.92 ms) or 0 (3.84 ms) alone.

The decay kinetics of tail currents were examined at higher resolution for a range of ratios (Fig. 9 A). Tail currents were fit with a single exponential function. The time constant of decay had a monotonic progression from fast to slow with increasing concentration of  subunits. This effect is summarized in Fig. 9, B and C. The hair cell with no  subunits had a relaxation rate of 1.12 ms¹ and the hair cell that has all ₀ channels relaxed with a rate of 0.26 ms¹. Intermediate rates fell along an exponential function that varied from fastest to the slowest upon increasing fractional saturation with  subunits. The oscillation frequencies produced by each of these model cells is plotted in Fig. 9 C as a function of the fraction of BK channels that were  modified. This result agrees qualitatively with the finding that the expression of slo- along the tonotopic axis of the cochlea can be described by a monotonically decreasing function with the highest levels at the (low frequency) apical region (Ramanathan et al., 1999).

View larger version (20K): [in this window] [in a new window]   FIGURE 9   Tail currents of model hair cells in voltage clamp and effect of  titration on hair cell tuning frequencies. (A) Tail currents of model hair cells expressing different ratios of 0 and 0 were examined in isolation. For a total of 1200 BK channels, tail currents for different saturation levels with  subunits show that they can all be fit with a single exponential curve (dashed lines). The voltage command was the same for all ratios of channels (deactivation from 10 to 50 mV). Because 0 channels had a higher affinity for calcium, they have a higher open probability under identical conditions of voltage and calcium and so a larger current is observed. The equation for monoexponential fits is given by I = Ib+I0exp(t/), where Ib is the baseline current, I0 is the maximum current amplitude, and  is the time constant of decay. The goodness of fit was greater than 0.99 for all single exponential fits, equivalent to that obtained by fitting double exponential functions for the tail currents. (B) Rate of decay of tail currents (inverse of time constants from A) is plotted as a function of  saturation of the channels. The rates are fit with a relationship given by r = rb + r0exp(fb/b), where rb is the rate of decay of 100%  saturated hair cell, r0 is the difference between fastest and slowest rates, fb is the fractional saturation with  subunits, and b is a fractional saturation analogous to a length constant. (C) Hair cell-tuning frequency plotted vs. fraction of 0 channels combined with  subunits. A hair cell expressing only 0 is tuned to 1133 Hz and addition of  subunits reduces the frequency proportionally until all the channels are -bound and the frequency becomes 424 Hz. The data points were fit with a straight line given by F0 = mx+Fc, where x is the fraction of total channels containing  subunits, m is the slope of the line, and Fc is the theoretical frequency when  subunits are absent.

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

A computational hair cell model based on the kinetic properties of cloned and expressed BK channel subunits was used to reconstruct electrical tuning. Graded variations in tuning frequency could be produced simply by altering the fraction of BK channels in each model cell that is modulated by accessory  subunits. These experiments were motivated by the observation that  subunit mRNA is expressed in a diminishing gradient from the low frequency end of the avian basilar papilla (Ramanathan et al., 1999).

Justification of the model

A variety of kinetic schemes have been used to describe BK channel gating, and these continue to evolve. Single BK channel recordings from skeletal muscle (Rothberg and Magleby, 1999; Nimigean and Magleby, 1999) and from transfected cells (Horrigan and Aldrich, 1999; Talukder and Aldrich, 2000) argue that the relatively simple allosteric model used here is an incomplete description of BK gating. These arguments include the observation of a multiplicity of open states, and evidence that channel opening can occur without charge movement in all four channel subunits. Nonetheless, the relatively restricted allosteric kinetic scheme is adequate for a quantitative description of the macroscopic currents generated by the cloned channels in HEK293 cells. The parameters of those fits provided a practical method to construct the excitability of model hair cells. Although such a channel model may not serve for insights into the mechanisms of BK channel gating per se, it enables a quantitative assessment of the whole-cell integration of multiple channel types as intended here.

We have used quantitative models based on the properties of cloned BK channels to show that a gradient of tuning frequencies among hair cells could be generated by the blending of kinetically distinct channel types. This effort arose from a desire to test the sufficiency of cochlear gene products to reconstruct hair cell tuning, and from the observation that  subunit mRNA appears in a gradient along the tonotopic axis of the avian basilar papilla when visualized with in situ hybridization (Ramanathan et al., 1999). Such a gradient in  expression was assumed to result in a heterogeneous population of BK channels in each hair cell, some fraction being "-modified," the remainder not. Because modulation by  subunits slows BK gating kinetics at least 10-fold (Jones et al., 1999; Ramanathan et al., 2000), we wished to determine whether such a pool of BK channels with widely variant gating kinetics could generate a single well behaved electrical resonance in a model hair cell.

An alternative approach would have been to construct heterotetrameric BK channels in which a variable fraction of the four  subunits were modified by combination with  subunits. Such a mechanism seems intrinsically better suited for fine gradations in channel properties, as might be expected for generating a continuum in hair cell tuning. Indeed, such heteromeric channels were proposed as part of an earlier model of tuning in turtle hair cells (Wu and Fettiplace, 1996). However, present evidence argues against  modulation occurring by such graded combination. Studies of coexpression in Xenopus oocytes showed that the effect of  subunits was all-or-nothing, irrespective of the stoichiometry of injected cRNAs (Jones et al., 1999). Likewise, cotransfection of HEK293 cells with different ratios of cDNAs never resulted in current with multiple kinetic components (Ramanathan and Michael, unpublished). Nonetheless, these experimental approaches do not completely resolve this issue, and further study is required to define the stoichiometry of association. If the all-or-none modulatory effect of  subunits is confirmed, the present work demonstrates that such an interaction does not prevent  modulation from serving as a means to tune hair cell BK channels.

Implications of the model

The hair cell models have shown that the gating properties of cloned and heterologously expressed cochlear slo gene products are sufficient to generate electrical tuning. Furthermore, coexpression with  subunits lowers the resonant frequency for model hair cells containing any one  subunit splice variant. Also, previously published effects of temperature on channel gating and conductance were used to alter the model and to recapitulate the temperature dependence of tuning in hair cells and afferent fibers. Finally, when model hair cells were composed of mixtures of -only and channels, monoexponential macroscopic current decays were generated, and intermediate tuning frequencies were produced. This unexpected blending of channel kinetics may reflect the combined effect of the  subunit on kinetics and the calcium/voltage sensitivity of the channels. For example, after a depolarizing voltage command, -only channels, with lesser calcium and voltage sensitivities, will contribute at early times, whereas the more sensitive channels will dominate at later times. Thus, not only the open probability of each channel type, but also the ratio between them changes exponentially as calcium and voltage decay. These results predict that electrically tuned hair cells should possess distinct populations of BK channels with different calcium- and voltage-sensitivities, reflecting the presence or absence of modulatory  subunits. This was not revealed in the original studies on turtle hair cells (Wu et al., 1995) but warrants revisiting in the avian papilla from which these gene products were obtained. Finally, it should be emphasized that this modeling effort does not provide evidence to the all-or-none effect of  subunits on slo- channels, nor does it eliminate other functional combinations. Thus, we can not conclude that other mechanisms such as subunit mixing do not also contribute to smooth gradients in electrical tuning frequencies along the basilar papilla.

Limitations of the model

As seen here, an expression gradient of  subunits by itself can not provide the entire range of electrical tuning. For example, the maximum change in tuning because of  modulation was from 1133 to 424 Hz for the ₀ splice variant, a factor of 2.67. Although the entire range of electrical tuning has not been established in chick, hearing spans a 100-fold range of frequencies, and electrical tuning by BK channels in turtle hair cells ranges ~50-fold, from 30 to 600 Hz (Wu et al., 1995). Additional range could be achieved by incorporating additional  splice variants; for example, ₆₁ has a resonant frequency of 307 Hz (at 40°C). Still other variants from the avian cochlea remain to be characterized (Navaratnam et al., 1997; Rosenblatt et al., 1997) and shorter  splice variants ('4, 26') from turtle hair cells can have more rapid gating kinetics than those described here (Jones et al. 1999). Alternative splicing of the  subunit may provide the intrinsic kinetic variability in gating that is further exaggerated by  modulation. Additional variability might be conferred by other modulatory subunits. Four  genes have been described in mammals (Xia et al., 1999; Wallner et al., 1999; Brenner et al., 2000; Uebele et al., 2000), with the avian  possibly representing a fifth. Most of these  genes produce somewhat different effects on BK channel gating. Alternative splicing of one of these  genes provides still further variability (Uebele et al., 2000). Other modulatory subunits such as Slob (Schopperle et al., 1998) and Slak (Joiner et al., 1998), and modification by phosphorylation (Tian et al. 2001) or other processes also require examination in the context of electrical tuning.