INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Hair cells of the internal ear detect mechanical stimuli by gating of mechanosensitive ion channels located in their stereociliary bundles. The common view of transduction is that force is delivered to the mechanically sensitive channels by extracellular tip links connecting the top of one stereocilium with the side wall of its taller neighbor (Pickles et al. 1984). Deflection of the bundle toward its taller edge transmits force via the tip links to open transducer channels attached at either end of the link (Denk et al. 1995). Hair cells, like other sensory receptors, possess an adaptation mechanism to reduce their sensitivity in the face of a sustained stimulus (Crawford et al. 1989; Eatock et al. 1987). Adaptation shifts the transducer activation curve, changing the range of displacements to which the channel is sensitive without diminishing the maximum response.

Transducer adaptation is regulated by changes in stereociliary Ca²⁺ concentration that reset the range of bundle displacements detected by the channel (Assad et al. 1989; Crawford et al. 1989; Ricci and Fettiplace 1997, 1998). One proposed mechanism for resetting sensitivity entails a force generator that adjusts the tension in the tip link by translating the tip link's attachment point along the side of the stereocilium (Howard and Hudspeth 1987). The force generator may be myosin I linking the transducer channel with the internal actin cytoskeleton (Hudspeth and Gillespie 1994). Ca²⁺ influx through open transducer channels is posited to inhibit the actomyosin interaction, causing the channel to slip down the stereocilium and relieve the stimulus to the channel. A difficulty with this mechanism is that adaptation can occur on a submillisecond time scale (Ricci and Fettiplace 1997), too fast for the kinetics of the full actomyosin cycle. It is conceivable that fast adaptation relies on another mechanism with kinetics swifter than achievable with actomyosin interactions.

To assess this hypothesis, we have characterized the time course of transducer adaptation in turtle auditory hair cells to look for fast and slow components identifiable with different mechanisms. To support experimental observations, we devised a model for adaptation in which Ca²⁺ entering through open transducer channels binds at two intracellular sites to trigger separate processes leading to channel closure. The model, incorporating diffusion of Ca²⁺ within the stereocilium in the presence of intracellular Ca²⁺ buffers, employs computational techniques introduced in a previous model of hair-cell calcium dynamics (Wu et al. 1996). Our model differs from previous theoretical schemes (Assad and Corey 1992; Lumpkin and Hudspeth 1998) in providing an explicit formulation of the role of stereociliary Ca²⁺ in transducer channel regulation. It takes for its background prior measurements of the channel's Ca²⁺ permeability (Ricci and Fettiplace 1998), and experimental data on the effects of extracellular Ca²⁺ and intracellular calcium buffers on adaptation in turtle hair cells (Ricci and Fettiplace 1997, 1998; Ricci et al. 1998). Both experimental and theoretical manipulations provide further information about the properties of fast adaptation.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

The preparation and techniques for hair-cell recording and stimulation in the intact basilar papilla were similar to those previously documented (Crawford and Fettiplace 1985; Ricci and Fettiplace 1997). Turtles (Trachemys scripta elegans, carapace length 100-125 mm) were decapitated, and the cochlear duct was dissected out and opened. After digestion in saline [composed of (in mM) 125 NaCl, 4 KCl, 2.8 CaCl₂, 2.2 MgCl₂, 2 Na pyruvate, 8 glucose, and 10 NaHEPES, pH 7.6] containing up to 0.1 mg/ml of protease (Sigma type XXIV), the hair bundles were exposed by removal of the tectorial membrane. The preparation was mounted, hair bundles uppermost, in a silicone elastomer (Sylgard) well of a recording chamber mounted on the stage of a Zeiss Axioskop FS microscope. The preparation was perfused with saline containing (in mM) 128 NaCl, 0.5 KCl, 2.8 CaCl₂, 2.2 MgCl₂, 2 Na pyruvate, 8 glucose, and 10 NaHEPES, pH 7.6. The upper surface of the hair-cell epithelium facing the endolymphatic compartment was separately and continuously perfused by a large pipette with an internal diameter of 100 µm introduced into the cochlear duct. Hair bundles were stimulated with a rigid glass pipette, fire-polished to a tip diameter of ~1 µm and cemented to a piezo-electric bimorph (Crawford et al. 1989). The bimorph was driven differentially with voltage steps, filtered with an eight-pole Bessel at 3 kHz and amplified through a high-voltage driver of 20-fold gain, to yield a fast stimulator with a 10-90% rise time of ~100 µs.

Whole cell currents were measured with a List EPC-7 amplifier attached to a borosilicate patch electrode. Patch electrodes were filled with an internal solution of composition (in mM) 125 CsCl, 3 Na₂ATP, 2 MgCl₂, and 10 CsHEPES, pH 7.2 to which various amounts of the calcium buffers bis-(o-aminophenoxy)-N,N,N',N'-tetraacetic acid (BAPTA; Molecular Probes, Eugene, OR) or EGTA (Fluka, NY) were added. Buffer concentrations of 0.1, 1, and 10 mM were used, and with the highest concentration, the CsCl was reduced to keep the osmolarity constant. After application of 50% series-resistance compensation, the electrode access resistance was 3-10 M, which gave a recording time constant of 45-150 µs. Transducer currents were measured at a holding potential that, after correction for the junction potential, was 90 mV. To inhibit intracellular myosin ATPases, sodium metavanadate (Aldrich Chemical Company, Milwaukee, WI) was added to the patch electrode solution and butanedione monoxime (Sigma Chemical, St. Louis, MO) was dissolved in the extracellular solution.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Fast and slow components of adaptation

Transduction in auditory hair cells, evoked by step deflections of the hair bundle, is characterized by rapid opening of the mechanically gated channels closely followed by an adaptation that, despite maintenance of the stimulus, causes the channels to shut again. Figure 1A shows a family of transducer currents measured experimentally in response to a range of bundle displacements. Over the entire dynamic range, the adaptive decline in the transducer current could be well described by two exponential components, one with a time constant of ~1 ms and the other an order of magnitude slower (Fig. 1B). The fast time constant, _fast, dominated for small displacements, but the slower time constant, _slow, became more conspicuous with increasing stimulus amplitude. The double-exponential fits in Fig. 1A were derived by determining the value of the _fast (= 0.7 ms) from small responses and then holding this value fixed while allowing the contribution of _slow to vary for larger stimulus amplitudes (Fig. 1). These fits showed that over much of the dynamic range, _slow had a constant value of 11 ms, but for the largest displacements, it increased to 70 ms.

View larger version (25K): [in this window] [in a new window]   Fig. 1. Two components of adaptation. A: transducer currents recorded in a high-frequency hair cell for step deflections of the hair bundle. Inward currents are produced by positive displacements toward the kinocilium. Whole cell recordings made with 2.8 mM external Ca2+, 0.1 mM intracellular calcium buffer bis-(o-aminophenoxy)-N,N,N',N'-tetraacetic acid (BAPTA), at a holding potential of 90 mV. Each trace is the average of 5-25 responses. The decays in current for positive stimuli have been fitted with double exponentials, giving values for the fast and slow time constants (f and s, respectively) and their respective amplitudes (Af and As). B: fast and slow time constants and the relative amplitude of the slow component obtained from the fits in A are plotted against bundle displacement. Note that both the magnitude and relative proportion of the slow time constant increase with displacement amplitude.

For stimuli that elicit less than half-maximal responses, adaptation is dominated by _fast, which we have used previously to assay the calcium sensitivity of the underlying process. Over a range of ionic conditions, the rate of adaptation (1/_fast) was proportional to Ca²⁺ influx (Ricci and Fettiplace 1998) and inversely proportional to the concentration of intracellular calcium buffer, BAPTA (Ricci and Fettiplace 1997). Application of two-exponential fits to transducer currents recorded with different concentrations of BAPTA showed that the buffer concentration had parallel effects on the limiting values of both _fast and _slow. Mean values for _fast were 0.74 ± 0.14 (SD) ms (n = 7) in 0.1 mM BAPTA, 1.32 ± 0.11 ms (n = 9) in 1 mM BAPTA, and 1.68 ± 0.13 ms (n = 9) in 10 mM BAPTA. The corresponding values for _slow in 0.1, 1, and 10 mM BAPTA respectively were 9.3 ± 1.4 ms (n = 7), 14.9 ± 2.6 ms (n = 9), and 19.5 ± 4.4 ms (n = 9). All these measurements were obtained with 2.8 mM external Ca²⁺. Thus increasing the BAPTA concentration from 0.1 to 10 mM roughly doubled the values of both fast and slow time constants. The buffer effects may be due to a reduction in amplitude and a slowing of the intracellular Ca²⁺ transient after opening of the transducer channel. The buffer results indicate that the mechanism underlying _slow also must be Ca²⁺ dependent.

Other features of the transducer responses, linked to adaptation, also depend on the stereociliary Ca²⁺ dynamics (Ricci and Fettiplace 1997). Increasing the concentration of intracellular calcium buffer diminished the extent of adaptation, defined as the reduction in current in the steady state relative to the initial peak (Fig. 2). Thus in 0.1 mM BAPTA, there was no steady-state response for small stimuli; this is equivalent to 100% adaptation. However, with 10 mM BAPTA, the extent of adaptation never exceeded 50%. The fraction of transducer current turned on at the hair bundle's resting position also varied with the concentration of intracellular calcium buffer (Fig. 2). This difference reflects a translation of the transducer's activation relationship along the displacement axis (Fig. 2). Previous experiments have indicated that two manifestations of adaptation, the fast adaptation time constant and the fraction of current activated at rest, are differentially sensitive to various experimental manipulations. These include changing the nature of the intracellular calcium buffer (Ricci et al. 1998) or treatment with cyclic AMP (Ricci and Fettiplace 1997). Such observations support the notion that different aspects of adaptation may be associated with distinct Ca²⁺-binding sites.

View larger version (37K): [in this window] [in a new window]   Fig. 2. Effects of intracellular BAPTA concentration on the mechanotransducer current. Families of currents in 3 hair cells recorded with internal solutions containing 0.1, 1, or 10 mM BAPTA; 1 mM external Ca2+, holding potential 90 mV. Ordinates denote transducer current I, scaled to its maximum values, Imax, for a large positive displacement. Each trace is the averages of 5-25 responses. For each cell the peak current, I, scaled to its maximum is plotted against bundle displacement (bottom right). BAPTA concentrations and values of Imax, were 0.1 mM, 1.14 nA; 1.0 mM, 1.06 nA; and 10 mM, 0.45 nA. Increasing internal BAPTA concentration shifts the relationship to the left. Smooth curves are double Boltzmanns relating the probability of opening, P, and displacement, x: P = {[1 + exp (axO  ax)] · [1 + exp (bxO  bx)]}1 where values of a, b, and xo are: 0.1 mM BAPTA, 6.5 µm1, 14 µm1, 0.15 µm; 1 mM BAPTA, 7.5 µm1, 14 µm1, 0.09 µm; and 10 mM BAPTA, 7.5 µm1, 14 µm1, 0.03 µm.

Effects of myosin ATPase inhibitors on fast adaptation

Previous experimental analysis of adaptation in turtle hair cells has focused on the fast component that dominates the responses (Ricci and Fettiplace 1997). The prevailing theory for the mechanism of adaptation involves operation of a myosin ATPase motor that adjusts the force delivered by the tip links to the transducer channel (reviewed in Hudspeth and Gillespie 1994). In support of this mechanism in frog saccular hair cells, agents that block the ATPase also inhibit adaptation (Yamoah and Gillespie 1996). We examined the effects on adaptation of two potential inhibitors of the myosin ATPase: the phosphate analogue, vanadate, and the membrane-permeable butanedione monoxime (BDM), an inhibitor of myosin II and myosin V ATPases (Cramer and Mitchison 1995). Vanadate (1 mM), introduced via the patch electrode solution, or 10 mM BDM perfused extracellularly had similar effects on the transducer currents (Fig. 3). Both agents shifted the current-displacement relationship to the right and decreased its slope. The positive shifts in the current-displacement relationship were 203 ± 32 nm (n = 5, vanadate) and 154 ± 52 nm (n = 3, BDM). Similar shifts produced by other ATPase inhibitors have been previously reported (Yamoah and Gillespie 1996). However, neither agent significantly diminished the fast component of adaptation (Fig. 3). The fast time constant, _fast, had mean values of 1.54 ± 0.12 ms (n = 4, control), 1.45 ± 0.14 ms (n = 3, BDM) and 1.65 ± 0.14 ms (n = 5, vanadate), all with 1 mM internal BAPTA. The time constant of the slow component, _slow, measured in the same cells was 7.5 ± 2.0 ms (control), 12.8 ± 1.6 ms (BDM), and 9.5 ± 1.3 ms (vanadate). Because vanadate was delivered via the patch electrode solution, it was not possible to obtain a good control in the same cell due to the "wash-in" of vanadate occurring over a similar time course to that of BAPTA. The controls therefore represent measurement on other cells. It should be noted that the effects of the ATPase inhibitors resemble qualitatively those produced by application of 8-bromo cyclic AMP (Ricci and Fettiplace 1997).

View larger version (24K): [in this window] [in a new window]   Fig. 3. Effects of myosin ATPase inhibitors on fast adaptation. A: before (control) and 10 min after extracellular perfusion of 10 mM butanedione monoxime (BDM). Displacement amplitudes and adaptation time constants, fast, were 140 nm and 0.8 ms (control) and 220 nm and 0.76 ms (BDM). B: current-displacement relationships before and after perfusing BDM. Holding potential, 80 mV. C: family of transducer currents recorded 15 min after attaining whole cell configuration with the patch electrode solution containing 1 mM sodium vanadate. D: current-displacement relationships for a control cell and 1 recorded with 1 mM vanadate in internal solution. Fits in B and D are single Boltzmanns.

Interpretation of the effects of ATPase inhibitors is complicated by the fact that they also block the CaATPase responsible for Ca²⁺ extrusion from hair cells (Tucker and Fettiplace 1995). Consistent with those results, both vanadate and BDM produced prolonged tail currents at the offset of depolarizing current steps due to sustained activation of the small-conductance Ca²⁺-activated K⁺ (SK) channels (see Fig. 6 of Tucker and Fettiplace 1995). Effects on the transducer current-displacement relationship therefore could be a combination of an elevation of stereociliary Ca²⁺ concentration and block of the slow component of adaptation. However, the lack of any significant effect on _fast argues that fast adaptation is unlikely to be mediated by a myosin ATPase.

The fast process of adaptation showed linear behavior for small displacement steps about the resting position of the bundle (Fig. 4). The linearity was most evident under conditions where the resting probability had been raised by lowering the concentration of external Ca²⁺ or by increasing the amount of intracellular calcium buffer. Thus in Fig. 4 recorded in 0.35 mM Ca²⁺, the responses for small positive and negative steps are mirror images of one another, and _fast has a similar value for stimuli in either direction. However, the same linearity held under other conditions provided the amplitude of the negative stimulus was sufficiently small not to turn off the transducer current during the initial peak of the response. Collected measurements of _fast for small positive and negative stimuli, obtained under a range of conditions, are plotted in Fig. 4B, which shows a good correlation between the adaptation time constants measured with the two stimulus polarities. This linearity implies that the reaction involved in generating the fast component of adaptation is a reversible one and contrasts with the behavior expected for a myosin-based motor in which the adaptation rate for positive stimuli is faster than for negative stimuli (Assad and Corey 1992). The fast positive rate was attributed to "slipping" of myosin's attachment to the actin cytoskeleton, whereas the slower negative rate was limited by myosin ascending on the actin core of the stereocilium.

View larger version (16K): [in this window] [in a new window]   Fig. 4. Linearity of fast adaptation. Left: averaged transducer currents recorded for small positive and negative displacement steps in 0.35 mM external Ca2+ and 3 mM intracellular BAPTA where a greater fraction of transducer current is turned on at rest. Note the symmetry of responses for small positive and negative stimuli. Maximum current, 1 nA. Right: collected measurements of the correlation between the fast adaptation time constants for small positive (pos) and negative (neg) stimuli; to obtain a range of time constants, measurements were made in a various concentrations of intracellular BAPTA (0.1-10 mM) and external Ca2+ (0.07-2.8 mM). Straight line has slope of 0.93 and correlation coefficient, r = 0.92.

Outline of the two-site model

The central tenet of the model is that the transducer channels respond to the difference (x  X_a) between an external stimulus, x, and an internal "set point" X_a. As the channels open, Ca²⁺ ions enter the stereocilium and bind to an intracellular site triggering a change in the set point that opposes the external stimulus. Ca²⁺ is thus part of a negative feedback loop. The sequence of events ensuing from an increase in the external stimulus x  x' can be summarized as follows: the channels open in response to the new stimulus (x'  X_a) promoting Ca²⁺ influx and binding to an intracellular site S; the proportion of S bound catalyzes a change in the set point, (X_a  X'_a), causing the channels to adjust their probability of opening in response to the new stimulus (x'  X'_a). To implement the Ca²⁺ feedback, a three-dimensional model of the stereocilium was constructed to simulate the diffusion of free Ca²⁺ ions and mobile Ca²⁺ buffers within the cytoplasm (see APPENDIX). Major components of the model are as follows: each turtle hair-cell stereocilium contains a small number of mechanoelectrical transducer channels (Ricci and Fettiplace 1997) represented as a diffuse Ca²⁺ source, 10 nm radius, located at the stereociliary tip (Jaramillo and Hudspeth 1992). Ca²⁺ influx was estimated from the transducer current per stereocilium and the proportion of the current carried by Ca²⁺ (Ricci and Fettiplace 1998). The time course of internal Ca²⁺ is determined by diffusion and binding to calcium buffers and by extrusion via a plasma membrane CaATPase known to occur in turtle hair cells (Tucker and Fettiplace 1995).

Ca²⁺ is assumed to interact with two classes of intracellular binding site, S1 and S2, associated with the fast and slow adaptation processes, respectively

(1)

where K_Di is the Ca²⁺ dissociation constant for the two sites, and the subscript, i = 1 or 2, corresponds to the fast and slow sites respectively. Both sites are diffusely distributed over cylindrical regions of radius 1.5 nm and of longitudinal extent 20-50 nm from the channel for S1 and either 50-100 nm or 150-200 nm for S2. S1 was positioned to encompass the "crossing points" of the Ca²⁺ gradients in different BAPTA concentrations (see Fig. 10 of Ricci et al. 1998). Because S2 is associated with the slower process, it initially was located further from the channel than S1; the effects of varying the position of S2 will be described in the following text. Owing to the steep Ca²⁺ gradient within the stereocilium dictated by the concentration of diffusible buffer, S1 must have a Ca²⁺ dissociation constant (K_D1 = 20 µM) higher than that of S2 (K_D2 = 0.5 µM). Previous studies (Ricci et al. 1998) indicate that the Ca²⁺ concentration may decline from several hundred micromolar near the channel to a few micromolar at a distance of 100 nm, so the dissociation constants were chosen appropriately for the location of the two sites within the gradient. The dissociation constants are within the range of values reported for calmodulin (Linse et al. 1991).

The fraction, f_SCa, of each calcium-binding site occupied catalyzes a change in X_a that takes place in two stages. First, a conformational transition is assumed to occur in a modulator molecule converting it from an inactive form M to an active form M*

(2)

Each binding site has its own modulator (M₁ and M₂) with distinct kinetics, the rate constants for the second site being 10 times slower than those for the first site.

Second, X_a is scaled linearly according to the concentration of the active form of each type of modulator

(3)

in which _i and _i are constants and the modulator concentration, M^*_i, is integrated over the regions specified for each binding site. The effects of the modulators are assumed to sum independently to control the set point X_a. Because the modulator concentration, M_i, takes values between 0 and 1, the constants _i and _i determine the dynamic range of the feedback. A restricted dynamic range is consistent with the limited extent of adaptation reported by Shepherd and Corey (1994).

Model transducer responses

The theoretical responses for three different intracellular BAPTA concentrations are given in Fig. 5. The simulations expressed as the probability of opening of the transducer channel have been inverted for easier comparison with the inward currents recorded experimentally. Comparison of the model with the experimental records in Fig. 2 shows a number of similarities in terms of the overall shape of the response and their sensitivity to BAPTA. Thus the extent of adaptation was comparable in the different conditions and was reduced with an increase in BAPTA concentration. The steady-state responses for small displacements in 0.1 mM BAPTA are all more closely grouped compared with 10 mM BAPTA, reflecting nearly a 100% adaptation in the low buffer concentration. Only a single external Ca²⁺ concentration (1 mM) is illustrated in Fig. 5, but model responses at other Ca²⁺ concentrations from 0.07 to 2.8 mM showed comparable agreement with the experimental transducer currents.

View larger version (26K): [in this window] [in a new window]   Fig. 5. Effects of intracellular BAPTA concentration on the model transducer responses. Theoretical families of transducer channel open probabilities in response to 50-ms stimulus steps, for 3 BAPTA concentrations in 1 mM external Ca2+. To compare with experimental responses (Fig. 2), increasing probability of opening is plotted downward. Bundle displacements (in µm) are: 0.5, 0.2, 0.05, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6 0.8, 1.0, and 1.2. Transducer activation curves for the 3 BAPTA concentrations are given in the bottom right, the peak probability of opening for each trace being plotted against bundle displacement. More points are plotted than shown in the theoretical responses. Increasing the intracellular BAPTA concentration shifts the relationship to the left as with the experimental results in Fig. 2.

As expected, the model responses exhibited two components of adaptive decay, one component, _fast, with time constant of 1 ms, and a second component, _slow, of 14 ms. Fitting of the decays with double exponentials indicated that the slow component became more pronounced with an increase in stimulus amplitude. Both _fast and _slow increased with intracellular BAPTA concentration in a similar manner to the experimental measurements. The mean values of _fast are plotted in Fig. 6, and the values for _slow were 11, 14, and 70 ms in 0.1, 1, and 10 mM BAPTA, respectively. It should be noted that the magnitude of _slow in the model responses remained approximately constant over the dynamic range in contrast to the experimental results where _slow increased at large displacements. This discrepancy indicates a nonlinearity in the slow process. The fraction of current activated at rest in the model, as in the experiments, also increased with buffer concentration due to a shift in the current-displacement relationship.

View larger version (24K): [in this window] [in a new window]   Fig. 6. Effects of external Ca2+ concentration on the parameters of adaptation for the theoretical responses. A: response onsets for 1 mM intracellular BAPTA and 0.07 and 2.8 mM external Ca2+ concentration; B: response onsets for 10 mM BAPTA and 0.07 and 2.8 mM external Ca2+ concentration. Note that in both A and B, the adaptation rate is slower and the fraction of current turned on at rest is higher in the lower Ca2+ concentration. C: fraction of total transducer current turned on at rest vs. external Ca2+ with different concentrations of intracellular BAPTA: 0.1 mM (); 1 mM (), and 10 mM (). D: time constant of fast adaptation vs. external Ca2+ with different concentrations of intracellular BAPTA, symbols as in A. Time constants were derived from the fits to small stimuli evoking less than a half-maximal open probability, a procedure applied to characterize experimental currents. Theoretical results show similar trends and ranges of values to those of the experimental transducer currents (see Fig. 3 of Ricci et al. 1998).

To complement the calcium buffer results, the effects of varying the external Ca²⁺ concentration also were examined. This was implemented by altering the fraction of current carried by Ca²⁺ in line with the values determined experimentally (Ricci and Fettiplace 1998). Reducing the external Ca²⁺ increased the fraction of current turned on at rest and slowed the adaptation time constant (Fig. 6), effects that agree qualitatively with the experimental observations (see Ricci et al. 1998). However, the Ca²⁺ sensitivity of the parameters, especially the adaptation time constant, was weaker in the model than in the experimental results. This defect might be corrected by making the sites bind multiple Ca²⁺ ions in a cooperative fashion as occurs with calmodulin-based receptors.

Properties of the second Ca²⁺-binding site

The model was useful for distinguishing the relative contributions of the two sites, a manipulation that is difficult to perform experimentally. The simulations were repeated in the absence of one or other site by setting the scaling constants,  and , for that site to 0. The responses are shown in Fig. 7 for the case of 1 mM internal BAPTA and should be compared with the equivalent simulations with both sites present in Fig. 5. Removing site 1 produced responses with slow adaptation with a time constant of ~14 ms that was independent of level. The removal of site 2 gave responses, characterized over most of the range solely by a fast time constant of 1 ms similar to that seen in the two-site model. Neither set of responses in Fig. 7 for a single Ca²⁺-binding site provided as good a match to the experimental results as did the two-site model.

View larger version (25K): [in this window] [in a new window]   Fig. 7. Contribution of the fast and slow components of adaptation to the theoretical transducer responses. Top: family of traces computed with only the 2nd, slower, site (S2). Bottom: family of traces calculated with only the 1st, fast site (S1). Stimulus monitor is shown above the theoretical transducer responses. Equivalent responses when both binding sites are present are shown top right in Fig. 3. All calculations performed for 1 mM external Ca2+, 1 mM intracellular BAPTA.

In the majority of simulations, S2 was placed further from the transducer channel (150-200 nm) than the Ca²⁺-binding site for the fast process (20-50 nm). However, if S2 was moved closer to the channels, similar responses could be achieved provided that the Ca²⁺-dissociation constant for the site (K_D2) was increased. With S2 at 50-100 nm from the channel, it was necessary to raise the Ca²⁺-dissociation constant, K_D2, from 0.5 µM (the standard value) to 3 µM. In contrast, it was not possible to alter significantly the range for S1 and still retain fast adaptation.

Damped oscillatory responses

A consistent feature of the model responses for the lower BAPTA concentrations was an under-damped oscillatory approach to the steady state (Fig. 5), a manifestation of resonance stemming from negative feedback control of the transducer channels. Such resonance theoretically could produce frequency tuning for sinusoidal stimuli (Crawford and Fettiplace 1981), with the transducer current being maximal at the resonant frequency. An expanded version of the smallest theoretical responses in 1 mM BAPTA from Fig. 5 are shown in Fig. 8A, where the oscillations are clearly evident at the onset and termination of the step. This type of resonance has been observed experimentally at frequencies ranging from 58 to 230 Hz (Ricci et al. 1998). Figure 8A includes an example of such experimental transducer currents recorded with 1 mM intracellular BAPTA. These currents exhibit damped oscillations at a similar frequency, 180 Hz, to the model responses.

View larger version (14K): [in this window] [in a new window]   Fig. 8. Damped oscillations in theoretical and experimental transducer responses. A: theoretical responses shown at top for small, ±0.05-µm displacement steps. Responses were calculated for the standard set of theoretical parameters. Bottom: examples of damped oscillatory behavior in experimental transducer currents exhibiting a resonant frequency comparable to that in the theoretical responses. For both model and experimental responses, the internal Ca2+ buffer was 1 mM BAPTA. Note that the resonance is more under-damped in the experimental than in the theoretical responses. B: effects on theoretical resonance of altering the kinetics of the modulator transition, M  M* for S1 (the fast component). In the top traces, both forward and backward rate constants were accelerated threefold, and for the bottom traces, the rate constants for the same site were slowed threefold. Oscillation frequencies for the stimulus offset are given beside the traces. Note that the resonant frequency increases as the modulator kinetics speed up.

The main parameter controlling the resonant frequency in the model responses was the speed of the fast adaptive process. Figure 8B shows the results of altering the rate constants for the modulator transition. A threefold increase in the rate constants from the standard value elevated the resonant frequency from 180 to 270 Hz. Conversely, a threefold decrease in the rate constants slowed the adaptation to the point where the resonance was not visible. Two conclusions may be drawn from these results: first, the resonant behavior stems from the operation of the fast adaptation process; second, some of the variability in the appearance of the oscillations may be caused by differences among cells in the kinetics of the fast adaptive feedback.

Effects of speed of stimulus onset on adaptation

An important experimental variable influencing the appearance of the fast component is the rate of onset of the displacement step. In the present experiments, the driving voltage to the piezoelectric stimulator was filtered with an eight-pole Bessel at 3 kHz. This yielded a 10-90% rise-time in the stimulating probe of ~0.1 ms, which is comparable with or less than the rise time of the transducer current (Crawford et al. 1989). When the driving voltage was filtered at 100 Hz, equivalent to a rise time of 3 ms, both the onset and adaptation time constants were slowed (Fig. 9). The example illustrated shows that the fast adaptation time constant, _fast, increased from 1.3 to 4.6 ms. The additional filtering also desensitized transduction (Fig. 9A), such that a larger stimulus was required to produce the same peak current amplitude. As a consequence, the current-displacement relationship for the stimulus filtered at 100 Hz was shifted to more positive displacements relative to that for the 3-kHz filtered stimulus.

View larger version (23K): [in this window] [in a new window]   Fig. 9. Effects of stimulus onset speed on the fast time constant of adaptation. A: averaged transducer currents for displacement steps for which the driving voltage to the piezoelectric element were shaped with an 8-pole Bessel filter set at 3,000, 500, 100, and 50 Hz. With the slower stimulus onsets, both the activation and adaptation of the transducer current were slowed, and the current magnitude was reduced for the same stimulus amplitude. Maximum current: 0.72 nA, 2.8 mM external Ca2+, 1 mM intracellular BAPTA. B: experimental transducer currents for small displacement steps filtered with an 8-pole Bessel filter at 3,000 Hz (left) or 100 Hz (right). To achieve a similar peak current, the stimulus amplitude was increased from 0.09 µm (left) to 0.22 µm (right). Fits to the current decays are smooth lines superimposed on the noisy experimental traces. Note that the predominant time constant of adaptation, , increases more than threefold as the stimulus onset is slowed. C: equivalent theoretical responses for submaximal stimuli shaped with a single-pole low-pass filter of time constant 0.1 ms (left) and 2 ms (right). To achieve a similar peak open probability, the stimulus amplitude was increased from 0.15 µm (left) to 0.4 µm (right). As with the experimental responses, the time course of adaptation is slowed by increased filtering of the stimulus. 2.8 mM external Ca2+, 1 mM intracellular BAPTA in both B and C.

In the computed responses, the stimulus onset was normally instantaneous, but filtering of the stimulus with a single pole filter of time constant 0.1 ms had no effect on _fast. However, when the filter time constant was raised to 2 ms, _fast increased from 0.7 ms to 5.6 ms (Fig. 9C). For theoretical as with the experimental responses, it was necessary to increase the stimulus amplitude with the more heavily filtered step to produce the same magnitude of response. An explanation for these changes is that with slower stimulus onsets, the rate of change and extent of the Ca²⁺ excursion at the first site are both reduced, which slows and diminishes the magnitude of fast adaptation. Both experimental and theoretical observations emphasize the importance of using a stimulus with a rapid attack to reveal the fast adaptive process.

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Two components of adaptation

Characterization of the time course of mechanoelectrical transducer currents showed that adaptation in turtle auditory hair cells proceeds with at least two time constants differing by an order of magnitude. To account for this observation, and other evidence summarized in Ricci et al. (1998), we constructed a model of adaptation of the transducer channels that involved two processes with different kinetics, each governed by stereociliary Ca²⁺ levels. The model reproduced several features of the experimental responses, including the sensitivity to the concentrations of external Ca²⁺ and intracellular calcium buffer, BAPTA, and a dependence on the onset speed of the stimulus. The model also mimicked the behavior of the turtle hair cell's transducer in its capacity to generate damped oscillatory responses. The resonant behavior depended on the kinetics of the mechanism responsible for the fast component of adaptation.

Models of hair-cell transducer adaptation assume that intracellular Ca²⁺ controls the range of bundle displacements detected by the mechanoelectrical transducer channel. This assumption is expressed in our model by the notion of the channel's "set point." One mechanism by which the set point might be altered invokes a myosin motor connected to both the transducer channels on the stereocilium's side wall and the internal actin cytoskeleton (reviewed in Hudspeth and Gillespie 1994). The speed of a myosin motor will be limited by the kinetics of myosin ATPase, which for fast skeletal muscle has a cycle time on the order of 50-100 ms at room temperature (Hibberd and Trentham 1986; Pollard et al. 1991). Although the most precise kinetic information is available for the skeletal muscle myosin II, the adaptation motor may depend on an unconventional myosin-I known to be present in hair-cell stereocilia (Hasson et al. 1997). The cycle time of myosin-I also may approach 50 ms (Pollard et al. 1991).

The properties of the fast component of hair-cell adaptation, its submillisecond kinetics, its symmetry for small positive and negative displacements, and its insensitivity to the ATPase-inhibitors vanadate and BDM, all argue that it does not rely on a conventional myosin-based motor. An alternative hypothesis is that fast adaptation is mediated by conformational rearrangements in the channel protein itself (Crawford et al. 1989) or in molecules directly connecting it to the cytoskeleton. A specific mechanism would be that the Ca²⁺-dependent modulator, M₁ (Eq. 2), is an auxiliary subunit of the transducer channel, the activation of which alters the gating kinetics of the channel stabilizing it in its closed configuration (Fettiplace et al. 1992). Activation of M₁ would result from association with Ca²⁺ bound to S1, which itself may be a separate Ca²⁺-binding protein like calmodulin or may be an integral part of M₁.

Location of the Ca²⁺-binding sites

Arrangement of the two Ca²⁺-binding sites along the stereociliary axis is convenient for a model constructed in cylindrical coordinates but may be physically unrealistic particularly with respect to the more distant second site. S1, positioned at 20-50 nm from the center of the transducer channel complex, is of dimensions only slightly greater than ion channels (~10 nm diam), which may be arranged in a cluster. Furthermore S1 does not need to be located directly on the axis, and its placement anywhere within a hemispherical shell centered on the channel complex would yield similar theoretical results. The spatial extent of S1 might represent the local cytoplasmic distribution of a Ca²⁺-binding protein like calmodulin, which has been suggested to mediate calcium's role in adaptation (Walker and Hudspeth 1996). The transducer channels were assumed to be entirely located at the apex of the stereocilium but channels may be present at both ends of the tip links (Denk et al. 1995). In the current model, channels placed on the side wall of the stereocilium were neglected due to the added geometric complexity incurred, which would have removed the radial symmetry and considerably lengthened the computations. Because S1 is located close to the channels, our model would still provide an adequate description of fast adaptation for channels on the side wall.

The location of S2 is more problematic because its distance from the transducer channels (150-200 nm in most calculations) was large relative to the size of the channel. However, we found that provided that the Ca²⁺-affinity of the site was adjusted, similar theoretical responses could be achieved with S2 positioned 50-100 nm from the transducer channels. Such distances are within the dimensions of the electron-dense plaques, representing cytoskeletal linking proteins or arrays myosin head groups, into which the tip links insert (Hudspeth and Gillespie 1994). Nevertheless, we do not feel it is possible from our results to derive a precise location for S2, and thus the coordinates for S2 in the model may not impose major limitations on its physical realization. In particular, the results neither establish nor eliminate a myosin motor as the mechanism of the slow component of adaptation.