Spontaneous Oscillation by Hair Bundles of the Bullfrog's Sacculus

Spontaneous hair bundle oscillation

When mounted in a two-compartment experimental chamber with artificial endolymph or NMDG endolymph bathing the apical surface and standard saline solution contacting the basolateral aspect, the saccular macula from a bullfrog remained healthy for several hours. In most preparations, hair bundles were observed by eye to undergo spontaneous oscillations. In occasional preparations, all of the dozen or so hair bundles within a microscopic field of view could be seen to oscillate. The movements could also be documented by video microscopy. Slow-motion replay revealed fast bundle strokes that looked like instant jumps for a video acquisition rate of 30 frames per second; after each stroke, the bundle remained nearly still before it leapt back in the opposite direction. Subtraction of successive video frames clearly revealed the two components of the motion of a hair bundle in each direction (Fig. 1).

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Figure 1.

Spontaneous oscillation of hair bundles from the bullfrog's sacculus. The center panel shows one frame from a video movie of a microscopic field encompassing two hair bundles; the arrowhead marks the kinociliary bulb of the upper bundle. Subtraction of this frame from that antecedent produced the false-color image shown at the left, in which it is apparent that the medium-sized bottom hair bundle moved during the 33 msec interval. A similar subtraction of the following frame yielded the right panel, which demonstrates abrupt motion of the large top hair bundle. Movement of a hair bundle toward its tall edge, and thus toward the kinocilium, to the right, is defined as positive in sign and is displayed as an upward deflection in all subsequent figures.

We monitored the movement of an individual hair bundle by attaching the distal end of a fine glass fiber to the top of its kinocilium and projecting an enlarged image of the tip of the fiber onto a dual photodiode. Although loading a hair bundle with a fiber affects the amplitude and frequency of the oscillation (see below), the use of a very flexible fiber minimized this interference and allowed accurate measurement of the dynamical behavior of a bundle.

Different bundles were observed to oscillate spontaneously at frequencies of 5–50 Hz with a peak-to-peak magnitude of motion as great as 80 nm but most commonly ∼25 nm (Fig. 2). There was no rigorous correlation between the frequency and magnitude of spontaneous movements by different hair bundles. Two hair bundles that oscillated at frequencies almost an order of magnitude apart displayed movements of similar magnitudes (Fig. 2A,G). The slow oscillations were often the largest, however, especially for a given hair bundle. We sometimes observed an oscillation that reversibly changed its behavior within short periods. A hair bundle that initially oscillated at ∼44 Hz, for instance, displayed movements at ∼28 Hz 1 min later before almost doubling its frequency to ∼53 Hz after another 1 min had elapsed (data not shown). Correspondingly, the amplitude of oscillation increased from 18 to 30 nm before returning to 16 nm. Some bundles produced movements of almost metronomic regularity (Fig. 2A,B); others moved sporadically and occasionally halted for variable intervals (Fig. 2C).

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Figure 2.

Varieties of spontaneous hair bundle oscillations. A, For a common pattern of regular low-frequency movement, in this instance at 6 ± 2 Hz, the rms magnitude was 18 nm. B, This bundle produced a highly regular, relatively high-frequency oscillationat 27 ± 3 Hz with an 18 nm rms magnitude. C, Some bundles showed variations in oscillation frequency or even occasional pauses in their motion. In this example, the oscillation occurred at 18 ± 9 Hz, and its rms magnitude was 13 nm. D, The spontaneous movement of this bundle was irregular both in frequency, 6 ± 3 Hz, and in waveform. Its rms magnitude was 12 nm. Even for a hair bundle capable of regular oscillation, a similar pattern could occur when the recording fiber was poorly attached to the bundle. E, Low-frequency oscillations, especially large ones, displayed two clear components during each phase of movement. In each half-cycle, a fast stroke of movement in one direction concluded with a slower component in the same direction. This oscillation was characterized by an exceptionally large rms magnitude of 30 nm and a frequency of 9 ± 4 Hz. F, Low-frequency oscillations could be relatively rectilinear. The rms magnitude of this oscillation was 16 nm, and the frequency was 11 ± 3 Hz. G, The waveform of a relatively high-frequency oscillation, here 53 ± 6 Hz, could not be parsed into fast and slow components. The rms magnitude of this oscillation was 15 nm.

Particularly for hair bundles that moved regularly and at relatively low frequencies, it was apparent that each cycle of spontaneous oscillation comprised movements on two time scales (Martin et al., 2000), a behavior typical of a relaxation oscillation (Strogatz, 1994). In each half-cycle, a rapid stroke was followed by a slow excursion in the same direction (Fig. 2E). The fast component generally consisted of a displacement lasting no more than 5 msec. The shape of the slower component varied from cell to cell. In some instances, the movement was nearly exponential (Fig. 2E). On other occasions, the bundle was almost stationary during this component (Fig. 2F). Finally, the trajectory was often more complex and sometimes even nonmonotonic (Fig. 2D,G). Moreover, the waveforms of the positively and the negatively directed slow components were often dissimilar (Fig. 2C,D).

Relation of channel open probability to spontaneous oscillation

By making tight-seal voltage-clamp recordings, we were able to measure changes in transduction current associated with the movement of hair bundles. In 10 spontaneously oscillatory hair cells, but not static ones, the current displayed well defined oscillations ∼100 pA in peak-to-peak magnitude. Like spontaneous hair bundle displacements, slow current oscillations alternated between rapid transitions and relatively static intervals (data not shown).

We were able to make simultaneous tight-seal electrical recordings and mechanical measurements from one hair cell and therefore to examine the relation of transduction channel gating to bundle movement. The transduction current was highly correlated with the phase of spontaneous bundle movement. Positive bundle motion corresponded to an increase in the inward transduction current, whereas negatively directed motion coincided with decreased current (Fig. 3A). Within the temporal resolution of the mechanical and electrical recording techniques, the rapid components of bundle movement and the quick steps in transduction current occurred simultaneously. Because the current oscillations were eliminated by saturating mechanical stimuli, they represented the activity of mechanoelectrical transduction channels.

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Figure 3.

Correlation of spontaneous hair bundle oscillation with channel open probability. A, Positive movements of a hair bundle (top record) were associated with inward transduction current, shown as a downward deflection (bottom record). This hair cell was voltage-clamped at -70 mV in artificial endolymph. At the relatively high oscillation frequency of ∼100 Hz, the bundle did not display distinct slow and fast phases of movement. B, The application of saturating mechanical stimuli defined the range of open probabilities during spontaneous bundle oscillation. Bundle movements were recorded in the absence of stimulation and during positive and negative offsets that were large enough to saturate the mechanoelectrical transduction process (top records). The corresponding records of transduction current (bottom records) demonstrate that the large stimuli completely closed or opened the transduction channels, thereby abolishing oscillation. The transduction current in the absence of stimulation corresponded to open probabilities ranging from 0.13 to 0.70. The base of the stimulus fiber, whose stiffness was 80 μN · m^-1, was displaced by ±1000 nm to produce the saturating deflections.

By applying large mechanical stimuli of both polarities to the hair bundle, we were able to determine the maximal transduction current with all channels open and the zero current with all channels shut (Fig. 3B). Comparison of these values with the extreme values of the transduction current measured during spontaneous oscillation revealed that the open probability varied between 0.13 and 0.70 in this spontaneously active cell.

On excitation with current pulses, a hair cell from the bullfrog's sacculus displays damped oscillations of its membrane potential, a phenomenon termed electrical resonance. Hair cells are also capable of spontaneous electrical oscillation in the absence of stimulation. Because changing the membrane potential of a hair cell evokes bundle movements (Assad and Corey, 1992; Denk and Webb, 1992; Ricci et al., 2000, 2002; Bozovic and Hudspeth, 2003), we were concerned that hair bundle oscillation might have been the result, rather than the source, of an electrical oscillation in the soma of the hair cell. Our observation that hair bundle oscillations remain under voltage-clamp circumstances demonstrates that electrical resonance is not involved in their production.

Effect of gentamicin on spontaneous oscillation

An oscillatory hair bundle evinces a peculiar mechanical feature: its displacement–force relation, obtained by measuring the external force required to move the bundle through various distances, displays a region of negative stiffness (Martin et al., 2000). According to the gating–spring model of mechanoelectrical transduction (Corey and Hudspeth, 1983) (for review, see Markin and Hudspeth, 1995; Hudspeth et al., 2000), direct mechanical gating of transduction channels is expected to reduce the stiffness of a bundle over a limited range of positions (Howard and Hudspeth, 1988). For suitable values of the parameters of the model, this gating compliance can be great enough to dominate the other elastic components of the bundle and to render the stiffness of the bundle negative (Denk et al., 1992) (for review, see Markin and Hudspeth, 1995). Aminoglycoside antibiotics are known to block transduction channels (Kroese et al., 1989). By preventing channel gating, these drugs abolish gating compliance (Howard and Hudspeth, 1988) and should thus eliminate the negative stiffness of spontaneously oscillating bundles.

A spontaneously active hair bundle characteristically displayed a region of negative stiffness encompassing a displacement range of ∼20 nm (Fig. 4A). When the solution bathing the apical hair cell surface was replaced by artificial endolymph containing 60 μm gentamicin, the bundle instead behaved as a Hookean spring throughout the range of deflections explored. The region of negative stiffness of the hair bundle was restored by exchanging the bath with gentamicin-free endolymph.

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Figure 4.

The effect of transduction channel blockage by the aminoglycoside antibiotic gentamicin. A, An oscillatory hair bundle initially displayed a region of negative stiffness in the displacement–force relation measured by displacement clamping (filled circles). When the solution bathing the hair bundle was replaced by artificial endolymph containing 60 μm gentamicin, the region of negative stiffness vanished; the displacement–force relation became linear throughout the range of deflections explored (open circles). This effect was reversible in that negative stiffness was restored by returning gentamicin-free endolymph to the bath (diamonds). The fiber was detached from the tip of the bundle to allow successive bath exchanges. B, During each of three iontophoretic applications of gentamicin to the same hair bundle, the spontaneous oscillation vanished transiently. Because gentamicin blocks transduction channels in the open position, the bundle initially moved in the positive direction. When the drug encountered a bundle that was lurching in the negative direction, the bundle reversed abruptly and moved in the positive direction (top record). In contrast, very little positive motion resulted from applying the drug when the hair bundle was on the verge of jumping in the negative direction (center record). When the iontophoretic current was small enough, the hair bundle ceased oscillating but displayed rapid, noisy bundle movements. This noisy motion likely corresponded to channel clatter as the gentamicin molecules alternately blocked and unblocked the channels. The holding current was -0.2 nA; the expulsion currents were +2 nA for the top two traces and +1 nA for the bottom trace. The 1 sec iontophoretic pulse is indicated below the experimental records.

Active mechanical biasing of a hair bundle into its unstable region of negative stiffness can explain spontaneous oscillations (Martin et al., 2000). By eliminating negative stiffness, aminoglycosides would be expected to remove one of the conditions necessary for spontaneous oscillation. On iontophoretic application of gentamicin near the top of an oscillatory hair bundle, we found that the movements reversibly disappeared (Fig. 4B). Gentamicin arrested the hair bundle in a positive position, where most transduction channels are likely to be open (Fig. 3). This observation thus confirms the previous inference that gentamicin blocks transduction channels in an open state (Denk et al., 1992; Jaramillo and Hudspeth, 1993). By blocking Ca ²⁺ entry through open transduction channels, gentamicin should also interfere indirectly with the myosin-based molecular motors that effect adaptation to sustained stimuli (Eatock et al., 1987; Hacohen et al., 1989). After the initial blockage, a hair bundle often moved slowly in the negative direction, a phenomenon attributable to mechanical adaptation when Ca ²⁺ entry was interrupted (Denk et al., 1992; Jaramillo and Hudspeth, 1993).

Effect of pharmacological agents on spontaneous oscillation

If spontaneous hair bundle oscillation involves the shape of the displacement–force relation of the bundle and the activity of adaptation motors, drugs that affect either would be expected to influence oscillation. Substances that interfere with the cAMP second messenger pathway evoke a shift of the transduction current–displacement curve (Ricci and Fettiplace, 1997; Géléoc and Corey, 2001) and are therefore suitable reagents with which to perturb oscillation. A wealth of information suggests that the molecular motors responsible for adaptation of the mechanoelectrical–transduction process are based on myosin molecules (for review, see Hudspeth and Gillespie, 1994; Gillespie and Corey, 1997; Eatock, 2000; Holt and Corey, 2000). Substances that interfere with force production by myosin would thus be expected to affect oscillation. An example is butanedione monoxime, which places myosin II molecules in a weakly bound state (Herrmann et al., 1992; Seow et al., 1997; Tesi et al., 2002). Although the effect of this substance on other myosin isozymes remains uncertain (Cramer and Mitchison, 1995; Ostap, 2002; Titus, 2003), butanedione monoxime lowers the open probability of transduction channels in hair cells (Wu et al., 1999) and may therefore affect adaptation motors.

Used at a concentration of 100 μm, forskolin, an activator of adenylate cyclase, consistently lowered the oscillation frequency in each of the four hair cells examined (Fig. 5A). Sp-adenosine 3′,5′-cyclic monophosphorothioate (Sp-cAMPS; 500 μm) and 8-bromo-cAMP (8-Br-cAMP; 100 μm), membrane-permeant analogs of cAMP that activate cAMP-dependent protein kinases, had similar effects in one and two cells, respectively (data not shown). 3-Isobutyl 1-methylxanthine (IBMX; 500 μm), an inhibitor of cAMP phosphodiesterases, also reduced the frequency of oscillation in each of the two cells studied (Fig. 5B). Rp-cAMPS (500 μm), a membrane-permeant inhibitor of cAMP-dependent protein kinases, had the opposite effect in each of three hair cells: in the presence of the drug, the oscillation frequency increased appreciably (Fig. 5C). Finally, okadaic acid, an inhibitor of some protein phosphatases, slowed oscillation in both of the cells exposed to the drug at a concentration of 5 μm (Fig. 5D). As expected, the myosin inhibitor butanedione monoxime (10 mm) suppressed oscillation altogether in all four cells tested (Fig. 5E). In the many instances in which we were able to record bundle movements after control endolymph solution had been restored, we found that the oscillation frequencies returned to nearly their initial values.

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Figure 5.

Effects of pharmacological agents on spontaneous hair bundle oscillation. In each instance, a control record is shown at the left, and a record in the presence of the drug is shown at the right; the corresponding recovery record is not illustrated. A, Forskolin lowered the frequency of spontaneous oscillation to one-third, from 33 ± 17 to 9 ± 4 Hz. The rms magnitude of the oscillation was relatively unaffected, falling from 11 to 10 nm. When the drug was removed, the frequency recovered to 24 ± 7 Hz with a magnitude of 9 nm. B, In the presence of isobutyl methylxanthine, the frequency of an oscillation fell from 23 ± 4 to 10 ± 3 Hz; the magnitude declined from 16 to 11 nm. C, Rp-adenosine 3′,5′-cyclic monophosphorothioate increased the oscillation frequency from 5 ± 2 to 10 ± 8 Hz, whereas the magnitude of oscillation declined from 18 to 14 nm. Restoring the control solution restored the frequency to 4 ± 3 Hz with a magnitude of 17 nm. D, The spontaneous oscillation of a hair bundle exposed to okadaic acid declined in frequency from 13 ± 3 to 5 ± 2 Hz; the magnitude fell from 20 to 16 nm. During recovery, the oscillation frequency rose to 12 ± 4 Hz with a magnitude of 7 nm. E, Oscillation at 7 ± 3 Hz with a magnitude of 13 nm was abolished by butanedione monoxime. All experiments were conducted with artificial endolymph.

Because each drug was applied by changing the solution contained in the upper compartment of the recording chamber, it was necessary to detach the stimulus fiber to prevent disruption of the recorded hair bundle by turbulence. This procedure might have affected an oscillation by damaging the bundle, offsetting its position, or affecting the strength or exact site of coupling between the fiber and bundle. However, although drugs other than butanedione monoxime could change the frequency of an oscillation by as much as threefold, the magnitude of oscillation during treatment remained within 30% of the control value. If this magnitude is determined by the shape of the displacement–force relation of the bundle (Martin at al, 2000), these observations suggest that hair bundles primarily retained their mechanical integrity throughout the experiment. In the cases of forskolin, 3-isobutyl 1-methylxanthine, and okadaic acid, the oscillation became slower and smaller on drug application. Because simply detaching and reattaching the fiber never produced these effects, the observed changes may safely be ascribed to the effects of the various drugs on the oscillation motor. Finally, to ensure that a bundle was not significantly offset after solution changes, we subjected most hair bundles to a range of static offsets both under control conditions and during drug exposure and compared the frequency ranges of the consequent oscillations. These control experiments confirmed in each instance that the effects of the drugs on oscillation frequency were legitimate.

Effect of Ca ²⁺ on spontaneous oscillation

Ca ²⁺ mediates both of the processes proposed to power mechanical amplification by hair bundles, activity of the adaptation motor (Eatock et al., 1987; Hacohen et al., 1989) (for review, see Hudspeth and Gillespie, 1994; Gillespie and Corey, 1997; Eatock, 2000; Holt and Corey, 2000) and rapid transduction channel reclosure (Howard and Hudspeth, 1988; Crawford et al., 1991; Benser et al., 1996; Ricci et al., 2000, 2002) (for review, see Hudspeth et al., 2000; Fettiplace et al., 2001). Ca ²⁺ also reversibly affects the stiffness of a hair bundle (Marquis and Hudspeth, 1997) and the open probability of transduction channels at rest (Corey and Hudspeth, 1983; Hacohen et al., 1989). Finally, Ca ²⁺ modulates the frequency of spontaneous otoacoustic emissions (Manley and Kirk, 2002). We therefore tested the effects of different extracellular Ca ²⁺ concentrations on spontaneous hair bundle motion.

Varying the Ca ²⁺ concentration by exchange of the solution bathing the apical surface of the saccular macula produced systematic changes in spontaneous oscillation. Raising the Ca ²⁺ concentration from the control value of 250 μm resulted in an increase in frequency and a decrease in amplitude of the oscillation of the bundle, whereas reducing the concentration had the opposite effect (data not shown). When the Ca ²⁺ concentration exceeded ∼1 mm, or when it fell below ∼100 μm, the oscillations disappeared. This effect was reversible, however, because reimposing a Ca ²⁺ concentration near 250 μm restored well defined bundle oscillations.

We performed additional experiments in which we used iontophoresis to rapidly raise or lower the Ca ²⁺ concentration near the stereociliary tips while keeping the glass fiber attached to the kinociliary bulb. Transiently elevating the concentration with a Ca ²⁺-containing electrode had three consistent and reversible effects: a hair bundle displayed a net movement in the negative direction; the amplitude of oscillation declined; and the frequency of oscillation increased (Fig. 6A). For small to moderate levels of iontophoresis, it was noteworthy that the increase in oscillation frequency involved principally a shortening of the slow component of movement in the positive direction. For a given bundle, the fastest oscillation that we could elicit by Ca ²⁺ iontophoresis occurred at approximately twice the frequency of the control movement. The oscillation amplitude was not as strongly affected, decreasing by ∼20%.

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Figure 6.

Effect of extracellular Ca ²⁺ concentration on spontaneous hair bundle oscillation. A, When a spontaneously oscillating hair bundle was exposed to the Ca ²⁺ expelled from an iontophoretic pipette, the increased Ca ²⁺ concentration offset the bundle by -7 nm and decreased the rms magnitude of the movement from 18 to 16 nm. The frequency of oscillation also rose from 17 ± 8 to 34 ±6Hz, primarily by shortening of the positive phase of motion. The 1 sec iontophoretic pulse is indicated below the experimental record in this and the subsequent records. The holding current was -15 nA, and the expulsion current was +10 nA. B, Although the largest Ca ²⁺ ejections suppressed rhythmic hair bundle activity, fast, erratic movements remained. These spikes lasted no more than a few milliseconds and were characterized by peak-to-peak magnitudes up to 80% those of the unperturbed bundle oscillation. Before application of the 0.7 sec iontophoretic pulse, the spontaneous oscillation of this bundle was characterized by an rms magnitude of 12 nm and a frequency of 42 ± 6 Hz. The pulse caused a static bundle deflection of -4 nm. The holding current was -15 nA, and the expulsion current was +30 nA. C, Ca ²⁺ expelled by a ramp of iontophoretic current initially accelerated and eventually suppressed spontaneous bundle oscillation. Fast, spiky movements of relatively large magnitudes remained. The 1.25 sec linear ramp of iontophoretic current commenced at the dot. The holding current was -15 nA, and the maximal expulsion current was +60 nA. The greatest hair bundle offset induced by Ca ²⁺ iontophoresis was -18 nm. D, An 0.7-sec iontophoretic pulse of ATP decreased the local Ca ²⁺ concentration around a hair bundle, causing a net movement of +5 nm and prolonging the positive phase of oscillation. The spontaneous oscillation of the bundle increased slightly in rms magnitude, from 18 to 20 nm, whereas its frequency fell nearly by half, from 17 ± 4 to 10 ± 3 Hz. The holding current was +5 nA, and the maximal expulsion current was -15 nA. E, A greater reduction of the Ca ²⁺ concentration, effected by the iontophoretic application of an 0.7 sec ATP pulse, further prolonged the slow component of positive bundle movement but left the negative phase unchanged. The holding current was +5 nA, and the expulsion current was -20 nA. F, Even greater reduction of the Ca ²⁺ concentration suppressed the oscillation, originally 14 nm in rms magnitude and 21 ± 4 Hz in frequency, and caused a bundle offset of +6.5 nm. The noisy displacement fluctuations that persisted had an rms magnitude of 7 nm. The holding current was +0.5 nA, and the expulsion current was -40 nA.

The greatest increases in Ca ²⁺ concentration attained by iontophoresis suppressed rhythmic hair bundle activity (Fig. 6B). However, erratic, rapid movements lasting no more than 1–2 msec persisted. Because these spikes were distinct from the background noise apparent under control conditions during the slow components of oscillation, they probably represented residual active bundle motion. At the conclusion of the iontophoretic pulse, the oscillation resumed progressively, further demonstrating the relation among movement amplitude, frequency, and the duration of the positive phase of bundle movement. The use of iontophoretic pulses of varying intensity or the application of a ramp of iontophoretic current disclosed that a progressive increase in Ca ²⁺ concentration evokes graded effects (Fig. 6C).

We were also able to lower transiently the local Ca ²⁺ concentration around a hair bundle by iontophoresis of a chelator. Because carboxylate Ca ²⁺ chelators damage the transduction process by breaking tip links (Assad et al., 1991; Crawford et al., 1991; Marquis and Hudspeth, 1997), we elected to use ATP to sequester Ca ²⁺ (for review, see Fabiato and Fabiato, 1979). The effects of applying this nucleotide were opposite those of Ca ²⁺ iontophoresis. For low to modest levels of iontophoresis, the bundle displayed an offset in the positive direction and a graded slowing of spontaneous oscillations dominated by protraction of their slow component of positive motion (Fig. 6D,E). The ATP ejected by stronger iontophoretic currents entirely suppressed oscillations, producing a significant bundle excursion in the positive direction (Fig. 6F).

Effect of bundle position on spontaneous oscillation

By applying step displacements at the base of a flexible stimulus fiber, we analyzed the effect on a spontaneously oscillating hair bundle of deflecting its top by distances up to ±200 nm. Large offsets completely but reversibly suppressed the oscillation (data not shown). For bundle displacements smaller than ∼150 nm in either direction, however, a bundle remained quiescent only transiently (Fig. 7). During this period, the hair bundle relaxed slowly in the direction of the stimulus with a time course characteristic of mechanical adaptation (Eatock et al., 1987; Hacohen et al., 1989). After the bundle position had attained a plateau and adaptation had presumably concluded, the oscillation eventually resumed.

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Figure 7.

Effect of offset on spontaneous hair bundle oscillation. The center record demonstrates an unperturbed bundle oscillation at a frequency of 7 ± 3 Hz with an rms magnitude of 27 nm. From top to bottom, the remaining records show the effect of offsetting the tip of the bundle by +140, +85, -95, and -160 nm. In each instance, after having been suppressed for a period of adaptation, the oscillation resumed. After the largest offsets, the magnitude of oscillation grew progressively from zero. Positive offsets shortened the negative phase of bundle movement, whereas negative offsets had the opposite effect. Offsets thus perturbed the symmetry of the oscillation. The stiffness of the fiber was 130 μN · m^-1.

The recovery of oscillation during a protracted displacement could be abrupt for small bundle offsets but was graded for larger ones. In the latter case, oscillation grew progressively over the course of a few cycles from zero amplitude to a steady-state level and correspondingly decreased its frequency. The recovery was not complete, however, in that the oscillation was consistently faster than at rest when the bundle was offset in the positive direction and slower when it was displaced in the negative direction. In contradistinction to the effect of changes in Ca ²⁺ concentration near the stereociliary tips, bundle offset affected primarily the slow component of negative movement. The symmetry of oscillation was modified in such a way that a bundle displayed spiky movements in one direction when it was offset in the other.

Effect of mechanical load on spontaneous oscillation

When attached to the flexible glass fibers used in this study, most hair bundles oscillated at frequencies near 10 Hz. These frequencies lie near the bottom of the range of 5–150 Hz that is characteristic of saccular nerve fibers (Koyama et al., 1982; Yu et al., 1991) and presumably of the corresponding hair cells. Although several aspects of in vitro recording might have perturbed the oscillation frequency, one condition whose effect could readily be tested was the elastic load against which the hair bundle operated.

While recording spontaneous bundle oscillations, we used varying degrees of negative feedback, or partial displacement clamping, to increase the effective stiffness of the stimulus fiber attached to a bundle. When the stiffness of the fiber rose, the magnitude of the oscillation of a bundle characteristically declined as the frequency of spontaneous oscillation increased (Fig. 8A). A sufficiently great increase in the effective stiffness of the stimulus fiber suppressed well defined bundle oscillations altogether (Fig. 8B).

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Figure 8.

Effect of mechanical load on hair bundle oscillation. As a bundle oscillated spontaneously, partial displacement clamping was abruptly activated during the 1.2 sec epoch at the center of each record. Because the movement of the base of the fiber opposed that of the hair bundle, activating the clamp system increased the effective impedance of the fiber above its actual value of 152 μN · m^-1. A, The hair bundle oscillated spontaneously at 5 Hz with an rms magnitude of 33 nm. The effective stiffness of the fiber during partial displacement clamping increased to 260 μN · m^-1 (top record), then to 490 μN · m^-1 (middle record), and finally to 690 μN · m^-1 (bottom record). The increasing load raised the frequency of oscillation successively to 9 ± 3, then to 12 ± 4, and finally to 18 ± 10 Hz. The rms magnitude of oscillation correspondingly decreased to 22, then to 14, and finally to 12 nm. B, Another hair bundle, contacted by the same fiber as in A, oscillated spontaneously at 21 Hz with an rms magnitude of 18 nm. During partial displacement clamping, the effective stiffness of the fiber rose to 750 μN · m^-1 (top record), then to 1070 μN · m^-1 (middle record), and finally to 1390 μN · m^-1 (bottom record). The oscillation frequency increased to 34 ± 9 and then to 41 ± 20 Hz, whereas the rms magnitude decreased to 10 and then to 9 nm. The greatest effective stiffness of the fiber suppressed well defined hair bundle oscillations, leaving noisy bundle movements with an rms magnitude of 7 nm.

The effect of mechanical load on the frequency and amplitude of spontaneous hair bundle oscillation bears on the status of unstimulated bundles in vivo. Like those in most acousticolateralis organs, a hair bundle of the bullfrog's sacculus is normally attached to an accessory structure. In the intact ear, the bulbous tip of the single kinocilium in the bundle is linked by numerous filaments to the compact layer of the otolithic membrane (Hillman and Lewis, 1971; Jacobs and Hudspeth, 1990; Kachar et al., 1990). The polycrystalline otoconia that surmount this structure constitute an inertial mass whose movement relative to the skull signals acceleration. Although the mechanical properties of the otolithic membrane are complex (Benser et al., 1993), the structure exhibits a steady-state stiffness of 1400 ± 800 μN · m^-1 for each of the ∼2500 attached hair bundles. The present results suggest that the stiffness of the otolithic membrane is comparable with the minimal load required to suppress spontaneous oscillation.

Role of the kinocilium in spontaneous oscillation

In addition to the clustered stereocilia that mediate mechanoelectrical transduction (Hudspeth and Jacobs, 1979), every hair bundle, at least during its development, includes a single kinocilium. Because it contains an axoneme, or 9 + 2 array of microtubules adorned with dynein motor molecules, a kinocilum is capable of performing mechanical work. A kinocilium can oscillate spontaneously (Bowen, 1931) and can move in response to electrical stimulation (Rüsch and Thurm, 1990). Because this organelle has been suggested to contain the motor molecules responsible for hair bundle oscillation (Camalet et al., 1999, 2000), we wished to determine its role in the process.

A kinocilium may be detached from the stereociliary cluster by microdissection with the tip of a microelectrode (Hudspeth and Jacobs, 1979). We selected hair bundles that displayed robust spontaneous oscillations in artificial endolymph solution. After the kinocilium had been detached from each of five such bundles, all continued their active movements (Fig. 9). Even flattening the kinocilium against the apical surface of the hair cell and holding it pointed away from the stereociliary cluster with the dissecting electrode did not arrest spontaneous movements. In most instances, the magnitude of the oscillation increased slightly after dissection. This response might indicate that a kinocilium imposes a load on the stereociliary cluster (Crawford and Fettiplace, 1985). The change after dissection might alternatively stem from repositioning of the stimulus fiber at the top of a bundle, slightly farther from the apical surface of the cell than the control point of attachment at the kinociliary bulb.

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Figure 9.

Persistence of spontaneous hair bundle oscillation after kinociliary detachment. A, A hair bundle initially displayed spontaneous oscillations at 10 ± 6 Hz with a magnitude of 16 nm (top record). After the kinocilium had been disconnected from the contiguous stereocilia and flattened against the epithelial surface with a glass microelectrode, the bundle continued to oscillate at 11 ± 4 Hz (bottom record). In this instance, the magnitude of oscillation increased to 26 nm. B, Another hair bundle, which originally oscillated at 9 ± 3 Hz with a magnitude of 12 nm (top record), survived microsurgery with oscillations at 10 ± 4 Hz and 16 nm in magnitude (bottom record).

Model for spontaneous hair bundle oscillation

To ascertain whether our understanding of the principal features of spontaneous hair bundle oscillation accords with the present experimental findings, we developed a model for oscillations and tested its performance in simulations of several experimental conditions. More specifically, we assembled equations to represent hair bundle mechanics, mechanoelectrical transduction and the flow of ionic current, adaptation of the transduction process, and Ca ²⁺-dependent channel reclosure. We did not include the effects of noise in the model. These equations, their origins and justifications, and other features of the model are described in the Appendix.

The model yields simulated oscillations resembling those observed experimentally. In particular, adjustment of parameter values within a range that accords with other measurements readily produces oscillations of frequencies extending throughout and beyond the observed range of 5–50 Hz and of peak-to-peak magnitudes up to 70 nm. The oscillation waveforms resemble those of actual bundle movements (Fig. 10A,B). Each phase of the slower oscillations is bipartite, with a rapid initial component followed by a slow relaxation. As for actual oscillations, the slow components may be approximately exponential, essentially flat, or even nonmonotonic.

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Figure 10.

Simulations of spontaneous hair bundle oscillations and effects of experimental procedures. Each record was produced by the model described in Appendix, with most of the parameter values shown in Table 1. Record D was produced with exactly the values in Table 1; the values that were altered for other records are provided in the captions below. All the records have a common distance scale, but the time scales vary so that each record may be compared with the corresponding experimental trace. Except when indicated, each record's the total length of each record is 2 sec. A, A high-frequency spontaneous oscillation produced by the model displays symmetry of its positive and negative phases (compare with Fig. 2 B). C_MAX = 0.31 μm · sec^-1; S_MAX = 420 km · sec^-1 · N^-1; k_OFF,M = 50 · 10³ sec^-1; k_OFF,RE = 75 · 10³ sec ^-1; ΔE^ø = 70 zJ; and X_SP = 272 nm. B, Each phase of a large low-frequency modeled oscillation comprises a fast component followed by a slow movement in the same direction (compare with Fig. 2 E). d = 9 nm; C_MAX = 0.15 μm · sec^-1; S_MAX = 390 km · sec^-1 · N^-1; k_ON,M = 1.1 · 10⁹ sec^-1 · M^-1; k_OFF,M = 80 · 10³ sec^-1; κ_RE,MAX = 1200 μN · m^-1; κ_RE,MIN = 200 μN · m^-1; k_OFF,RE = 75 · 10³ sec^-1; ΔE^ø = 65 zJ; and record length = 500 msec. C, The response of the model to an elevation of the extracellular Ca ²⁺ concentration was mimicked by evaluating the response to a 10 nA rectangular pulse of iontophoretic current, indicated beneath the record, with a transfer number of 0.1 for Ca ²⁺. Ca ²⁺ accelerates the oscillations but renders them smaller (compare with Fig. 6 A). C_MAX = 0.22 μm · sec^-1; S_MAX = 220 km · sec^-1 · N^-1;k_OFF,M =50 · 10³sec^-1;κ_RE,MIN =200 μN · m^-1;ΔE^ø=54 zJ; and X_SP = 178 nm. D, The response to iontophoretic application of a Ca ²⁺ chelator, indicated below the trace, was modeled by reducing the extracellular Ca ²⁺ concentration. The removal of Ca ²⁺ slows oscillations and slightly augments their amplitude (compare with Fig. 6 D). E, Modeling an increase in the load imposed by the stimulus fiber from 152 to 490 μN · m^-1 during the middle portion of the record produces smaller and faster oscillations (compare with Fig. 8 A, middle record). C_MAX = 0.06 μm · sec^-1; S_MAX = 110 km · sec^-1 · N^-1; k_OFF,M = 80 · 10³ sec^-1;κ_RE,MIN = 200 μN · m^-1; k_OFF,RE = 75 · 10³ sec^-1; ΔE^ø = 68 zJ; X_SP = 251 nm; and record length = 2.5 sec.

Changing the extracellular Ca ²⁺ concentration affects simulated oscillations as it does actual bundle movements. In particular, an abrupt increase in Ca ²⁺ concentration, similar to that during iontophoretic application of the ion, accelerates oscillations by shortening the slow component of positive motion and reduces their magnitude (Fig. 10C). A comparable concentration decrease, which mimics the effect of iontophoretic ejection of a Ca ²⁺ chelator, slows oscillations by prolonging the slow component of positive movement and augments their size (Fig. 10D).

The formulation of the model for the mechanism of Ca ²⁺-dependent channel reclosure was meant to encompass the experimental observation that the amplitude of oscillation changes in response to alterations of the extracellular Ca ²⁺ concentration. If each cycle of oscillation represents a trajectory around the displacement–response relation (Martin et al., 2000), its amplitude can be changed only by adjusting the shape of the relation. In the present model, Ca ²⁺ entry into the stereociliary cytoplasm relaxes a component of the gating spring, thus reducing the magnitude of negative stiffness and shortening the oscillatory trajectory. The response of the model to simulated Ca ²⁺ exposure captures this aspect of the experimental result.

The model also recapitulates the effects of increasing the mechanical load on a hair bundle. As the stiffness of the fictive stimulus fiber increases, oscillation becomes faster and smaller (Fig. 10E) until it is wholly suppressed.

Modeling readily simulates the effects of pharmacological reagents on spontaneous hair bundle oscillation. Reducing the rate of adaptation motor climbing lowers the frequency of oscillation by prolonging the negative phase of movement. Such a change would occur in response to substances that promote protein phosphorylation, such as forskolin, IBMX, 8-Br-cAMP, Sp-cAMP, and okadaic acid. Increasing the climbing rate of the motor recapitulates the effect of Rp-cAMP, which presumably reduces phosphorylation, by accelerating oscillation. Not surprisingly, eliminating or greatly reducing the motility of a myosin-based motor, as might be effected by butanedione monoxime, suppresses oscillations (data not shown).

Modeling affords an opportunity to gain insight into complex physiological processes by examining the behavior of parameters whose values cannot be measured experimentally. For an oscillation 78 nm in peak-to-peak magnitude (Fig. 11A), the model suggests that the Ca ²⁺ concentration rises as high as 10–20 μm at the adaptation motor and reclosure element. In the same instance, the total excursion of the insertion of a gating spring as a result of adaptation is predicted to be ∼7 nm. The model implies that the mechanical twitch observed in standard saline solution when a bundle moves abruptly in the positive direction (Howard and Hudspeth, 1987a, 1988; Benser et al., 1996) stems from the rapid, Ca ²⁺-dependent relaxation of an intracellular component of the gating spring whose softening fosters channel closure. In the steady state, the displacement–force relation and displacement-open probability relation of a bundle are symmetrical with respect to the bundle position at which half the channels are open (Fig. 11). In the case of an oscillating bundle, however, the model predicts an asymmetry between the positive and negative phases of bundle movement and of the corresponding trajectory of channel open probability. When the bundle moves in the positive direction, the opening of transduction channels admits Ca ²⁺, which binds to the reclosure element and lowers the stiffness of the bundle. During the negative phase of motion, however, Ca ²⁺ entry diminishes and the bundle becomes stiffer. As a result, the displacement–force and displacement–open probability relations of the bundle continuously adjust their shapes during the oscillation, causing the trajectories of the bundle in the two directions to display hysteresis. This effect potentially explains the measured asymmetry of the open probability oscillation (Fig. 3).

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Figure 11.

Model for spontaneous hair bundle oscillation driven by the adaptation motor. A, The displacement–force relation at the top left (green curve) shows the instantaneous stimulus fiber force (F_SF) necessary to hold the hair bundle at a given position (X), as set by a displacement-clamp apparatus. This relation is specified by the steady-state solution of Equation 5 with x_A = 0. The region of negative stiffness corresponds to most of the range of bundle positions in which transduction channels are moving between their closed and open states, as revealed by the dependence of p_Oon bundle displacement represented by the continuous line at the bottom left. The adaptation motor strives to position the hair bundle near the middle of the displacement–force relation, in this instance at p_O = 0.35. Because it is unstable in the region of negative slope stiffness, however, the hair bundle is destined to oscillate. The red arrows delineate the trajectory of bundle movements corresponding to the oscillation at the top right. Any perturbation that nudges the bundle into to the right part of the negative-stiffness region obliges the bundle to lurch in the positive direction (step 1). As shown in the plot of open probability against time at the bottom right, this movement raises the open probability of the transduction channel to ∼1. Slipping adaptation then ensues, causing a slow component of movement (step 2) and lowering the open probability of the transduction channels to a value of ∼0.8. When mechanical instability prompts another fast component of movement in the negative direction (step 3), the open probability of the channel falls to ∼0. Climbing adaptation then raises the open probability to ∼0.1 while completing a cycle of oscillation (step 4). Note that the plots of force and open probability against displacement represent steady-state relations with a fixed value for the gating spring stiffness, 1050 μN · m^-1, characteristic of an open probability of 0.35. During the positive phase of each cycle of oscillation, however, the effect of Ca ²⁺ entry on there closure element decreases the gating spring stiffness from a maximal value of 1143 μN · m^-1 to a minimum of 808 μN · m^-1. As a consequence and in agreement with the experimental results (Fig. 3), the simulated excursion of open probability in the time domain is asymmetrical. The displacement–force relation and oscillation correspond to the values of Table 1 but with K_SF = 0 μN · m^-1. The range of open probabilities from 0 to 1 is calibrated by the extremes of the steady-state relation of open probability to displacement. B, When the bundle is attached to a stimulus fiber of stiffness K_SF = 150 μN · m^-1, its oscillation follows a trajectory whose fast components parallel the load line of slope -K_SF (blue line). The resultant bundle movement, which is smaller than that in A and occurs at a greater frequency, corresponds to the values in Table 1. The displacement–open probability relation in this instance is identical to that in A. C, By altering the shape of the displacement–force relation, Ca ²⁺ also affects the magnitude, frequency, and waveform of spontaneous oscillation. Bundle movements become smaller and faster when the single-channel gating force, z = γκ_GSd, is decreased by a diminution in either the maximal stiffness of the reclosure element, here κ_RE,MAX = 860 μN · m^-1 (oscillation at right and dotted green curve at left), or the gating distance, here d = 5.2 nm (dashed green curve at left). Note that in the first case, both the static stiffness and maximum negative stiffness of the bundle are reduced, whereas in the second case, only the negative stiffness is affected (compare B, C). The remaining parameter values correspond to those in Table 1, again with K_SF = 0 μN · m^-1. The displacement–open probability relations for these circumstances are shown at the bottom left in A; the curve for diminished stiffness is dotted, and that for reduced gating distance is dashed. The identical vertical and horizontal distance calibrations apply to the abscissas of all graphs as well as to the simulated oscillations.

A few features of the experimental results are not represented in the simulations. In response to an increased or decreased Ca ²⁺ concentration, for example, hair bundles display gradual changes on a time scale of hundreds of milliseconds (Fig. 6). These responses may reflect stereociliary Ca ²⁺ homeostasis (Lumpkin and Hudspeth, 1998; Ricci et al., 1998; Yamoah et al., 1998), which has been omitted from the model. Perhaps for the same reason, the model cannot reproduce the slow, progressive recovery of oscillations after large bundle offsets (Fig. 7).

Relation of oscillation to transduction channel gating

The negative stiffness of a hair bundle originates from gating compliance, a reduction in stiffness associated with the mechanical gating of transduction channels (Howard and Hudspeth, 1988) (for review, see Hudspeth et al., 2000). It therefore stands to reason that bundle oscillation is abolished by blockage of transduction channels with gentamicin, a procedure that arrests channel gating and thereby eliminates negative stiffness (Fig. 4).

Our electrical recordings also concur with theory by associating the positive phase of bundle movement with channel opening and the negative with closure. The results are consistent with the hypothesis that channel gating provides one of the forces that generate hair bundle motion.

The magnitude of oscillation depends on the trajectory that an active hair bundle executes with respect to its displacement–force relation (Fig. 11). In principle, the size of an oscillation depends on the magnitude of the gating compliance of a bundle relative to the combined stiffness of the other elastic components of the bundle and the load. Increasing the elastic load applied to the bundle would be anticipated to diminish the oscillations until it suppresses them altogether. Both the experimental result (Fig. 8) and the simulation (Fig. 10E) accord with this expectation. Alternatively, reduced gating compliance would flatten the displacement–force relation and yield smaller oscillations.

Participation of myosin in oscillation

Adaptation by the mechanoelectrical transduction process of the bullfrog's hair cells is mediated by myosin molecules, in particular myosin Ic (for review, see Hudspeth and Gillespie, 1994; Gillespie and Corey, 1997; Eatock, 2000). The success of the model at simulating most of the present results reinforces the previous inference (Martin et al., 2000) that the combination of adaptation motors and negative bundle stiffness underpins spontaneous hair bundle oscillation.

Phosphorylation of proteins associated with transduction channels, including the channels themselves, could explain the effects of drugs that interact with the cAMP signaling pathway. In particular, cAMP analogs and other substances that affect the activity of protein kinase A, and the phosphorylation of its targets might influence myosin-based adaptation motors, whose activity contributes to setting the operating point of transduction (Hacohen et al., 1989). In many cells, Ca ²⁺ interacts with calmodulin to activate adenylyl cyclase, producing cAMP that stimulates protein kinase A. This enzyme then phosphorylates the target proteins that constitute the effectors of the pathway. In the present instance, it is attractive to hypothesize that the operation of adaptation motors is impeded by the phosphorylation of myosin Ic at a consensus target site, such as serine 701 of the bullfrog's myosin Ic (Metcalf et al., 1994; Solc et al., 1994; G. Géléoc, P. Gillespie, and D. Corey, personal communication).

Butanedione monoxime blocks spontaneous hair bundle oscillation. Because this substance interferes with the ATPase cycle of myosin II (Herrmann et al., 1992; Seow et al., 1997; Tesi et al., 2002), this effect may reflect the arrest of myosin-based adaptation motors. This inference must be qualified, however, because the substance affects other physiological processes as well. Butanedione monoxime broadens the displacement–transduction current relation of a bundle (Wu et al., 1999), suggesting that gating compliance is diminished, so the blockage of oscillation might reflect a decrease in the negative stiffness of the bundle. The compound can also block ion pumps (Tucker and Fettiplace, 1996) that might influence stereociliary Ca ²⁺ homeostasis.

Effects of Ca ²⁺ on oscillation

Altering the Ca ²⁺ current that enters the stereociliary cytoplasm through transduction channels has an immediate effect on oscillation: Ca ²⁺ affects the duration of the slow component of positive movement (Fig. 6). Because this component is thought to be mediated by slippage of myosin-based adaptation motors down the stereocilia, our observations buttress previous evidence that an increased Ca ²⁺ concentration facilitates detachment of the myosin molecules from actin (Hacohen et al., 1989; Assad and Corey, 1992). The activity of the motor is relatively unaffected by extracellular Ca ²⁺ during the negative slow component of oscillation, when most of the channels are closed.

The decrease of the oscillation magnitude (Fig. 6A,B) in response to Ca ²⁺ iontophoresis may readily be modeled by rendering the stiffness of some component of the gating spring sensitive to Ca ²⁺ (Fig. 10C). Removal of extracellular Ca ²⁺ with a chelator has the opposite effect (Figs. 6D,E,10D).

The shift of the mean position of a bundle when the extracellular Ca ²⁺ concentration is elevated probably results from an adaptive response to a change in the average Ca ²⁺ concentration at the adaptation motor. Because this treatment reduces the open time of the channels, Ca ²⁺ iontophoresis may lower the average intracellular Ca ²⁺ concentration. The adaptation motor would then ascend the stereocilia, increasing the gating spring tension and thus evoking a negative shift. Because our model does not include a description of Ca ²⁺ homeostasis, however, our simulation does not reproduce this effect.

When the Ca ²⁺ concentration is either too high or too low, a hair bundle becomes quiescent. At these concentrations, the displacement–force relation of the bundle might not display an unstable region of negative stiffness (Howard and Hudspeth, 1988). Alternatively, the channel open probability at which the adaptation motors reach their steady state might lie outside the region of instability. If the open probability lies on the brink of the negative-stiffness region, however, noise would occasionally push the bundle into this region, triggering arrhythmic twitches (Fig. 6B). Because the magnitude of these twitches remains comparable with that of control oscillations, the negative-stiffness region probably persists at higher Ca ²⁺ concentrations.

Adjustment to an Hopf bifurcation

Active hair bundle motility appears to operate near a Hopf bifurcation that separates the quiescent and spontaneously oscillatory regimens of a bundle (Martin and Hudspeth, 2001). Because a dynamical system provides optimal amplification and frequency discrimination when operating near the bifurcation (Choe et al., 1998; Eguíluz et al., 2000), it might be expected that a negative-feedback mechanism holds it there (Camalet et al., 2000). The response of an oscillatory bundle to protracted displacements indeed reveals the operation of such feedback. After having been transiently suppressed by these stimuli, spontaneous oscillation progressively resumes (Fig. 7). During recovery, the oscillation magnitude grows from zero, primarily by increasing the size of the rapid components of bundle movement. Because these components correspond to jumps across the negative-stiffness region of the displacement–force relation of the bundle (Martin et al., 2000), their magnitude depends on the breadth of this region of instability. To accord with our observations, this region must expand as the oscillation recovers. Negative stiffness arises when gating compliance dominates other elastic components in the hair bundle, so our results suggest that the gating force can be regulated.

The myosin molecules underlying adaptation could themselves be regulators of the bifurcation. A few dozen of these, apparently myosin Ic molecules (Gillespie et al., 1993; Holt et al., 2002), are thought to link each transduction channel to the actin core of the stereocilia. This ensemble of motors provides a dynamical elastic structure that could be part of the gating spring. Its effective stiffness would depend both on the elasticity of each element and on the fraction of motors attached to actin filaments at any time. Either of these parameters is potentially sensitive to Ca ²⁺: the duty cycle of adaptation motors might be regulated through the cAMP second-messenger pathway (Ricci and Fettiplace, 1997), or Ca ²⁺ binding to calmodulin might affect the stiffness of the myosin neck (Howard and Spudich, 1996; Gillespie and Corey, 1997).

Matching the negative stiffness of a hair bundle to the positive stiffness of the load would allow the bundle to operate at a Hopf bifurcation. Because the myosin-based adaptation motors would approach a steady state under these circumstances, Ca ²⁺-mediated reclosure of transduction channels might then power spontaneous oscillations (Choe et al., 1998). Although Ca ²⁺ was originally posited to affect the state of transduction channels directly, channel gating could also be controlled by allowing Ca ²⁺ to regulate the stiffness of gating springs. This arrangement would endow the gating spring with an intracellular component sensitive to Ca ²⁺.

In the frog's sacculus, where each hair bundle is attached to an otolithic membrane, precise matching of the negative stiffness of the bundle to the positive stiffness of the membrane is improbable. The hair cell may thus require the capacity to adjust the stiffness of the bundle over a limited range. The responses shown here demonstrate that hair bundles in fact possess mechanisms for responding to various perturbations. The hair bundles of most receptor organs are attached to an accessory structure such as a tectorial or an otolithic membrane, a sallet, or a cupula. The capacity to adjust hair bundles to operate at a Hopf bifurcation may therefore prove to be a general characteristic of hair cells that ensures optimal performance.

Hair bundle mechanics

The hair bundle is represented as an elastic structure with a specific mass, moving at a low Reynolds number through a viscous medium under the influence of force delivered by an external stimulus fiber and of forces internal to the bundle. At any time, the balance of forces acting at the tip of the kinocilium, where the fiber is attached, requires that the force exerted by the stimulus fiber, F_SF, be: 5 in which X represents the position of the top of the bundle as well as that of the tip of the fiber (Howard and Hudspeth, 1988; Benser et al., 1996). Stimuli are applied through a fiber of stiffness K_SF, whose base is displaced by a distance Δ with the stimulator. m_HB and m_SF are, respectively, the masses of the bundle and of the relevant part of the attached stimulus fiber with the entrained fluid. In the overdamped regimen germane to the present low-frequency measurements, these masses are unimportant; they are included for the sake of completeness and to allow extension of the model to higher frequencies of stimulation. m_HB and m_SF are the corresponding drag coefficients. Because the base of the stimulus fiber was stationary during most measurements of spontaneous hair bundle motion, we have neglected an additional drag coefficient attributable to fluid flow caused by the base movements of the fiber (Martin et al., 2001). The N_GS transduction elements, each comprising a gating spring of stiffness κ_GS attached to the gate of a single transduction channel, are assumed to lie in parallel with one another and to experience identical mechanical inputs (Howard et al., 1988; Jacobs and Hudspeth, 1990; Iwasa and Ehrenstein, 2002). The exact distribution of channels between stereocilia and their positions with respect to the ends of tip links are immaterial to the model. The geometrical gain γrelates the shearing motion between contiguous stereocilia, and hence gating spring extension, to movement at the top of the bundle. In the initial state of the hair bundle, the gating spring experiences an extension of x_C when the channel is closed (Jaramillo and Hudspeth, 1993). Opening of the channel shortens the gating spring by a distance d; positive adaptation shortens it by an amount x_A. At any instant, the open probability of the channel is p_O; the total extension of the gating spring is γX - x_A + x_C - p_Od. The bending of each stereocilium at its base is opposed by the elasticity of the actin fascicle that extends as a rootlet into cuticular plate (Crawford and Fettiplace, 1985; Howard and Ashmore, 1986). The stiffness of the ensemble of these stereociliary pivots is K_SP (Howard and Hudspeth, 1988; Marquis and Hudspeth, 1997). When the tip links are severed, the hair bundle relaxes to the equilibrium position of those pivots, X_SP (Howard et al., 1988; Assad et al., 1991).

Mechanoelectrical transduction and Ca ²⁺ entry

The gating of the transduction channels in a hair bundle is described by the gating spring model (Corey and Hudspeth, 1983) (for review, see Howard et al., 1988; Hudspeth, 1992; Markin and Hudspeth, 1995; Hudspeth et al., 2000). The open probability of each transduction channel is: 6 The intrinsic internal energy change associated with channel gating, ΔE^℘, reflects the increase in energy content of a channel when it moves from the closed to the open state in the absence of a gating spring. k is the Boltzmann constant, and T is the temperature. Because of the rapidity of transduction on the time scale of the oscillations studied here (Corey and Hudspeth, 1983), a description of the kinetics of gating is unnecessary, and this equilibrium formulation of the open probability suffices.

The very low cytoplasmic concentration of Ca ²⁺ virtually precludes outward current carried by that ion, so it is unrealistic to describe the Ca ²⁺ current through an open transduction channel, I_Ca, in terms of a conductance and driving force. We instead use the Goldman–Hodgkin–Katz current equation (Hille, 1992): 7 Here P_Ca is the Ca ²⁺ permeability, z_Ca is the valence of the ion, e is the electron charge, F is the Faraday constant, [Ca ²⁺] is the Ca ²⁺ concentration in endolymph, and V_M is the membrane potential.

Ca ²⁺ that enters the stereociliary cytoplasm through transduction channels is subject to buffering and extrusion (Lumpkin and Hudspeth, 1998; Ricci et al., 1998; Yamoah et al., 1998). The mean time to capture by buffer molecules, however, is great enough that buffering has little effect within ∼100 nm of a channel (Roberts, 1993, 1994). Because an insertional plaque is only ∼50 nm in diameter, we may assume that the Ca ²⁺-binding sites responsible for regulating motor activity are essentially exposed to Ca ²⁺ diffusing from a point source, the channel, into a semi-infinite volume. For a motor at a distance r_M from the channel, the steady-state Ca ²⁺ concentration is then (Berg, 1993): 8 in which D_Ca is the diffusion coefficient of Ca ²⁺. This steady-state concentration is mostly achieved within a few tens of micro-seconds (Lumpkin and Hudspeth, 1998).