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The Journal of Neuroscience, June 1, 2003, 23(11):4533-4548

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Spontaneous Oscillation by Hair Bundles of the Bullfrog's Sacculus

Pascal Martin, D. Bozovic, Y. Choe, and A. J. Hudspeth

Howard Hughes Medical Institute and Laboratory of Sensory Neuroscience, The Rockefeller University, New York, New York 10021-6399

	Abstract

Top Abstract Introduction Materials and Methods Results Discussion Appendix Adaptation References

 
One prominent manifestation of mechanical activity in hair cells is spontaneous otoacoustic emission, the unprovoked emanation of sound by an internal ear. Because active hair bundle motility probably constitutes the active process of nonmammalian hair cells, we investigated the ability of hair bundles in the bullfrog's sacculus to produce oscillations that might underlie spontaneous otoacoustic emissions. When maintained in the normal ionic milieu of the ear, many bundles oscillated spontaneously through distances as great as 80 nm at frequencies of 5–50 Hz. Whole-cell recording disclosed that the positive phase of movement was associated with the opening of transduction channels. Gentamicin, which blocks transduction channels, reversibly arrested oscillation; drugs that affect the cAMP phosphorylation pathway and might influence the activity of myosin altered the rate of oscillation. Increasing the Ca ²⁺ concentration rendered oscillations faster and smaller until they were suppressed; lowering the Ca ²⁺ concentration moderately with chelators had the opposite effect. When a bundle was offset with a stimulus fiber, oscillations were transiently suppressed but gradually resumed. Loading a bundle by partial displacement clamping, which simulated the presence of the accessory structures to which a bundle is ordinarily attached, increased the frequency and diminished the magnitude of oscillation. These observations accord with a model in which oscillations arise from the interplay of the hair bundle's negative stiffness with the activity of adaptation motors and with Ca ²⁺-dependent relaxation of gating springs.

Key words: adaptation; amplification; auditory system; mechanoelectrical transduction; negative stiffness; vestibular system

	Introduction

Top Abstract Introduction Materials and Methods Results Discussion Appendix Adaptation References

 
The ear is mechanically active. Whether spontaneous or evoked by sound, otoacoustic emissions constitute the most striking evidence that energy-consuming elements within the inner ear can produce work. The ear's ability to deliver metabolically powered forces results in mechanical amplification: for sound stimuli near threshold, the active process augments vibrations in the mammalian cochlea by >100-fold, thus countering the dissipation caused by viscous drag in the fluid of the inner ear. Amplification of faint stimuli is closely associated with sharp frequency selectivity, because each portion of a receptor organ acts as a highly tuned mechanical resonator that responds preferentially at a natural frequency. Finally, the response of the ear to stimuli of increasing magnitude grows nonlinearly; the basilar membrane of the chinchilla, for example, represents six decades of sound pressure by only two orders of magnitude of vibration (Ruggero et al., 1997). Otoacoustic emissions, amplification, frequency selectivity, and compressive nonlinearity represent four essential characteristics of the active process that enhances detection of mechanical stimuli by the vertebrate inner ear (for review, see Manley, 2000, 2001).

Although amplification in the mammalian cochlea is widely believed to involve membrane-based electromotility by outer hair cells (reviewed in Dallos, 1992; Nobili et al., 1998), the receptor organs of amphibians, reptiles, and birds are not endowed with electromotile cells (He et al., 2003). The ears of nonmammalian tetrapods nevertheless display all four characteristics of the active process (for review, see Manley and Köppl, 1998; Manley, 2000, 2001). The alternative mechanism proposed to underlie the active process in those animals, and perhaps in mammals as well, is active hair bundle motility (for review, see Hudspeth, 1997; Hudspeth et al., 2000; Fettiplace et al., 2001). This process displays the four hallmarks of the active process (Martin and Hudspeth, 2001). A bundle can oscillate spontaneously (Crawford and Fettiplace, 1985; Howard and Hudspeth, 1987a; Denk and Webb, 1992; Benser et al., 1996; Martin and Hudspeth, 1999, 2001; Martin et al., 2000, 2001), a behavior that might underlie spontaneous otoacoustic emissions. The power expended by an oscillating bundle can be funneled into a weak sinusoidal stimulus to amplify the input (Martin and Hudspeth, 1999, 2001). The sensitivity of a bundle is tuned, with the greatest responsiveness at its natural frequency (Martin et al., 2001). Finally, the response of a hair bundle at its natural frequency displays a compressive nonlinearity (Martin and Hudspeth, 2001).

The four characteristics that define the aural active process are signatures of a dynamical system operating near a Hopf bifurcation (Choe et al., 1998; Camalet et al., 2000; Eguíluz et al., 2000; Jülicher et al., 2001; Martin et al., 2001). Such an oscillatory instability may emerge from the interplay of the negative stiffness of a hair bundle (Martin et al., 2000) with the molecular motors responsible for mechanical adaptation (for review, see Hudspeth and Gillespie, 1994; Eatock, 2000; Holt and Corey, 2000). To strengthen the evidence that this mechanism can mediate active hair bundle movements, we have examined the effects of treatments that affect the mechanical properties of a bundle and the adaptation motor on the ability of a hair cell to oscillate spontaneously.

	Materials and Methods

Top Abstract Introduction Materials and Methods Results Discussion Appendix Adaptation References

 
Experimental preparation. Experiments were performed at a room temperature of 21°C on saccular hair cells from the bullfrog Rana catesbeiana. Each internal ear was dissected in oxygenated standard saline solution containing (in mM): 110 Na ⁺, 2 K ⁺, 4 Ca ²⁺, 122 Cl ^-, 3 D-glucose, and 5 HEPES. After dissection from the labyrinth, the saccular macula was mounted over a 1 mm hole in a plastic coverslip and secured around the edges of its basal aspect with tissue-compatible acrylic adhesive (Isodent; Ellman International, Hewlett, NY). While the basolateral surface remained in contact with standard saline solution, the otolithic membrane was loosened by a 30 min exposure of the apical surface to 40 µg/ml protease type XXIV in N-methyl-D-glucamine (NMDG) endolymph containing (in mM): 2 Na ⁺, 3 K ⁺, 0.25 Ca ²⁺, 110 NMDG, 111 Cl ^-, 3D-glucose, and 5 HEPES. The otolithic membrane was then lifted from the hair bundles, and the coverslip was secured in a two-compartment experimental chamber (Martin and Hudspeth, 1999).

During most experiments, the standard saline solution in the lower compartment was not exchanged. The upper compartment ordinarily contained oxygenated NMDG endolymph; except when specifically indicated, the results presented here were obtained in the presence of this solution. Some experiments instead used artificial endolymph containing (in mM): 2 Na ⁺, 118 K ⁺, 0.25 Ca ²⁺, 118 Cl ^-, 3 D-glucose, and 5 HEPES. Each solution had a pH of 7.3 and an osmotic strength of 230 mmol/kg.

Enzymes, drugs, and other chemicals were obtained from Sigma (St. Louis, MO).

Kinociliary dissection. After control recordings had been made, kinocilia were detached from individual hair bundles with a horizontally mounted glass microelectrode (Hudspeth and Jacobs, 1979). The tip of the electrode was situated about halfway up the bundle and insinuated between the kinocilium and the remainder of the bundle. Moving the electrode upward then severed the filamentous connections between the kinociliary bulb and the five tallest stereocilia (Hillman and Lewis, 1971; Jacobs and Hudspeth, 1990). While additional mechanical recordings were made with the stimulus fiber resting atop the stereociliary cluster, the electrode was used to hold the loosened kinocilium flat against the epithelial surface.

Iontophoresis of Ca²⁺ and drugs. Iontophoresis was used to apply several substances to hair bundles during measurement of their spontaneous oscillation and stiffness. In each instance, a coarse microelectrode, whose resistance would have been 5 M if filled with 3 M KCl, was fabricated with a patch-electrode puller (L/M-3P-A; List Electronic, Darmstadt, Germany). After having been bent through an angle of 45° in its tapered region, the electrode was filled with 2.5 M CaCl₂, 350 mM disodium ATP, or 500 mM gentamicin sulfate. The tip of the pipette was then situated over the hair cell under investigation, 3 µm above the top of the bundle, and oriented toward the bundle. An appropriate holding current was used in each instance to minimize diffusive release of the contents of the electrode under control conditions.

Tight-seal electrical recording. We used the perforated patch recording technique with a voltage-clamp amplifier (Axopatch 200B; Axon Instruments, Union City, CA) to measure transduction currents in spontaneously oscillating hair cells. Tight-seal electrodes were drawn with a pipette puller (L/M-3P-A, List Electronic), bent to permit an orthogonal approach to the apical surface of a cell, and heat-polished to give resistances of 5–10 M. The tip of each recording electrode was positioned with a Huxley micromanipulator on the apical surface of a hair cell opposite the kinocilium (Holton and Hudspeth, 1986).

Each recording pipette was tip-filled for 1 sec with a solution containing (in mM): 110 Cs ⁺, 3 Mg ²⁺, 20 spermine, 114 Cl ^-, 40 SO₄ ^2-, and 5 HEPES. The pipette was then back-filled with an identical solution supplemented with 260 µM amphotericin B that had been dissolved in DMSO, whose concentration in the internal solution was 2% (v/v). When used in the presence of artificial endolymph, this internal solution produced a liquid–junction potential of 0 mV. A polycationic molecule, spermine, was included in the internal solution because of the difficulty in creating tight seals on the apical membrane of hair cells (Holton and Hudspeth, 1986). The presence of spermine permitted the formation of tight seals within several seconds of applied suction. Like its precursor spermidine, this polyamine occurs in cytoplasm at a concentration of 1 mM as an important counterion to RNA (Igarashi and Kashiwagi, 2000). The transduction currents measured in the presence of spermine did not differ in magnitude from those recorded from isolated hair cells (Jaramillo and Hudspeth, 1991; Assad and Corey, 1992; Shepherd and Corey, 1994; Walker and Hudspeth, 1996; Lumpkin and Hudspeth, 1998), for which no special treatment is required to make tight seals. Moreover, control recordings earlier demonstrated no untoward effects of lower concentrations of spermine and spermidine (Holton and Hudspeth, 1986). Because the membrane potential of a hair cell was held either at the resting potential ascertained under current-clamp conditions or at -70 mV, transduction currents were always inward and it is improbable that a cation such as spermine would have significantly blocked transduction channels.

Microscopic apparatus. Experiments were conducted under an upright microscope (MPS; Zeiss, Jena, Germany) equipped with a 100 W mercury illuminator, an infrared-reflecting hot mirror (K43-842; Edmund Industrial Optics, Barrington, NJ), and a broadband green interference filter (500 ± 40 nm, half-width at half-maximal transmittance; K46-157; Edmund Industrial Optics). The image formed by a 40x water immersion objective lens of numerical aperture 0.75 was further magnified by a 1.6x relay lens. Observations by eye and video microscopy were made with differential interference contrast optics. To increase the signal reaching the photodiodes, the analyzer was relocated from the microscope tube to the eyepiece assembly; moreover, during measurements of hair bundle motion, the polarizer was removed.

Because spontaneous hair bundle movements were sometimes too small or too fast to be detected directly by eye, we also used video microscopy to locate spontaneously active bundles. The image provided by the microscope was relayed through a projection eyepiece of 125 mm focal length or by a 1.5x telescope to a charge-coupled–device camera whose field of view encompassed several hair bundles. Its output was directed to a video image processor (Argus-20; Hamamatsu Photonics, Hamamatsu City, Japan). To highlight hair bundle oscillations, we digitally subtracted from each frame the running average of several consecutive frames.

Mechanical stimulation. Mechanical stimuli were delivered by a flexible glass fiber whose tip was attached to the kinociliary bulb of an individual hair bundle. Fibers were fabricated from borosilicate capillaries of 1.2 mm diameter (TW120-3, World Precision Instruments). Each capillary was first reduced with an electrode puller (P-80/PC; Sutter Instruments, Novato, CA) and then pulled finer in a direction perpendicular to its shank with a 120 V solenoid (Howard and Hudspeth, 1988). We found that a fiber of 500 nm in diameter was best for attachment to the kinociliary bulb. The fiber was trimmed with iridectomy scissors to a length of 100–400 µm. To enhance optical contrast, we coated the fiber with an 100 nm layer of gold-palladium (Hummer VI; Anatech, Alexandria, VA). The stiffness and drag coefficient of the fiber were, respectively, 80–400 µN · m^-1 and 40–110 nN · sec · m^-1, as determined by power spectral analysis of Brownian motion of the tip of the fiber in water. The fiber behaved as a first-order low-pass mechanical filter with a cutoff frequency of 0.2–1.6 kHz.

The fiber was secured by its base to a stack-type piezoelectric actuator (P835.10; Physik Instrumente) driven by a matched power supply (P-870; Physik Instrumente); this actuator provided displacements up to ±1 µm with a bandwidth of 5 kHz. The piezoelectric actuator was in turn mounted on a Huxley micromanipulator, which allowed fine positioning of the tip of the fiber with a submicrometer resolution. The fiber was used both to apply stimuli at the top of a hair bundle and to report bundle movements. Holding the base of the fiber at a fixed position permitted accurate measurement of the spontaneous motion of the bundle.

Displacement monitor. The tip of the fiber was imaged at a magnification of 1000x on a dual photodiode (UV-140-2; EG&G Electro-Optics, Salem, MA). A pair of preamplifiers directly attached to the photodiodes provided current-to-voltage conversion with a gain of 10⁷. After the difference between the electrical signals from the two photodiodes had been computed, the displacement monitor yielded an output linearly proportional to the displacement of the tip of the fiber with a resolution of 1 nm. To allow centering of the image of the fiber on the detector, the dual photodiode and preamplifier were mounted on a two-axis linear stage equipped with a stepping motor microdrive and controller (B-05 and PMC100; Newport, Irvine, CA). The photometric apparatus was placed on an independent platform mounted above the air table holding the microscope.

For calibration purposes, the photometric system was additionally secured to a piezoelectric stimulator. Each experimental record was calibrated by imposing a 20 µm offset pulse on the dual photodiode. Because of the optical magnification of the imaging system, the resultant output signal from the photodiodes was equivalent to that attributable to a 20 nm displacement of the stimulus fiber. The photodiode offset system was in turn calibrated with a heterodyne interferometer (OFV3001; Polytec GmbH, Waldbronn, Germany).

Displacement-clamp measurements. We used negative feedback to ensure that the bundle position, X(t), matched a command position, X_C(t) (Jaramillo and Hudspeth, 1993; Benser et al., 1996). The feedback signal, obtained by comparing the position of the bundle to the command signal, provided the input to the piezoelectric stimulator that moved the base of the fiber by a distance (t). It follows that:

(1)

in which () defines the gain of the clamp circuit at the angular frequency , and tildes denote the Fourier transforms of variables. In addition to proportional gain, which does not depend on frequency, we added integral gain to counter slow drifts in the position of the bundle and differential gain to dampen high-frequency components of the movement of the bundle. To prevent instabilities, the feedback signal sent to the piezoelectric stimulator was further filtered using a single-pole low-pass filter. The complex gain function thus assumes the form:

(2)

in which the scaling factor A 10, and the corner frequencies ₁ = 400 sec^-1; ₂ = 3 sec^-1; ₃ = 10,000 sec^-1; and ₄ = 1000 sec^-1. Before each experiment, the magnitudes of the proportional gain G_P, the integral gain G_I, and the derivative gain G_D were set manually to bring the spontaneous oscillation of a bundle to a complete halt; the maximal values of the respective gains were G_P = 12.5, G_I = 77, and G_D = 50. We then applied a 100 nm test command pulse and adjusted the gain of the clamp to achieve the bundle movement that reflected the command most faithfully. For the partial-displacement-clamp experiments, the gain of the clamp was controlled electrically by the computer. The clamp circuit was able to track command steps with a time constant of 1 msec.

For measurements of the negative stiffness of a hair bundle, we recorded the forces necessary to displace the bundle through various distances under displacement-clamp control and plotted the relation between displacement and applied force (Martin et al., 2000).

We used partial displacement clamping to study the effect of varying elastic loads on the spontaneous oscillation of a bundle. In this procedure, the movement imposed on the base of the stimulus fiber opposed but did not completely arrest that of a spontaneously oscillating bundle. More specifically, the signal sent to the piezoelectric stimulator consisted of only a fraction of the feedback required to clamp the bundle completely. Neglecting viscous components, the force, F_SF, exerted by the stimulus fiber against the hair bundle was proportional to the deflection of the fiber:

(3)

in which K_SF represents the stiffness of the stimulus fiber. In combination with Equation 1 for X_C = 0, the Fourier transform of Equation 3 yields:

(4)

in which _EFF() = K_SF[1 + ()] defines the impedance of the fiber at angular frequency . Because|_EFF| exceeds K_SF at any frequency, the clamp circuit effectively stiffened the fiber in the frequency range over which the feedback circuit operated. Although the impedance of the fiber depended on frequency, it remained approximately constant at frequencies from a few hertz to 50 Hz with the parameter settings used in these experiments. Because most of the spectral power of the oscillation of a bundle lay within this frequency band, the fiber effectively provided a linear stiffness.

Data collection and analysis. Stimulation and recording were performed under the control of a computer (P6400 GX1; Dell Computer Corp., Round Rock, TX) running LabVIEW software, version 5.0 (National Instruments, Austin, TX). Stimulus commands and experimental control signals were provided by a dedicated interface (AT-AO-10; National Instruments). Before sampling, responses were low-pass-filtered with an eight-pole Bessel antialiasing filter adjusted to a half-power frequency of 1 kHz. A multipurpose interface card (PCI-MIO-16E-1; National Instruments) conducted signal acquisition and analog-to-digital conversion with a precision of 12 bits and a sampling rate of 2.5 kHz.

To characterize the spontaneous oscillation of a hair bundle, we computed its power spectrum and fitted it with a Lorentzian function (Martin et al., 2001). The peak frequency of the best fit defined the frequency of the oscillation, and the half-width of the Lorentzian at half its maximal value described the extent of frequency fluctuation.

Data were analyzed with Mathematica, version 4.0 (Wolfram Research, Inc., Champaign, IL) and Matlab, version 6.0 (The MathWorks, Natick, MA).

	Results

Top Abstract Introduction Materials and Methods Results Discussion Appendix Adaptation References

 
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).

View larger version (121K): [in this window] [in a new window]   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).

View larger version (40K): [in this window] [in a new window]   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.

View larger version (23K): [in this window] [in a new window]   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.

View larger version (20K): [in this window] [in a new window]   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.

View larger version (27K): [in this window] [in a new window]   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%.

View larger version (41K): [in this window] [in a new window]   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.

View larger version (32K): [in this window] [in a new window]   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).

View larger version (43K): [in this window] [in a new window]   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.

View larger version (29K): [in this window] [in a new window]   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.

View larger version (52K): [in this window] [in a new window]   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). CMAX = 0.31 µm · sec-1; SMAX = 420 km · sec-1 · N-1; kOFF,M = 50 · 103 sec-1; kOFF,RE = 75 · 103 sec -1; E ø = 70 zJ; and XSP = 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; CMAX = 0.15 µm · sec-1; SMAX = 390 km · sec-1 · N-1; kON,M = 1.1 · 109 sec-1 · M-1; kOFF,M = 80 · 103 sec-1; RE,MAX = 1200 µN · m-1; RE,MIN = 200 µN · m-1; kOFF,RE = 75 · 103 sec-1; E 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). CMAX = 0.22 µm · sec-1; SMAX = 220 km · sec-1 · N-1;kOFF,M =50 · 103sec-1;RE,MIN =200 µN · m-1;E ø=54 zJ; and XSP = 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). CMAX = 0.06 µm · sec-1; SMAX = 110 km · sec-1 · N-1; kOFF,M = 80 · 103 sec-1;RE,MIN = 200 µN · m-1; kOFF,RE = 75 · 103 sec-1; E ø = 68 zJ; XSP = 251 nm; and record length = 2.5 sec.

 

View this table: [in this window] [in a new window]   Table 1. Representative parameter values of the model for spontaneous hair-bundle oscillation

 
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 (