Abstract
The hair bundle is the universal mechanosensory organelle of auditory, vestibular, and lateral-line systems. A bundle comprises mechanically coupled stereocilia, whose displacements in response to stimulation activate a receptor current. The similarity of stereociliary displacements within a bundle regulates fundamental properties of the receptor current like its speed, magnitude, and sensitivity. However, the dynamics of individual stereocilia from the mammalian cochlea in response to a known bundle stimulus has not been quantified. We developed a novel high-speed system, which dynamically stimulates and tracks individual inner-hair-cell stereocilia from male and female rats. Stimulating two to three of the tallest stereocilia within a bundle (nonuniform stimulation) caused dissimilar stereociliary displacements. Stereocilia farther from the stimulator moved less, but with little delay, implying that there is little slack in the system. Along the axis of mechanical sensitivity, stereocilium displacements peaked and reversed direction in response to a step stimulus. A viscoelastic model explained the observed displacement dynamics, which implies that coupling between the tallest stereocilia is effectively viscoelastic. Coupling elements between the tallest inner-hair-cell stereocilia were two to three times stronger than elements anchoring stereocilia to the surface of the cell but were 100–10,000 times weaker than those of a well-studied noncochlear hair bundle. Coupling was too weak to ensure that stereocilia move similarly in response to nonuniform stimulation at auditory frequencies. Our results imply that more uniform stimulation across the tallest stereocilia of an inner-hair-cell bundle in vivo is required to ensure stereociliary displacement similarity, increasing the speed, sensitivity, and magnitude of the receptor current.
SIGNIFICANCE STATEMENT Generation of the receptor current of the hair cell is the first step in electrically encoding auditory information in the hearing organs of all vertebrates. The receptor current is shaped by mechanical coupling between stereocilia in the hair bundle of each hair cell. Here, we provide foundational information on the mechanical coupling between stereocilia of cochlear inner-hair cells. In contrast to other types of hair cell, coupling between inner-hair-cell stereocilia is weak, causing slower, smaller, and less sensitive receptor currents in response to stimulation of few, rather than many, stereocilia. Our results imply that inner-hair cells need many stereocilia to be stimulated in vivo to ensure fast, large, and sensitive receptor currents.
Introduction
Vertebrate hearing organs are spectrum analyzers, breaking down complex sounds into their fundamental frequency components and conveying information about the timing, intensity, and frequency of each component to the CNS (Peng et al., 2011; Köppl et al., 2014; Corey et al., 2017; Ó Maoiléidigh and Ricci, 2019). How the sound signal is transmitted to the hearing organ, how it responds, and the frequency range over which it operates depend on species-specific specializations. The sensory hair cell is common to each hearing organ, named for its apically protruding mechanosensitive organelle, the hair bundle. Each hair bundle converts its mechanical deflection into a receptor current.
Hair bundles have a pseudo-hexagonal array of actin-filled stereocilia, increasing in height across the bundle like a staircase (Peng et al., 2011; Corey et al., 2017; Miller et al., 2021). Although most hair bundles also have a microtubule-based kinocilium, mammalian cochlear hair bundles lose their kinocilium during development (Wang and Zhou, 2021). All bundles, however, have rows of stereocilia of similar height (Ó Maoiléidigh and Ricci, 2019).
Stereocilia are coupled by the fluid between them and by a variety of extracellular linkages (Kozlov et al., 2011; Richardson and Petit, 2019; Miller et al., 2021). Tip links couple neighboring stereocilia of differing height (Pickles et al., 1984). Hair-bundle displacement toward the taller stereocilia tenses the tip links, which transmit tension forces to open mechanoelectrical transduction (MET) channels located near the tips of the shorter stereocilia (Beurg et al., 2009). Tip links contribute to the gating springs regulating MET-channels, through which the receptor current flows (Corey and Hudspeth, 1983; Howard and Hudspeth, 1988; Assad et al., 1991).
The role of a second class of link, connectors that are not attached to channels, is less clear. Connectors that mechanically couple stereocilia may ensure stereocilia move with similar angular displacements in response to direct stimulation of few (<10%) stereocilia (nonuniform stimulation) (Kozlov et al., 2011). This idea is supported by observations in frog saccular and turtle auditory hair bundles, in which stereocilia move with similar angular displacements in response to nonuniform stimulation (Crawford and Fettiplace, 1985; Kozlov et al., 2007; Karavitaki and Corey, 2010). Stereocilium displacements being similar may ensure that forces exerted by tip links are similar in time and magnitude, maximizing coordination of MET-channel gating and thus maximizing the receptor current (Crawford and Fettiplace, 1985; Kozlov et al., 2007; Karavitaki and Corey, 2010; Kozlov et al., 2012; Ó Maoiléidigh and Ricci, 2019). In contrast, mammalian cochlear hair bundle stereocilia do not move similarly in response to a nonuniform stimulus, decreasing receptor currents for nonsaturating stimuli (Langer et al., 2001; Karavitaki et al., 2013; Nam et al., 2015; Scharr and Ricci, 2018). Connectors between stereocilia in the mammalian cochlea may be weaker than in other organs, explaining the dissimilar displacements (Nam et al., 2015). Previous observations of individual stereocilia have, however, been of their static displacements in vitro, whereas auditory stereocilia move dynamically in vivo (Langer et al., 2001; Karavitaki et al., 2013). Coupling through damping forces, which are absent for static stimuli, may ensure similar stereociliary displacements (Kozlov et al., 2011). Alternatively, a more uniform stimulus in situ may be required to ensure similar stereociliary displacements.
The goal of the present work is to measure the dynamic displacements of individual mammalian cochlear inner-hair-cell (IHC) stereocilia in response to a well-specified bundle stimulus and to quantify the mechanical coupling between IHC stereocilia in the tallest row (row 1). We present a novel method for stimulating row 1 stereocilia and measuring their dynamical displacements in rat IHCs. A mathematical model explains our observations, quantifies mechanical coupling within row 1, and allows us to calculate displacements versus auditory stimulus frequency.
Materials and Methods
Animals and dissection
Sprague Dawley rats of both sexes, age postnatal day 7–10, were killed by decapitation using methods approved by the Stanford University Administrative Panel on Laboratory Animal Care. Organs of Corti from the 1–6 kHz apical region were dissected and their tectorial membranes removed as previously described (Müller, 1991; Beurg et al., 2009). The tissue was placed into a recording dish and immobilized using dental floss fibers, while ensuring the hair cell bundles were oriented vertically (Peng and Ricci, 2016; Effertz et al., 2017).
Electrophysiology
Whole-cell patch-clamp electrophysiology was used to monitor MET currents elicited from IHCs as described previously (Ricci et al., 2005). The tissue was perfused at rates of 0.35–0.5 ml/min with an external solution containing the following (in mm): 140 NaCl, 2 KCl, 2 CaCl2, 1 MgCl2, 10 HEPES, 2 creatine, 2 pyruvate, and 2 ascorbate, pH 7.4, at 300–315 mOsm, as measured by freezing point depression (Model 3320 Micro-Osmometer, Advanced Instruments). In addition, to protect hair bundles from the internal solution, which damages MET because of its low Ca2+ concentration, the apical surface of the tissue was perfused with external solution (at rates of 0.07–0.1 ml/min) using pipettes with tip sizes of 100–200 µm (Ricci and Fettiplace, 1997; Ricci et al., 2005). The whole-cell patch-clamp configuration was achieved using an Axopatch 200b amplifier (Molecular Devices) with thick-walled borosilicate patch pipettes with <4 MΩ pipette resistance after filling with an intracellular solution containing the following (in mm): 125 CsCl, 3.5 MgCl2, 3 ATP, 5 creatine phosphate, 10 HEPES, 1 EGTA, 4 ascorbate, pH 7.2, at 280–290 mOsm. Experiments were performed at 18–22°C. Whole-cell currents were filtered at 10 kHz, sampled at 100 kHz using a Digidata 1440A digitizer (Molecular Devices) controlled by jClamp (SciSoft) software. Cells were held at a potential of −84 mV after being corrected off-line for liquid junction potentials (∼4 mV). Whole-cell membrane capacitance was 10 ± 0.75 pF, uncompensated series resistance was 11 ± 3.5 MΩ, and leak current was 104 ± 51 pA, n = 15 cells, at the holding potential.
Mechanical stimulus probes
Borosilicate pipettes were fire polished to create tip sizes of 1–2 µm for narrow probes and 7–10 µm for wide probes (Ricci et al., 2005). Probe tips were shaped to blunt rectangles, which for wide probes, approximated the shape of the front of the IHC bundle. Probes were secured in an aluminum holder affixed to the front end of a piezoelectric stack (catalog #AE0505D08F, Thorlabs) with marine epoxy. The stack was mounted on a tungsten rod as described previously (Peng et al., 2013). The input to the probe was first filtered using an eight-pole Bessel filter (900 Tunable Active Filter, Frequency Devices) at 5–10 kHz before being sent to a custom-designed high-voltage/high-current amplifier to drive the piezoelectric stack. When stimulating with wide probes, each cell was required to have a peak current of at least 600 pA for inclusion. For narrow probes, there was no minimum current requirement.
Visualization and imaging
The tops of the row 1 stereocilia were visualized using 5×/0.15 NA dry and 100×/1.00 NA water immersion lenses (Olympus) on an Olympus BX51WI microscope (Fig. 1). Cells were recorded during high-speed imaging experiments with a Phantom VEO 640S (Vision Research) camera at 25,000–50,000 frames per second (fps) controlled by jClamp software (SciSoft; Peng et al., 2021; Wang et al., 2021). To prevent vibration, the internal fan of the camera was silenced for at least 3 s before the start of the protocol until the end of data capture. The exposure time of each frame was 20–30 µs. A Lambda TEDled+ (Sutter Instruments) light source, 530 nm, was used to provide enough light to resolve the image, given the short exposure time. The Lambda TEDled+ outputs 90 mW of light at full power from the LED. Using a light meter, the output measured at the camera was 0.5 mW.
Experimental protocols
Narrow versus wide probe activation-step protocol
Initially, MET currents were recorded without individual stereocilium imaging, to characterize the current in response to different size probes (Fig. 2; Scharr and Ricci, 2018). In these experiments, the hair cell was stimulated for 50 ms with step displacements ranging from −400 to 1300 nm for wide probes and from −500 to 1400 nm for narrow probes (positive values indicate displacement toward row 1). Wide-probe bundle displacements were determined from probe movement, as calibrated with a reticle, because the bundle moves with the probe, and movement is linearly dependent on the voltage (Ricci et al., 1998; Peng et al., 2013; Effertz et al., 2017). Probe displacement was confirmed using our displacement detection algorithm described below.
High-speed imaging displacement-step protocol
Stereociliary displacement during narrow-probe stimulation was recorded using high-speed imaging. The stimulating probe was placed in the recording dish above a bundle whose row 1 was aligned horizontally (X-axis) in the image. The stiff probe stimulator was aligned to move in the vertical (Y-axis) direction of the image, ensuring any displacement in the X direction was <25 nm. This alignment was tested by taking a video of a 300 nm displacement step of the probe alone while monitoring displacement with the edge detection and tracking analysis function in jClamp software. To stimulate a hair bundle, the tip of the glass probe stimulator was placed near the tops of its row 1 stereocilia. The image was focused on row 1 stereocilia tops, which appear white when in focus. The hair bundle was then stimulated for 25–50 ms using step displacements, filtered at 5–10 kHz, with sizes ranging from 25 to 600 nm, toward row 1. Each stimulus was repeated four times to determine the average response. The maximum displacement was limited to ∼600 nm because larger displacements often caused the probe to pass through the bundle, disrupting the links between stereocilia and preventing further experimentation. In addition, the probe sometimes detached from the stereocilia on retracting the probe, slowing the stereocilium motion (see Fig. 4A, arrowheads). Because detachment was variable, we did not analyze the off displacements in detail.
Image analysis
Displacement tracking of individual row 1 stereocilia was performed using the TrackMate plug-in for ImageJ. Contrast enhanced video images of row 1 tops were used to track stereocilia (Fig. 1A,B). TrackMate creates displacement trajectories by first independently identifying spots in each frame and then linking spots between frames (Fig. 1C). The algorithm used for spot detection was the Laplacian of Gaussian (LoG) method with an estimated spot diameter of 400 nm and with subpixel localization enabled. Subpixel localization was performed using a quadratic-fitting scheme around the local maximum of the identified spot. Spots were inspected by eye and filtered based on their X and Y locations, local maximum intensity, SD of intensities, and total or median intensities, to exclude as many nonstereocilium spots as possible. Remaining spots were then linked between frames using the Linear Assignment Problem (LAP) framework (Jaqaman et al., 2008). The LAP framework calculates a linking cost, a number that quantifies the likelihood a spot in the next frame belongs to the same track as a given spot, based on the distance between that given spot and all other spots in the subsequent frame, and then chooses the link with the smallest cost. Track merging and splitting was not allowed, and the maximum possible displacement was set to the size of the displacement step. Tracks were then filtered both by eye, only keeping tracks that passed through stereocilia, and by the number of spots in the track, only keeping tracks where the location of a stereocilium was determined for the entire video. The displacement of the probe was determined by applying the TrackMate algorithm to the bright tip of the probe, using the method described above but with an estimated spot diameter equal to the diameter of the probe.
Image-analysis validation
To validate the displacement-tracking method and to determine inclusion criteria for TrackMate-generated displacement trajectories, a control experiment was performed in which the recording dish was displaced sinusoidally at 20 Hz by a piezoelectric stack mounted to a tungsten rod to have peak-to-peak displacements of 25–500 nm while videos were taken of row 1 stereocilia (Fig. 1D–F). The stereocilia were then analyzed using TrackMate, as described above, to create displacement trajectories (Fig. 1G). The displacement SD of each stereocilium before stimulation (Pre-SD) quantified its intrinsic noise. The noise of each stereocilium was also quantified as the SD of the difference between its displacement trajectory and the average of all the stereociliary trajectories for a given stimulus (Intra-SD; Fig. 1H,I). Pre-SD was proportional to Intra-SD, implying that stimulation did not create additional noise because of tracking errors (Fig. 1J,K). A SD cutoff of 15 nm ensured that the noise was sufficiently small to track each stereocilium in every video frame. The lack of tracking errors during stimulation and the cutoff also applied to probe-displaced stereocilia because the Pre-SD distributions of probe-displaced and dish-displaced stereocilia were similar (Fig. 1L). For inclusion in subsequent analyses, experimental displacement trajectories were required to have a SD of <15 nm in both the prestimulus period and during the steady-state portion of the step stimulus.
Displacement trajectory analysis
TrackMate trajectories were analyzed using custom code written in R software (release 3.6.0; https://www.r-project.org/foundation/). Trajectories were measured relative to their initial X and Y positions, and the measurements presented in the results were taken from these trajectories in both X and Y directions.
The dynamics of the reverse displacement was determined by fitting the first 25 ms of the trajectory after the peak displacement with both single and double exponentials. The single exponential is given by the following:
Stereocilium numbering
To determine how force was transmitted from stereocilia directly stimulated by a narrow probe to other stereocilia, stereocilia of a given bundle wing (row 1 stereocilia on one side of the probe) were numbered relative to the point of stimulation. The most lateral stereocilium touching the edge of the probe was numbered 0, and stereocilia farther from the probe were sequentially numbered 1, 2, 3, …, n. These numbers defined the stereocilium positions.
Analysis groups
Stereocilia were grouped by stimulation size. Stimulation size was determined by the largest displacement of the stereocilium touching the probe. Stimulation sizes ranged from under 50 nm to ∼600 nm. Stimulation sizes were grouped into bins of 50 nm between 0 and 450 nm, and those >450 nm formed one group. To control for variability, groups with fewer than five data points were not analyzed.
Experimental design and statistical analysis
Differences between groups were determined by one- or two-way ANOVA using custom code written in R and followed, if appropriate, by Tukey's HSD post hoc test for paired differences. If measurement values did not differ between the step-size groups (p >0.05), as determined by ANOVA, then the groups were combined. Measurement values that decreased or increased exponentially with increasing stereocilium position (Y/Y0, X/X5, the normalized maximum Y speed
Modeling description
Modeling approach
To model the mechanics of a hair bundle in 3D requires knowledge of tens of parameters including stereociliary heights, widths, and pivot positions; the stiffness, damping, and unloaded states of pivots, gating springs, and connectors; MET channel properties such as the gating swing and the channel transition rates; and adaptation time constants and extent (Fig. 3A shows some of these elements; Nam et al., 2015; Ó Maoiléidigh and Ricci, 2019). The present dataset allows us to reliably quantify only six types of mechanical response data, X and Y onsets, X and Y steady states, Y onset speeds, and Y relaxation time constants. Because fitting a model to data when there are more model parameters than types of data (overfitting) does not have a unique solution, we did not use a detailed 3D model. There are additional measurements from the literature that could be used to constrain a detailed 3D model, but there are measurement uncertainties, technical differences in experimental approach, and uncertainties about mammalian cochlear hair-bundle biology; for example, the mechanisms contributing to adaptation remain under debate (Kennedy et al., 2003, 2005; Ó Maoiléidigh and Hudspeth, 2013; Peng et al., 2013; Nam et al., 2015; Caprara et al., 2020; Miller et al., 2021). These uncertainties and differences would cause uncertainties in fitting a 3D model to our data. Even a 2D model describing X and Y motion would require more parameters (at least 10 if only effective viscoelastic elements were included—X and Y pivot stiffnesses, unloaded deformations, and damping coefficients; link stiffnesses, unloaded deformations, and damping coefficients; and the stimulus onset rate) than the six data types we measured. However, we show below that a model describing only the row 1 Y motion can be reduced to four independent parameters, which can then be fit to four types of measurements—Y onsets, steady states, onset speeds, and relaxation time constants. The present work does not determine how much viscoelastic, MET channel, or adaptation elements contribute to stereocilium displacements but rather quantifies the combined effects of these elements using effective parameters based on our experimental data. The ability of this model to reproduce the experimental data demonstrates its utility.
Modeling assumptions
The model assumes that all row 1 stereocilia have identical heights, are coupled by identical mechanical links, and are anchored to the hair-cell apex by identical mechanical elements (Fig. 3B,C). We also assume that the displacement of a stereocilium in contact with the probe is clamped by the probe so that this displacement can be used as input to the model. The model describes the stereocilia in a row 1 wing, a single contacted stereocilium, and the stereocilia to the left or right of that stereocilium. To eliminate needing to know the unloaded deformations, we assume that the Y motions are linearly related to the stimulus and find the displacement changes from rest
Model parameters
In row 1, N stereocilia in a wing are coupled by links of stiffness
Displacement responses to step stimuli
In response to a stimulus, we calculate the linear change in the displacement of each stereocilium relative to its resting displacement
Consider an instantaneous step-displacement stimulus on a single stereocilium of magnitude
Onset and steady-state displacements
The Laplace solution allows us to find explicit expressions for the onset and steady-state displacements and constraints on the time dependence, which yield insights into how the displacements depend on the parameters and which we can directly fit to our measurements. The onset displacement is given by the following:
In other words, the explicit expression Equation 7 shows that the onset depends on the ratio
The ratio between the onsets of adjacent stereocilia is as follows:
If a stereocilium peaks near the onset, the difference between the onset and the steady state yields the reverse as follows:
The reverse and reverse to onset ratio can be found from this expression (see Fig. 9F,G). This explicit expression also yields a necessary but not sufficient condition for a stereocilium to reverse. For an instantaneous step stimulus, a stereocilium reverses only if
Displacement time dependence
In general,
The dynamics of each stereocilium is determined by
Using one of Vieta's formulas, we calculate the geometric mean of the time constants to be the following:
To model noninstantaneous step stimuli, we use the following:
In summary, the explicit expressions Equations 7–15 provide useful insights into the dependence of the Y displacement on the parameters of the model. Responses to instantaneous steps depend on the parameter combinations
Predicted sinusoidal displacements
Equations 4 can also be solved explicitly for sinusoidal stimuli in the Fourier domain. The Fourier transform of the displacement of stereocilium n in response to the sinusoidal displacement of stereocilium 0,
This expression shows that in response to sinusoidal displacement of stereocilium 0, the other stereocilia respond at the same frequency with smaller displacements (see Fig. 13). The amplitudes of the responses relative to stereocilium 0 are then given by the following:
Responses relative to stereocilium 0 depend on the ratio
In other words, the explicit expression Equation 19 shows that the high-frequency amplitude depends on the ratio
Quantifying the contributions to the effective parameters
Here, we discuss how the stereocilium pivots, gating springs, connectors, fluid coupling, MET channel gating, and adaptation might contribute to the effective parameters and their ratios. The damping ratio
Because MET gating and adaptation potentially cause mechanical effects by deforming elements in series with the gating spring, the combined effective damping and stiffness of all these elements may be decreased by MET gating or adaptation. Thus,
Damping and stiffness ratios from prior work
We calculated what the anchor and link properties of the IHC would be if they were based on the mechanical properties of a well-studied bundle. In prior work on frog saccular hair bundles, the translational pivot damping coefficient was estimated to be
In prior work on IHC bundles, the connector stiffness within rows was taken to be
Data accessibility
Experimental data analysis code is available at https://github.com/scharr/movement_analysis. Model analysis and fitting were done using Mathematica 12.0–13.0. Modeling code is available at https://github.com/dmelody9/ScharrNarrowProbe.
Results
Wide- and narrow-probe receptor currents
We stimulated IHC bundles with wide and narrow probes and recorded the receptor currents (Figs. 1, 2; see above, Materials and Methods). The probe displacement direction was as close as possible to the direction of highest sensitivity (Y direction), and the X direction was defined to be orthogonal to the Y direction. Unlike an ideal stimulus, neither wide nor narrow probes were expected to stimulate the row 1 stereocilia perfectly uniformly: at the same time, the same amount, and the same direction (Fig. 2A–C). Wide probes may also have stimulated the shorter stereocilia of rows 2 and 3, whereas narrow probes stimulated only row 1 stereocilia, like an ideal stimulus (Fig. 2D–F). For the same probe displacement, wide-probe receptor currents had larger near-onset peaks and steady states and decreased more quickly and to a greater extent than narrow-probe receptor currents, in agreement with prior data (Fig. 2G,H; Scharr and Ricci, 2018). This decrease in receptor current during a step stimulus is known as receptor-current adaptation. Because narrow probes directly stimulated only a few stereocilia rather than many row 1 stereocilia, narrow-probe displacement-current activation curves were broader than wide-probe activation curves (Fig. 2I). The receptor current is driven by stereociliary displacement. Therefore, the receptor-current differences evidentially stemmed from the wide probes causing more similar stereociliary displacements than the narrow probes. However, IHC stereociliary-displacement similarity has been quantified to a very limited degree, and the extent and regulation of displacement similarity was unknown (Karavitaki et al., 2013; Nam et al., 2015). In principle, stereociliary-displacement similarity depends on the dynamics and the mechanical coupling between stereocilia. We therefore sought to quantify the dynamics of individual stereocilium displacements and the mechanical coupling to better understand how they shape the receptor currents.
Stereocilium identification and displacement detection. A, A cross section through the top of the tallest row of an IHC bundle (row 1) is shown during stimulation with a narrow glass probe. The image is one frame of a high-speed video taken at 100×, 99 nm pixels, and 25,000 frames per second. The narrow probe directly displaced two row 1 stereocilia, whose tips appear as bright spots. The X direction was aligned with the row, and the probe was displaced in the Y direction. B, A contrast enhanced version of A is shown. Enhancement using a Laplacian of Gaussian filter facilitated stereociliary spot detection. C, The stereociliary spots were identified using the TrackMate plug-in for ImageJ (yellow circles). D–F, Images like A–C are shown in which the dish containing a different bundle and its surrounding tissue was displaced by a piezoelectric stack. A–F, Error bars represent 1 μm. G, The Y displacements of row 1 stereocilia of a bundle (different colors) are shown in response to sinusoidal dish displacement (12.5 nm amplitude, 20 Hz, piezoelectric stack voltage time course shown above). The row 1 stereocilia moved with amplitudes similar to that of the dish (gray dashed lines) and followed the dish ringing after the piezoelectric stack voltage stopped oscillating. The displacement SD (standard deviation) of each stereocilium before dish displacement was calculated over the time interval labeled Pre-SD. H, The average Y displacement of the row 1 stereocilia is shown. I, The difference between displacement of an individual stereocilium (G) and the average (H) is shown. The SD of the difference during dish stimulation was calculated over the time interval labeled Intra-SD. J, For each stereocilium of G, the Pre-SD and Intra-SD are similar, implying stimulation does not cause stereociliary spot detection artifacts. K, For a population of bundles (n = 488 stereocilia), the Pre-SD and Intra-SD are similar (solid-line proportional fit, slope = 1.03, R2 = 0.952). Stereocilia with Pre-SDs larger than 15 nm (dashed line) were not included in further analyses because TrackMate did not detect their spots at all times. L, The Pre-SD distribution of the dish-displaced population (n = 488 stereocilia) was similar to that of the narrow-probe displaced population (n = 1357 stereocilia). The dashed line indicates the 15 nm cutoff for inclusion.
MET currents differ for wide and narrow probes. A, B, Diagrams of row 1 stereocilium tips depict wide- and narrow-probe placement. C, A diagram of row 1 stereocilium tips depict an ideal stimulus in which the row is stimulated perfectly uniformly; each row 1 stereocilium is stimulated the same amount, at the same time, and in the same direction (arrows). D, E, Diagrams of a hair-bundle column depict wide- and narrow-probe placement. A wide probe can stimulate all three rows of stereocilia, whereas a narrow probe stimulates only row 1. F, A diagram of a hair-bundle column shows that an ideal stimulus directly stimulates row 1 at its tip and does not stimulate rows 2 and 3. D–F, Stereocilium pivots (blue dots) are at the hair-cell apex (gray box). Gating springs (green lines) with MET channels at their base (brown dots) and connectors (purple lines) link neighboring stereocilia. G, MET currents of a bundle (bottom) are shown in response to step displacements (top) of a wide probe in the Y direction. The currents peak and adapt to the stimuli. H, MET currents of a different bundle (bottom) are shown in response to step displacements (top) of a narrow probe in the Y direction. I, Activation curves are shown for different bundles in which the peak MET current is shown versus the Y displacement of wide (red, n = 13 bundles) and narrow (blue, n = 16 bundles) probes. The examples from G and H are indicated by circles.
We proceeded to use narrow probes because wide probes have several limitations. First, individual stereociliary displacements could not be measured using wide probes because they obscured the stereocilia. Thus, neither stimulus uniformity nor stereocilium displacement similarity for wide probes could be quantified. Second, although wide probes provided a more uniform stimulus, this stimulus was not completely uniform (Fig. 2A; Nam et al., 2015). Third, many row 1 stereociliary displacements in wide-probe experiments are clamped to equal that of the probe. Thus, the mechanical properties of a bundle could not be determined from its row 1 displacements. Finally, the stimulus applied to each stereocilium in wide-probe experiments is unknown. It remains unclear how to determine which stereocilia are displacement clamped, the extent to which row 2 stereocilia are displacement clamped, and how the other stereocilia are stimulated (Fig. 2A,D) (Indzhykulian et al., 2013). Narrow probes circumvent these difficulties by allowing us to measure row 1 stereociliary displacements, to measure the dynamic unclamped displacements of many stereocilia, to avoid contacting row 2 and 3 stereocilia (because narrow-probe diameters are smaller than the height difference between rows 1 and 2), and to determine the displacement-clamped stimulus applied to the stereocilia contacting the probe (Figs. 1, 2E; George et al., 2020). Narrow probes cause nonuniform stimulation that differs from the in situ stimulus, but they allow us to perform biophysical experiments to determine bundle properties, much like voltage-clamp experiments allow us to characterize the biophysical properties of neurons.
Individual stereocilium displacements
To determine individual row 1 stereocilium displacements, we recorded high-speed brightfield videos (≥25K fps) of row 1 stereociliary displacements in response to step-displacement stimuli with a narrow glass probe, while holding the cells at −84 mV using whole-cell voltage clamp (Fig. 1). The inset in Figure 4A illustrates how stereocilium positions were defined relative to the stimulus probe. Stereocilia at position 0 were directly stimulated by the probe, and stereocilia to the left and right of the probe formed two wings for each bundle. Because of mechanical coupling between the stereocilia, the other row 1 stereocilia were indirectly stimulated by the probe. Stereocilia that were not well tracked because their displacement traces had a high level of noise were excluded from further analysis. In Figure 4, the left-wing stereocilium at position 3 was excluded from further analysis because the SD of its prestimulus displacement was >15 nm (see above, Materials and Methods). We stimulated the bundle toward its tall edge because according to measurements in IHCs and other types of hair cells this is its direction of highest sensitivity (Shotwell et al., 1981; Kros et al., 1992). We displaced the probe predominantly in the Y direction (probe X displacements were <25 nm, such that X/Y ≪ 1) and analyzed stereociliary displacements in the X and Y directions independently. Figure 4A shows the trajectories of individual stereocilia in the Y direction versus time. The displacement of the probe and the directly stimulated stereocilia are colored in blue. The directly stimulated stereocilia (positions 0) moved less in the Y direction than the probe because these stereocilia also moved in the negative X direction, owing to their leaning away from vertical at rest and sliding on the surface of the probe (Miller et al., 2021). Y displacement decreased as stereocilium position relative to the probe increased. In contrast, X displacement magnitude increased to a constant value with distance from the probe, and the indirectly stimulated stereocilia moved toward the probe in the X direction (Fig. 4B). Y displacements exceed X displacement magnitudes near the probe, but X displacement magnitudes sometimes exceeded Y displacements far from the probe (Fig. 4B). Directly stimulated stereocilia sliding on the probe surface affected the Y motions little but had a larger impact on the X motions (e.g., right-wing stereocilia 1 and 2 in Fig. 4B). However, the X motions moving toward the probe were consistent with mechanical coupling between stereocilia pulling the indirectly stimulated stereocilia toward the probe.
The Y displacements of many indirectly stimulated stereocilia peaked near the onset of the step stimulus and reversed direction before attaining their steady-state values (Fig. 4C). The onset displacement and steady-state displacements were quantified as the mean displacements within time windows near the onset and offset of the stimulus. In contrast to Y displacements, X displacements did not peak and reverse direction.
Onset displacement position dependence
If row 1 stereocilia are similar and are coupled by links with similar mechanical properties, then the displacements of indirectly stimulated stereocilia would only depend on their positions relative to the probe. To test this hypothesis, we stimulated some bundles at both their centers and sides (Fig. 5). Figure 5A shows images from two high-speed videos of stereociliary displacement when the bundle was stimulated at its center and when the bundle was stimulated at its side. The stereocilia are numbered relative to the center of the bundle and relative to the probe. Like Figure 4, the Y displacements decreased as stereocilium position relative to the probe increased. We quantified the Y onset displacement normalized to the displacement of the directly stimulated stereocilia (YON/Y0ON) for bundles stimulated both at their centers and sides (Fig. 5C; n = 4 cells). For the population of cells, the normalized displacement, YON/Y0ON, decreased exponentially with stereocilium position with similar space constants for center (1.8 ± 0.07) and side stimulation (1.7 ± 0.06). The same functional decrease and similar space constants show that the Y displacements depend mostly on stereocilium positions relative to the probe and agree with preliminary prior work for static displacements (Karavitaki et al., 2013; Nam et al., 2015). These results support the conclusion that row 1 stereocilia and their mechanical coupling differ little within a given bundle.
We next asked whether onset displacements depended on the stimulus size in a larger population of cells (15 cells and 139 step displacements ranging in size from under 25 nm to almost 600 nm). Y onset displacements decreased with stereocilium position for all bundles, individual stimulus sizes, and stimulus-size groups (Fig. 6A,B). To determine whether the Y onset displacement dependence on stereocilium position differed between step-size groups, we quantified the normalized displacement YON/Y0ON versus stereocilium position for each step-size group (Fig. 6C). YON/Y0ON did not differ between step-size groups (two-way ANOVA, F(1,51) = 0.22, p = 0.641), but did differ between stereocilium positions (two-way ANOVA, F(1, 51) = 53.63, p < 1 × 10−8). No differences between step-size groups implies that the Y onset displacements were linearly related to the probe stimulus. This linear relationship implies that mechanical nonlinearities determining the Y onset displacements within row 1 were small. Because there was no difference between normalized displacements between step-size groups, YON/Y0ON, step-size groups were combined. Like Figure 5, the normalized displacement for all bundles and step sizes decreased exponentially with a space constant of 2.11 ± 0.09 stereocilia (Fig. 6C).
For all hair bundles and step sizes analyzed, indirectly stimulated stereocilia moved toward the probe in the X direction (Fig. 4B). To account for directly stimulated stereocilia sometimes sliding on the probe surface causing stereocilium 0 X motions (X0ON is nonzero; Fig. 4B), we determined the X onset displacements relative to X0ON. Across step sizes, the X onset displacement magnitude, |XON|, increased with distance from the probe and plateaued after one to two stereocilium positions (Fig. 6D). When |XON| were grouped by step size, it became clear that larger step sizes caused larger |XON| (Fig. 6E). For stereocilia near the stimulus probe, |XON| were much smaller than Y-onset displacements. The group of largest stimuli (>450 nm) caused |XON|< 100 nm on average. |XON| exceeded the Y-onset displacement, however, for some stereocilia far from the probe.
To determine whether the |XON| dependence on stereocilium position differed between step-size groups, we quantified the normalized displacement |XON|/|X5ON| versus stereocilium position for each step-size group (Fig. 6F). Position 5 was the farthest position for which there were at least five observations for each step-size group. |XON|/|X5ON| did not differ between step-size groups (two-way repeated-measures ANOVA, F(1,51) = 0.419, p >0.5). No differences between step-size groups implies that the X onset displacements were linearly related to the probe stimulus. This linear relationship implies that mechanical nonlinearities determining the X onset displacements within row 1 are small. Therefore, we combined step-size groups and fit the data to a single-exponential function (Fig. 6F). The space constant (1.3 ± 0.2) was about half of that for the normalized Y displacement (2.11 ± 0.09).
Steady-state displacement position dependence
Having quantified the onset displacements, we next asked whether the steady-state displacements had similar dependencies on stereocilium position. Y steady-state displacements decreased with stereocilium position for all bundles, individual stimulus sizes, and stimulus-size groups (Fig. 7A,B). To determine whether the Y steady-state displacement dependence on stereocilium position differed between step-size groups, we quantified the normalized displacement YSS/Y0SS versus stereocilium position for each step-size group (Fig. 7C). YSS/Y0SS did not differ between step-size groups (two-way ANOVA, F(4,372) = 0.93, p = 0.45), but did differ between stereocilium positions (two-way ANOVA, F(1,372) = 795, p < 1 × 10−16). No differences between step-size groups implies that the Y steady-state displacements were linearly related to the probe stimulus. This linear relationship implies that mechanical nonlinearities determining the steady-state displacements within row 1 are small. Because there was no difference between normalized displacements between step-size groups, YSS/Y0SS, step-size groups were combined. The normalized displacement for all bundles and step sizes decreased exponentially with a space constant of 1.73 ± 0.06 stereocilia (Fig. 7C).
Like the onset displacements, for all hair bundles and step sizes analyzed, indirectly stimulated stereocilia moved toward the probe in the X direction. To account for directly stimulated stereocilia sometimes sliding on the probe surface causing stereocilium 0 X motions (X0SS is nonzero; Fig. 4B), we determined the X displacements relative to X0SS. Across step sizes, the X steady-state displacement magnitude, |XSS|, increased with distance from the probe and plateaued after one to two stereocilium positions (Fig. 7D). When |XSS| were grouped by step size, it became clear that larger step sizes caused larger |XSS| (Fig. 7E). For stereocilia near the stimulus probe, |XSS| are much smaller than Y-steady-state displacements. The group of largest stimuli (>450 nm) cause |XSS|< 100 nm on average. |XSS| exceeds the Y-steady-state displacement, however, for some stereocilia far from the probe because the Y-steady-state displacements decrease with increasing stereocilium position (Fig. 7B).
To determine whether the |XSS| dependence on stereocilium position was statistically significant and if it differed between step-size groups, we quantified the normalized displacement |XSS|/|X5SS| versus stereocilium position for each step-size group (Fig. 7F). |XSS|/|X5SS| did not differ between step-size groups (two-way repeated-measures ANOVA, F(5,32) = 0.337, p >0.89) but did differ between stereocilium positions (two-way repeated-measures ANOVA, F(1,38) = 31, p < 1 × 10−5). Thus, |XSS|/|X5SS| groups were combined and fit versus stereocilium position. However, an exponential fit of |XSS|/|X5SS| versus stereocilium position was not statistically significant because of the large variability in |XSS|/|X5SS|. Because directly stimulated stereocilia sometimes slide on the probe surface and because of small magnitudes and greater sensitivity to the exact stimulus direction of X displacements, X displacements were much more variable than Y displacements, and were thus not analyzed further (Figs. 4, 6, 7).
Reverse displacement during stimulation
The Y displacements of indirectly stimulated stereocilia often peaked and reversed direction during stimulation, whereas their X displacements did not (Figs. 4, 8A). Focusing on the Y displacements near the onset of the probe shows that the reverse depended on the stereocilium position (Fig. 8B). We quantified reverse displacements as the difference between the Y onset and steady-state displacements. To exclude spurious reverse displacements because of noise, we included reverse displacements in our analysis only if they exceeded twice the SD of the steady state. The percentage of stereocilia that reversed direction depended on stereocilium position and the stimulus step size (Fig. 8C). Directly stimulated stereocilia did not reverse displacement, and indirectly stimulated stereocilia were more likely to reverse direction as step size increased. Step sizes <51 nm rarely resulted in clear reverse displacements because small reverse displacements were more likely to be excluded from analysis. For step sizes greater or equal to 51 nm, the percentage of stereocilia that reversed direction was maximized at stereocilium positions 3–4. The decrease in the percentage of stereocilia that reversed direction after the maximum was likely caused by a decrease in the reverse magnitude with stereocilium position, which caused them to be excluded from analysis. Thus, we next characterized the reverse magnitudes versus stereocilium position.
The reverses had a large range (6–75 nm) in comparison to their median (20 nm; n = 285; Fig. 8D). Seventy-five percent of the reverses are below 28 nm, however, and only 11 were >50 nm, corresponding to very large stimuli. There was a difference in the reverse because of step size (two-way repeated-measures ANOVA, F(1,55) = 18.71, p < 1 × 10−4) but not because of stereocilium position (two-way repeated-measures ANOVA, F(1,55) = 1.83, p < 0.182). This difference was driven by the largest step sizes because when we excluded step sizes >450 nm, there was no difference in the reverse because of step size (two-way repeated-measures ANOVA, F(1,47) = 1.013, p = 0.3). Grouping all step sizes <450 nm, we found a difference in the reverse because of stereocilium position (one-way ANOVA, F(1,250) = 21.75, p < 1 × 10−5). There was a statistical difference between positions 2–6, 2–7, and 2–8 (post hoc Tukey's HSD test; p = 0.012, 0.002, and 0.018). Excluding the very largest step sizes, the reverse had a small maximum but generally decreased as stereocilium position increased. This decrease made it more likely that a reverse was excluded from our analysis based on its small size, contributing to the decrease in the percentage of stereocilia that reversed direction with stereocilium position (Fig. 8C).
Because a given step size caused a range of Y onset and steady-state displacements, we also quantified the reverse normalized to the onset displacement (Figs. 6A–C, 7A–C, 8E). The normalized reverse increased as stereocilium position increased for all step sizes (one-way repeated-measures ANOVA, F(1,56) = 194.6, p < 1 × 10−15). The reverse displacement accounted for a larger fraction of the Y displacement as the stereocilium position increased. Dividing the data into groups with step sizes ≤250 nm and >250 nm, we found that the normalized reverses were smaller for step sizes >250 nm than for those ≤250 nm (two-way repeated-measures ANOVA, F(1,55) = 17.74, p < 1 × 10−4).
In summary, once the stimulus was sufficiently large, the Y displacement of many indirectly stimulated stereocilia peaked and reversed direction. To determine why the onset and steady-state displacements decreased exponentially with stereocilium position, why Y displacements peaked and reversed, and the reverse dependence on stereocilium position, we developed a model of effectively viscoelastic row 1 stereocilia coupled by effectively viscoelastic links.
Effectively viscoelastic stereocilia and links explained stereocilium displacements
We took three sources of information into account to constrain our model and avoid overfitting. First, our data implied that row 1 displacements depended mostly on stereocilium positions relative to the stimulus probe, row 1 stereocilia and the coupling between them differed little within each bundle, and row 1 stereocilium mechanics and coupling between stereocilia were approximately linear (Figs. 5–7). Second, because variability in the X displacements hindered additional analysis, we focused on modeling the much less variable Y displacements (Figs. 4, 6, 7). Third, to reduce uncertainties and constrain the model parameters using our measurements, we created a model that depended on only four parameter combinations—the same number of data types that we measured (Fig. 3; see above, Materials and Methods). The model consisted of a wing of identical row 1 stereocilia displaced in the Y direction by narrow-probe stimulation of a single stereocilium (Fig. 3C; see above, Materials and Methods). The stereocilia were anchored to the apex of the hair cell by identical linear viscoelastic elements (springs and dashpots) and coupled by identical linear viscoelastic links. These viscoelastic parameters were effective parameters that potentially had contributions from the stereocilium pivots, gating springs, connectors, fluid between stereocilia, MET channels, and adaptation elements (Fig. 3; see above, Materials and Methods). However, it was not necessary to specify these contributions for the model to explain the experimental observations.
A mathematical model describes row 1 Y displacements. A, A schematic map of IHC stereocilium pivot locations and the links between stereocilia is shown. Stereocilia (blue dots, associated with translational pivot stiffness
Stereociliary responses to narrow-probe displacement. A, For the stereocilia tracked in Figure 1C, their Y displacements are shown in response to narrow-probe displacement. The time course of the piezoelectric stack voltage driving the probe (black) is shown above the measured probe displacement (blue). The probe directly displaced two stereocilia (blue). Inset, The position of a stereocilium is defined relative to the probe in which the directly displaced stereocilia have position 0. The left (right) wing of the bundle consists of stereocilia to the left (right) of the probe (long arrows). Retraction of the probe sometimes detached the directly stimulated stereocilia from the probe, slowing their displacements (arrowheads). B, Using the same format as in A, the X displacements are shown in response to narrow-probe displacement. C, Magnified views of X and Y displacements of a stereocilium are shown (A, B, dashed box). Near the onset of the probe stimulus, the Y displacement peaked. Then the stereocilium reversed direction, its Y displacement decreased and relaxed to its steady state. The onset peak, reverse, and steady state are indicated by red arrows. The onset and steady state equal the means over the time intervals indicated by red bars (onset, 0.15–0.3 ms after stimulus start; steady state, 1–0 ms before stimulus end). The X onset and steady state displacements were determined using the same time intervals.
The response of a stereocilium is determined by its position relative to the probe. A, Images of a bundle are shown, which was stimulated by a narrow probe at its center (left) or its side (right). The directly stimulated stereocilia are circled. A, B, Colored numbers indicate stereocilium positions relative to probe. White numbers identify stereocilia relative to the center of the bundle. Error bars represent 1 μm. B, Stereociliary Y displacements are shown in response to narrow-probe displacement at the center of the bundle (top) and side (bottom). C, For four bundles (n = 4 bundles, different colors), the Y onset displacement versus stereocilium position is shown for center (left) and side (right) stimulation. In each case, the displacement is normalized to the displacement of stereocilium 0, YON/Y0ON. The Y onset displacement decreases exponentially with increasing stereocilium position, The black lines are fits (nonlinear least-squares regression) to
Onset responses to narrow-probe stimulation. The onset responses are shown versus stereocilium position (Fig. 4) and are grouped/colored according to probe step size. A, For n = 15 bundles and 139 stimuli, the Y onsets are shown versus stereocilium position and wing identity. B, The mean Y onsets per step-size group are shown versus stereocilium position (n = 5–19 stereocilia for each position and group). C, The means of YON/Y0ON per step-size group are shown versus stereocilium position (n = 5–16 stereocilia for each position and group). YON/Y0ON was not statistically different between groups (two-way repeated-measures ANOVA, F(1,51) = 0.22, p = 0.641). YON/Y0ON was statistically different between stereocilium positions (two-way repeated-measures ANOVA, F(1,51) = 53.63, p < 1 × 10−8). Step-size groups were combined, and the data were fit using nonlinear least-squares regression to an exponential [Fig. 5; C = 0.11 ± 0.01 (p < 1 × 10−15), Λ = 2.11 ± 0.09 (p < 1 × 10−15)]. D, For n = 12 bundles and 41 stimuli, the X onset magnitudes, |XON|, are shown versus stereocilium position. E, The mean |XON| per step-size group are shown versus stereocilium position (n = 6–17 stereocilia for each position and group). F, The means of |XON|/|X5ON| per step-size group are shown versus stereocilium position because X0ON was approximately zero, and the greatest position for which there were data in all groups with step sizes ≥51 nm was position 5 (n = 5–16 stereocilia for each position and group). |XON|/|X5ON| was not statistically different between groups (two-way repeated-measures ANOVA, F(1,51) = 0.419, p > 0.5). |XON|/|X5ON| was statistically different between stereocilium positions (two-way repeated-measures ANOVA, F(1,35) = 93, p < 1 × 10−11). Step-size groups were combined, and the data were fit using nonlinear least-squares regression to an exponential [
We first analyzed model responses to instantaneous step stimuli because in this case the onsets and steady states can be analyzed separately and yield independent constraints for, respectively, the damping coefficients and stiffnesses (see above, Materials and Methods). In response to a step stimulus with an instantaneous onset, the model stereocilium displacements peaked at the onset of the stimulus and reversed direction before relaxing to their steady state (Fig. 9A). Onset and steady-state displacements decreased with stereocilium position because each link transmitted a smaller force to each successive stereocilium, owing to forces also being transmitted to the anchors. These decreases were approximately exponential and depended on the number of stereocilia within each wing (see above, Materials and Methods). The onset displacements depended on the ratio of the anchor to link damping coefficients (λA/λL), which we determined by fitting the experimental onsets to the model (Fig. 9B). In contrast, the steady-state displacements depended on the ratio of the anchor to link stiffnesses (KA/KL), which we determined by fitting the experimental steady-state displacements to the model (Fig. 9C). By fitting all the experiment wings with at least five stereocilia to the model, we found that anchors were weaker than links, as evidenced by the damping and stiffness ratios being <1 (Fig. 9D,E). The model showed that the onset and steady-state displacements had the same approximate exponential dependence on stereocilium position and differed because the damping and stiffness ratios differ. Smaller damping (0.21 median) than stiffness (0.28 median) ratios caused onset displacements to decrease more slowly with stereocilium position than steady-state displacements, explaining the larger space constants for onsets (2.11 ± 0.09) in comparison to steady states (1.73 ± 0.06; Figs. 6, 7, 9B–E; see above, Materials and Methods).
Steady-state responses to narrow-probe stimulation. The steady-state responses are shown versus stereocilium position (Fig. 4) and are grouped/colored according to probe step size. A, Y steady states are shown versus stereocilium position and wing identity. (B) The mean Y steady states per step-size group are shown versus stereocilium position (n = 5–16 stereocilia for each position and group). C, The means of YSS/Y0SS per step-size group are shown versus stereocilium position (n = 5–14 stereocilia for each position and group). YSS/Y0SS was not statistically different between groups (two-way repeated-measures ANOVA, F(4,372) = 0.93, p = 0.45). YSS/Y0SS was statistically different between stereocilium positions (two-way repeated-measures ANOVA, F(1,372) = 795, p < 1 × 10−16). Step-size groups were combined, and the data were fit using nonlinear least-squares regression to an exponential (Fig. 5; C = 0.03 ± 0.01, p = 0.001; Λ = 1.73 ± 0.06, p < 1 × 10−15). D, X steady-state magnitudes, |XSS|, are shown versus stereocilium position. E, The means |XSS| per step-size group are shown versus stereocilium position (n = 5–14 stereocilia for each position and group). F, The means of |XSS|/|X5SS| per step-size group are shown versus stereocilium position because X0SS was approximately zero, and the greatest position for which there were data in all groups with step sizes ≥51 nm was position 5 (n = 5–10 stereocilia for each position and group). |XSS|/|X5SS| was not statistically different between groups (two-way repeated-measures ANOVA, F(5,32) = 0.337, p >0.89). |XSS|/|X5SS| was statistically different between stereocilium positions (two-way repeated-measures ANOVA, F(1,38) = 31, p < 1 × 10−5). Step-size groups were combined, and the data were fit using nonlinear least-squares regression to an exponential [
Reverse displacements occurred when the onset displacements were larger than the steady-state displacements, which required the damping ratios to be smaller than the stiffness ratios (see above, Materials and Methods; Fig. 9F–H). The reverse displacements were a consequence of the effective viscoelasticity of a bundle; damping forces determined the onset displacements, whereas stiffness forces determined the steady-state displacements. In agreement with experimental observations, the model showed that the reverse had a maximum but generally decreased as stereocilium position increased and that the reverse normalized to the onset increased with position (Figs. 8D,E, 9F,G). Unlike the experimental data, the percentage of stereocilia that reverse in the model does not depend on stereocilium positions >0 because the model lacks the stochastic fluctuations that confound counting small reverses in the experimental data (Fig. 8C). Slowing the stimulus onset sufficiently eliminated the peaks (Fig. 9I). Because the model predicted that our experimental observations depended on the stimulus onset speed, we next turned our attention to the experimental dynamics of stereocilium onset displacements.
Reverse displacement increases with the stimulus. A, Inset, For the right wing of a bundle (prestimulus image), the Y responses to narrow-probe displacement are shown. The time course of the probe driving voltage is shown at position 0. The error bar represents 1 μm. B, Magnified views of the Y displacements near the stimulus onset show that stereocilia 1–6 peak and reverse displacement (Fig. 4). C, For each step-size group, the percentage of stereocilia that reverse is shown versus stereocilium position (n = 5–15 stereocilia for each position and group). D, Colored by step-size group, the magnitude of the reverse is shown versus stereocilium position (n = 13–49 stereocilia for each position). Box plots indicate the median (thick line) and first and third quartiles (box). For positions 1–8, the median magnitudes were 20, 22, 24, 23, 20, 19, 17, and 17 nm. E, Colored by step-size group, the reverse-to-onset ratio is shown versus stereocilium position. The ratio was statistically different between stereocilium positions (one-way repeated-measures ANOVA, F(1,56) = 194.6, p < 1 × 10−15). The mean ratios for step-sizes >250 nm (black line) are smaller than those for step-sizes ≤250 nm (gray line; two-way repeated-measures ANOVA, F(1,55) = 17.74, p < 1 × 10−4).
A mathematical model explains Y onset and steady-state responses to narrow-probe stimulation. A, Y displacements of a wing with 8 stereocilia are shown in response to a step stimulus with instantaneous onsets and offsets (black);
Dynamics of onset displacements
In experiments, indirectly stimulated stereocilia responded to narrow-probe step stimuli very quickly and almost simultaneously (Fig. 10A,B). To avoid errors caused by noise, we determined the time relative to stereocilium 0 to reach a 50 nm displacement in our population of bundles (Fig. 10C,D). Only the stereocilia closest to the probe reached 50 nm, and the time to reach 50 nm increased with stereocilium position. There was no statistical difference in the time to 50 nm because of step size (two-way ANOVA, F(1,116) = 2.53, p = 0.11), but there was a difference because of stereocilium position (F(1,116) = 163, p < 1 × 10−15). The mean times to 50 nm at different positions were statistically different from each other (except 3–4), as determined by Tukey's post hoc analysis (p < 0.001 for all combinations except for 2–3, in which p = 0.013). Consistent with the increasing time to 50 nm with stereocilium position, the speed of the onset displacements decreased with stereocilium position, although noise caused fluctuations in the speeds of the smallest displacements (Fig. 10E). In our population of bundles, the maximum speed increased with step size and decreased with stereocilium position (Fig. 10F). There were statistical differences in maximum speeds because of step size (two-way ANOVA, F(1,143) = 12.9, p < 0.001) and stereocilium position (two-way ANOVA, F(1,143) = 70.1, p < 1 × 10−14). Normalizing the maximum speed to that of stereocilium 0 eliminated the step-size differences (two-way ANOVA, F(1,121) = 0.009, p = 0.9) but not the stereocilium-position differences (F(1,121) = 409, p < 1 × 10−15). We therefore combined data across step sizes and determined that the space constant for the normalized maximum speeds (2.0 ± 0.28) was like that for the onset displacements (2.11 ± 0.09; Fig. 10G).
Stereocilium onset speed decreases with position. A, Y displacements near the onset of a 275 nm stimulus step are shown for the bundle numbered in B. The gray box highlights the time period in which stereocilia reached 50 nm. B, A prestimulus image of a bundle is shown. The number colors match those in A, C, and E. The error bar represents 1 μm. C, A magnified view is shown of the time period highlighted in A. The horizontal line shows that the stereocilia farthest from the probe had Y displacements <50 nm in size. D, For the stereocilia that had Y displacements >50 nm in a population of bundles (positions 0–4, colored by step-size group), the time to 50 nm relative to stereocilium 0 is shown versus position (n = 39, 33, 28, 14, and 5 stereocilia for each position). Stereocilia farther from the probe took more time to reach 50 nm (two-way ANOVA, F(1,116) = 163, p < 1 × 10−15), but there was no statistical difference between step-size groups (two-way ANOVA, F(1,116) = 2.53, p = 0.11). The mean times (combined step-size groups) to 50 nm between positions were statistically different from each other except for 3–4 (Tukey's post hoc analysis, p = 0.013 for 2–3; p < 0.001 for all other combinations). The mean times are shown versus position (black). E, The Y displacement speed S (slope of each line in C) is shown versus time. The maximum Y speed (SMAX) is indicated for stereocilia 0–4 (black dots). F, For a population of bundles, the mean SMAX per step-size group is shown versus position (colored by step-size group; n = 5–14 stereocilia for each position and step-size group). There was a statistical difference in SMAX because of step size (two-way ANOVA, F(1,143) = 12.9, p < 0.001) and stereocilium position (two-way ANOVA, F(1,143) = 70.1, p < 1 × 10−14). G, The ratio
To further quantify stereocilium response timing, we determined the delay between their maximum-speed time and the maximum-speed time of stereocilium 0. The maximum-speed delay did not change with stimulus step size (two-way ANOVA, F(1,134) = 1.046, p = 0.3) or stereocilium position (two-way ANOVA, F(1,134) = 0.009, p = 0.9; data not shown). In fact, 96% of stereocilia achieved their maximum speed within one video frame of stereocilium 0. Therefore, stereocilia responded within 40 µs of each other, our temporal detection limit. Evidently, the time to 50 nm increased with stereocilium position because the stereocilium speed decreased with stereocilium position and not because stereocilia took longer to reach their maximum speed.
In summary, all row 1 stereocilia responded almost simultaneously, although only a few stereocilia in the center of the bundle were directly stimulated by the narrow probe. Substantial slack in the links between row 1 stereocilia would cause large maximum-speed delays between their displacements. The small maximum-speed delays we measured implied that there was little slack in the links.
Dynamics of reverse displacements
The dynamics of reverse displacements depends on the mechanical properties of the stereocilia and the links between them. To quantify these dynamics in a population of stereocilia (n = 281 stereocilia), we fitted reverse displacements versus time with single and double exponentials (see above, Materials and Methods; Fig. 11A). The distributions of single-exponential and double-exponential fast time constants overlapped and had maxima at 0.25 ms (Fig. 11B). In contrast, the distribution of double-exponential slow time constants was irregular and had a large range, from ∼1 ms to over 20 ms. Thus, we focused on the small single-exponential and double-exponential fast time constants. We used a threshold of 1.5 ms to define the small-time constants that were included in subsequent analysis (n = 251).
Reverse speeds up with increasing position. A, For a stereocilium at position 2, its Y response is shown 25 ms after its peak (Fig. 8B). A single exponential fit the response [red,
Small time constants did not differ because of step size (two-way ANOVA, F(1,52) = 2.0, p >0.1) and depended little on the onset magnitudes (Fig. 11C). Therefore, we combined step-size groups, revealing that the small time constants decreased with increasing stereocilium position (one-way ANOVA, F(1,242) = 7.0, p < 0.01; Fig. 11D). Moreover, the space constant (2.17 ± 0.7) was similar to that of the onsets (2.11 ± 0.09) and the normalized maximum speeds (2.0 ± 0.28). The lack of dependence of the small time constant on step size and onset magnitude was consistent with stereocilium responses being linearly related to the stimulus. To understand these data and the onset dynamics further, we returned to our effective viscoelastic model.
Effective viscoelastic stereocilia and links explained stereocilium dynamics
We studied stereocilium dynamics using our model and the stiffness and damping ratios corresponding to one experimental wing (Fig. 9). We chose a step stimulus with a maximum speed matching the experimentally-observed maximum speed of stereocilium 0 (Figs. 10, 12A–C). By setting the ratio of the link damping coefficient to stiffness (λL/KL = 0.8 ms), we found that we could match the experimental onset and reverse dynamics (Figs. 10, 11, 12; see above, Materials and Methods). With increasing stereocilium position, the times to 50 nm increased, and the maximum speeds decreased (Fig. 12D,E). Moreover, only the stereocilia closest to the probe reached 50 nm, and the stereocilia achieved their maximum speed within 2 µs of each other (Fig. 12C).
The mathematical model explains the onset and reverse dynamics. Y displacements near the stimulus onset are shown for the orange wing from Figure 9. To match the experimental timescales of Figure 10, the stimulus rate parameter σ = 50 kHz, and
Although the number of exponentials describing the reverse dynamics in the model was one less than the number of stereocilia, reverse dynamics were well fit by a single exponential (see above, Materials and Methods). Like the experimental small time constants, the single-exponential time constants of the model decreased with increasing stereocilium position (Figs. 11D, 12F). To determine how the stimulus speed affected reverse dynamics, we also calculated the reverse time constants corresponding to stimulus steps with instantaneous onsets (Fig. 12F). Time constants for experimentally determined and instantaneous steps differed little, implying that the speed of the experimental stimulus did not change the reverse dynamics substantially.
The extensive quantitative agreement between the experiments and the model supported the conclusion that row 1 stereocilia were nearly identical and coupled by nearly identical linear elements.
Stereociliary links within row 1
In situ, sound-induced oscillatory stimuli are applied to all row 1 stereocilia, although we do not precisely know the magnitude, timing, or direction of the stimulus on each row 1 stereocilium (Prodanovic et al., 2015; Sasmal and Grosh, 2018; Wang et al., 2021). In the future, it may be possible to use narrow probes to apply precisely known oscillatory stimuli to 2–3 row 1 stereocilia and measure the displacements of the indirectly stimulated stereocilia. Thus, we used the model to predict the responses of stereocilia to sinusoidal narrow-probe stimulation of a single stereocilium (Fig. 13). The phase delay between stereocilia was small (<6 µs), but responses decreased the farther a stereocilium was from the probe (Fig. 13A). If the damping ratio was smaller than the stiffness ratio (the case for most of our observations; Fig. 9H), the response amplitudes relative to the probe increased with increasing stimulus frequency (Fig. 13B). Conversely, if the damping ratio was larger than the stiffness ratio, the response amplitudes relative to the probe decreased with increasing stimulus frequency (data not shown). Whether the response amplitudes relative to the probe increased or decreased with stimulus frequency, they were much less than unity at low and high frequencies for models based on our population of experimental wings (Figs. 9D,E, 13C,D). For our experimentally based population of damping and stiffness ratios, the high-frequency responses were up to three times greater than those at low frequencies (Figs. 9D,E, 13E). The low-frequency amplitudes depended on the stiffness ratio in the same way as the high-frequency amplitudes depended on the damping ratio (Fig. 13F; see above, Materials and Methods).
The mathematical model predicts the responses to sinusoidal narrow-probe stimulation. A, The responses to a 4 kHz displacement of a narrow probe (black) are shown versus time. The response amplitudes decrease the father a stereocilium is from the probe. Parameters are the same as in Figure 12. Inset, A schematic shows stereocilium positions relative to the probe. B, The response amplitudes relative to the probe are shown versus the stimulus frequency. Amplitudes increase with stimulus frequency. Parameters are the same as in Figure 12. C, The low-frequency amplitude of stereocilium 5 relative to the probe is shown for the 31 wings fit in Figure 9. D, The high-frequency amplitude of stereocilium 5 relative to the probe is shown for the 31 wings fit in Figure 9. E, The ratio of the high-frequency to low-frequency amplitude is shown for stereocilium 5 for the 31 wings fit in Figure 9. F, The amplitudes of the responses relative to the probe at either low or high frequencies are shown versus the anchor-to-link strength ratio, which equals the stiffness ratio
The range of ratios determined from our experimental data implies the links were not sufficiently strong to ensure that the stereocilia would move with similar amplitudes if row 1 in an IHC bundle was nonuniformly stimulated at high or low frequencies with a narrow probe (Figs. 9D,E, 13F). In contrast, the ratios for an IHC bundle estimated using the stiffness and damping values from bundles in a noncochlear vibration-detection organ, the bullfrog sacculus, were much smaller than those we found for IHC bundles (see above, Materials and Methods; Fig. 13F; Koyama et al., 1982; Kozlov et al., 2011). Consequently, if IHC bundles had links and anchors like those of the bullfrog sacculus, then the links would have been sufficiently strong relative to the anchors to ensure similar amplitudes in response to nonuniform stimulation with a narrow probe (Fig. 13F). This implies that IHC bundle links relative to anchors are much weaker than those of bundles in the bullfrog sacculus.
Discussion
For nonuniform stimulation of IHC bundles, Karavitaki et al. (2013) demonstrated that static stereocilium displacements were not similar, Nam et al. (2015) showed that fewer channels open than for a more uniform stimulus, and A.L.S. and A.J.R. (Scharr and Ricci, 2018) quantified the effects on MET channel activation and adaptation kinetics. These data motivated the present work, in which we nonuniformly stimulated IHC bundles and measured the dynamics of individual stereocilia.
Differences between wide- and narrow-probe receptor currents
Narrow-probe peak and steady-state receptor currents were smaller than those of wide probes because, respectively, the row 1 onset and steady-state stereociliary displacements decreased almost-exponentially with distance from the narrow probe (Nam et al., 2015; Scharr and Ricci, 2018). The slow dynamics of the indirectly stimulated row 1 stereocilia likely slowed the receptor current dynamics and increased the adaptation timescale.
Previous observations, using slow nonuniform stimulation (>5 ms rise time) and calcium imaging, suggested that MET channels farther from the stimulus opened for less time and were more delayed in opening (Nam et al., 2015). Our results, using fast nonuniform stimulation (<40 μs rise time), suggest that MET channels opening less was caused by a decrease in stereocilium displacement and speed with increasing distance from the stimulus and that the delay in opening was caused by slow calcium dye kinetics rather than stereocilium motion delay.
Row 1 inner-hair-cell stereocilia are effectively viscoelastic and coupled by effective viscoelastic links
By determining the stimulus on and measuring the displacements of individual row 1 stereocilia, and using a model explaining these displacements, we found that row 1 stereocilia and the links between them behaved effectively as viscoelastic elements. The fast onsets we observed imply there was little slack in the links. We found that anchor and link effective damping and stiffness changed little across row 1 and determined that the anchors had smaller effective damping and stiffness than the links. For most of the bundles we investigated, the anchor-to-link stiffness ratio exceed the damping ratio, causing the onsets of indirectly stimulated stereocilia to exceed their steady states, and thus the stereocilia reversed direction.
In IHCs, anchor-to-link ratios primarily reflect the effects of elements within bundle columns relative to the effects of connectors and interstereociliary fluid within row 1 (Fig. 3; see above, Materials and Methods). The ratios we determined differ from those of a previous IHC bundle model (see above, Materials and Methods; compare present work with Nam et al., 2015,
Our model explains the reverse using the effective viscoelasticity within and between rows, whereas a prior model suggests that stereocilium creep in the stimulus direction can be explained using only the effective viscoelasticity between rows (Peng et al., 2021). These models do not necessarily disagree but instead capture different aspects of bundle mechanics.
MET channel gating and adaptation effects on stereociliary displacements
For nonuniform narrow-probe stimulation, we found nonlinear MET channel gating had a small effect on stereociliary displacements because they were linearly dependent on stimulus size (Howard and Hudspeth, 1988; Russell et al., 1992; Martin and Hudspeth, 2001; Ricci et al., 2002; Kennedy et al., 2005). The linearity of IHC displacements in comparison to the nonlinearity of displacements in other types of bundles may be related to the weaker connectors we found here in IHCs (Howard and Hudspeth, 1988; Ricci et al., 2002; Kozlov et al., 2011; Ó Maoiléidigh and Ricci, 2019). Strong connectors in frog saccular and turtle auditory hair bundles may be required for the strong MET-gating effects on stereocilium displacements seen in these bundles (Howard and Hudspeth, 1988; Martin and Hudspeth, 2001; Ricci et al., 2002). Here, a decrease of stereociliary coordination in IHC bundles may have obscured MET channel gating effects. The effects of MET channel gating on IHC bundle mechanics could be quantified in future work by blocking the MET channels and determining the changes in the effective stiffness and damping of the anchors and links.
There are additional components in hair bundles that have an impact on their mechanical properties, which have been best described in noncochlear bundles. Fast adaptation correlates with a fast negative bundle motion (rebound, notch, twitch), whereas slow adaptation correlates with slow positive bundle motion (Eatock et al., 1987; Howard and Hudspeth, 1987; Assad et al., 1989; Hacohen et al., 1989; Assad and Corey, 1992; Benser et al., 1996; Ricci et al., 2000; Cheung and Corey, 2006). Slow adaptation can, however, be modulated independently of bundle motion (Caprara et al., 2020). Intracellular and extracellular calcium have an impact on the adaptation-associated bundle motion. There is a voltage-dependent bundle movement that requires tip links but not current flow through the MET channels (Ricci et al., 2000; Cheung and Corey, 2006). Finally, there is evidence for a negative bundle motion (sag) that does not have an impact on MET channel gating (Ricci et al., 2002; Cheung and Corey, 2006). The processes underlying these observations in noncochlear bundles may all exist to some degree in IHC bundles. How these processes affect IHC bundle motion depend on IHC stereocilium coupling, which we show is weaker in IHCs than in well-studied frog saccular bundles. Narrow-probe stimulation, high-speed imaging, and the modeling framework will allow us to assess the impact of each of these processes on IHC bundles with unprecedented resolution.
In situ tectorial-membrane stimulation uniformity
In situ, shear motion between the tectorial membrane and the hair-cell apex is expected to stimulate all row 1 IHC stereocilia (>25% of the stereocilia). The distance between the tectorial membrane and the hair-cell apex is less than the viscous boundary layer, implying that in situ fluid forces are laminar and are predominantly applied to row 1 stereocilia rather than rows 2 and 3 (Prodanovic et al., 2015; Sasmal and Grosh, 2018). Other complex in situ stimuli have been proposed in addition to shear motion, but more direct evidence is needed to support these proposals (Steele and Puria, 2005; Nowotny and Gummer, 2006; Smith and Chadwick, 2011; Guinan, 2012; Prodanovic et al., 2015). The extent to which row 1 stereocilia, whose attachment to the tectorial membrane is under debate, are uniformly stimulated in situ is unknown (Hakizimana and Fridberger, 2021). The traveling wave wavelength (>200 µm) greatly exceeds the diameter of a hair bundle (<10 µm), implying that tectorial-membrane motion is uniform over a single hair-bundle diameter (Ren, 2002; Sasmal and Grosh, 2018). However, the stimulus on each row 1 stereocilium depends on the distance between its tip and the tectorial membrane, which may vary across a bundle because of tectorial membrane irregularities, row 1 stereocilium height variability, or row 1 stereocilium leaning angle variability (Sasmal and Grosh, 2018; Miller et al., 2021).
Noncochlear auditory stereocilia move similarly in response to nonuniform stimulation (<10% of the stereocilia), are nonuniformly stimulated in situ, and, we find, have stronger connectors than IHC bundles (Crawford and Fettiplace, 1985; Kozlov et al., 2007; Karavitaki and Corey, 2010; Kozlov et al., 2011; Strimbu et al., 2012; Ó Maoiléidigh and Ricci, 2019). Strong IHC connectors may not be needed between IHC row 1 stereocilia because weaker connectors and the more uniform tectorial-membrane stimulus are sufficient to ensure similar stereociliary displacements. Observations of IHC stereociliary motion in situ show stereocilia have more similar displacements than narrow probes, but these displacements still differ in magnitude, timing, and direction (Wang et al., 2021). Whether these differences are caused by incompletely uniform tectorial-membrane stimuli or variability within each IHC bundle remains to be determined.
Effective viscoelastic coupling between row 1 stereocilia is not sufficient to ensure their displacements are similar
Prior work in IHCs and outer hair cells showed that elastic coupling between row 1 stereocilia was not sufficient to ensure their displacements were similar in response to nonuniform static stimuli (Langer et al., 2001; Karavitaki et al., 2013; Nam et al., 2015). However, IHCs and outer hair cells respond to dynamic oscillatory stimuli at auditory frequencies in vivo. For example, the characteristic frequency range in our sample is 1–6 kHz (Müller, 1991). At these frequencies, viscous coupling is large and might have been sufficient to ensure similar stereociliary displacements, as proposed for frog saccular bundles (Kozlov et al., 2011; Miller et al., 2021). We show here that this is not the case for IHCs.
Stronger links relative to anchors, such as those found in noncochlear bundles, would have ensured similar displacements, but neither IHC link stiffness nor damping were sufficiently large (Kozlov et al., 2011). The mean IHC bundle anchor-to-link stiffness ratio (0.26 ± 0.0003) is four orders of magnitude larger than for the bullfrog sacculus (0.000037), and the mean damping ratio (0.18 ± 0.0006) is two orders of magnitude larger than for the bullfrog sacculus (0.00122)! Although more work is needed to determine the molecular bases for these vast differences, we know that IHC and frog saccular bundles are covered by proteins, sugars, and lipids that may contribute to damping and stiffness of their connectors (LeBoeuf et al., 2011; Richardson and Petit, 2019; Wu et al., 2019). The functional benefit associated with weak row 1 links in IHCs remains unclear. However, given that narrow-probe high-frequency responses are dictated by λA/λL, whereas low-frequency responses are dictated by KA/KL, it is possible that stereocilium coupling contributes to frequency tuning in IHC bundles.
Footnotes
This work was supported by National Institutes of Health–National Institute Deafness and Other Communication Disorders Grant DC0003896 to A.J.R., National Science Foundation Predoctoral Fellowship to A.L.S., and the Stanford Initiative to Cure Hearing Loss supported by the Oberndorf Foundation.
The authors declare no competing financial interests.
- Correspondence should be addressed to Dáibhid Ó Maoiléidigh at dmelody{at}stanford.edu or Anthony J. Ricci at aricci{at}stanford.edu
