Coherent motion of stereocilia assures the concerted gating of hair-cell transduction channels
CCoherent motion of stereocilia assures the concerted gatingof hair-cell transduction channels
Andrei S. Kozlov , Thomas Risler & A. J. Hudspeth The hair cell's mechanoreceptive organelle, the hair bundle, is highly sensitivebecause its transduction channels open over a very narrow range ofdisplacements. The synchronous gating of transduction channels also underliesthe active hair-bundle motility that amplifies and tunes responsiveness. Theextent to which the gating of independent transduction channels is coordinateddepends on how tightly individual stereocilia are constrained to move as a unit.Using dual-beam interferometry in the bullfrog's sacculus, we found that thermalmovements of stereocilia located as far apart as a bundle's opposite edgesdisplay high coherence and negligible phase lag. Because the mechanicaldegrees of freedom of stereocilia are strongly constrained, a force appliedanywhere in the hair bundle deflects the structure as a unit. This feature assuresthe concerted gating of transduction channels that maximizes the sensitivity ofmechanoelectrical transduction and enhances the hair bundle's capacity toamplify its inputs. Howard Hughes Medical Institute and Laboratory of Sensory Neuroscience, The Rockefeller University,1230 York Avenue, New York, New York 10021, USA. These authors contributed equally to this work. Present address: Laboratoire Physicochimie Curie UMR 168, Institut Curie section de Recherche, 26 rued'Ulm, F-75248 Paris cedex
05, France. Correspondence should be addressed to A.J.H.([email protected]).
The high sensitivity of sensory systems requires an efficient use of the energy in stimuli to biasthe open probability of ion channels. For a hair cell of the inner ear, mechanical forces directlygate transduction channels atop stereocilia, the rod-like constituents of the mechanosensitive hairbundle . A hair bundle's sensitivity is determined by the relation between the applied force andthe number of channels opened: the narrower the force range over which gating occurs, thegreater the sensitivity. The coordinated gating of transduction channels is also thought tounderlie active hair-bundle motility, a component of the active process that amplifies and tunesthe responses of hair cells .Because mechanical stimuli are ordinarily applied at the tall edge of a hair bundle,channel gating depends upon the propagation of mechanical force across the array of stereocilia.Each stereocilium possesses a basal rootlet of actin filaments that tends to hold the processupright; as measured at the hair bundle's tip, the combined stiffness of these stereociliary pivotsis about 200 µ N·m –1 (ref. ). In addition, the successive stereocilia in each file are joined by tiplinks that are thought to represent the gating springs attached to transduction channels at one orboth ends. For large bundle deflections and in the presence of a physiological concentration ofCa , the combined stiffness of these gating springs is typically 1000 µ N·m –1 (ref. .The stereocilia of a hair bundle appear at first glance to be connected in a series-parallelconfiguration such that a force applied to the tallest stereocilium in each file would first deflectthat process alone ( Fig. ). Movement of the tallest stereocilium would then tighten the tip linkand perhaps other filaments connecting it to the second, deflecting that process; the secondstereocilium would in turn pull on the third, and so on across the hair bundle. If stereocilia wereto operate in such a configuration, the magnitude of their deflection would diminishprogressively from the tallest to the shortest and the open probability of the transductionchannels would therefore vary with their location ( Fig. ). This arrangement would broaden the relation between force and channel open probability and thereby reduce the sensitivity ofmechanoelectrical transduction. The ensemble of mechanosensitive channels would functionmore efficiently if the stereocilia were instead to adopt a parallel arrangement, in which anapplied force is shared equally among them and deflects all to a similar extent ( Fig. ). Thisarrangement would prevail if the stereocilia were somehow constrained to remain in closecontact during excitation.Visual observations and video measurements have suggested that a hair bundle movesas a unit when subjected to relatively large, low-frequency stimuli. However, a conclusiveexamination of the propagation of mechanical forces across a bundle requires simultaneousmeasurements of the positions of two stereocilia with a sub-nanometer spatial and asub-millisecond temporal resolution, for these are the scales typical of stereociliary movementsduring hearing. Because interferometry can detect motions with the requisite precision, weconstructed a dual-beam, differential laser interferometer and used it to examine the correlationsbetween the thermal motions of individual stereocilia both in quiescent and in spontaneouslyoscillating hair bundles ( Fig. ). Our experiments showed that the movements of stereocilialocated as far apart as the bundle's opposite edges displayed high coherence and zero phase lagover a wide range of frequencies. This result implies that stereocilia are strongly constrained tomove in unison, a feature that promotes the high sensitivity of mechanoelectrical transductionand fosters the coordinated gating that underlies active hair-bundle motility. RESULTS
The morphological organization of the hair bundle is highly conserved throughout thevertebrates. The experiments reported in this study were therefore performed on hair cells fromthe bullfrog's sacculus, a preparation whose mechanical and electrical properties have beenextensively studied and which was used successfully in earlier experiments employing laserinterferometery . For each hair bundle, we measured the time-dependent positions of stereociliawith three distinct configurations of the dual-beam interferometer. In the first arrangement, the two laser beams were focused together on one of the bundle'sedges, for example at the same point on the tip of the longest stereocilium (
Fig. ). Asexpected, the resultant records were closely similar to one another ( Fig. ). For a quiescent hairbundle, the autocorrelation of each signal declined exponentially from its peak value with a timeconstant of approximately 1 ms ( Fig. ). This correlation time represents the quotient of thebundle's drag coefficient, circa nN·s·m -1 (ref. µ N·m -1 including the gating compliance near the resting position . In 38 measurements from 18 hairbundles, the peak value of the cross-correlation of the two signals was 0.93 ± ± Fig. ).In the third and critical set of measurements, we estimated the degree of common motionbetween stereocilia across a hair bundle by positioning the laser beams on its two opposite edges.The individual records were highly similar in quiescent hair bundles ( Fig. ). Theautocorrelations and cross-correlation from each cell were almost identical ( Fig. ). For 29measurements from 18 quiescent hair bundles, the peak value of the averaged cross-correlationwas 0.92 ± phase lag between two signals and whose modulus, called the coherence, represents the qualityof this phase estimate . For signals of infinite duration, the coherence is bounded at eachfrequency by zero for uncorrelated signals and by unity for perfectly correlated ones. For 18quiescent hair bundles, and except for a few deviations caused by laser noise, the averagecoherence obtained in 29 measurements from opposite hair-bundle edges remained high at allsampled frequencies and exceeded 0.88 between 100 Hz and 5 kHz (
Fig. ). When the twolaser beams were positioned together on one edge of the bundle, the average coherence in 38measurements was only 2% greater ( Fig. ). Phase-lag estimates were close to zero over thewhole spectrum for all the quiescent hair bundles. Between 100 Hz and
5 kHz, the standarddeviation was smaller than 0.23 radian for determinations from the bundles' opposite edges andbeneath 0.18 radian for measurements at the same edges. The coherence and phase estimateswere limited at low frequencies by independent drifts in the two laser beams and at highfrequencies by the diminishing signal power, which eventually approached the background noiselevel. The gating compliance of healthy hair bundles lowers their effective stiffness to a valuecomparable to that of a damaged bundle. To ensure that our data reflected the activity offunctional hair bundles, we repeated the measurements with nine oscillatory bundles, whosespontaneous movements required the normal gating of transduction channels by intact tip links(
Fig. ). In this instance, the average cross-correlation obtained in 29 measurements fromopposite hair-bundle edges peaked at a value of 0.92 ± Fig. ). The associated coherenceexceeded 0.87 between 100 Hz and 5 kHz, (
Fig. ), as compared to the value of 0.93 obtainedin 17 measurements with the lasers focused on the same bundle edge ( Fig. ). In everyinstance, the estimated phase lag was negligible over the whole spectrum: the standard deviationwas less than 0.25 radian for measurements at the opposite edges of the oscillating hair bundles,and below 0.12 radian for same-edge measurements.The striking concordance in the movements of the stereocilia across a hair bundle mightreflect the transmission of forces through the tip links or other filamentous connections between stereocilia. Alternatively, the geometrical arrangement of the hair bundle might constrain thestereocilia to move together. To distinguish between these possibilities, we treated cells with5 mM BAPTA, a procedure known to disrupt tip links and certain other connections . Asexpected because tip links contribute significantly to hair-bundle stiffness, treated bundlesdisplayed both a greater root-mean-square displacement (10.2 ± nm, n = versus ± nm, n = Fig. ) and an increased correlation time (2.2 ± ms, n = versus ± ms, n = Fig. ). The coherence between 100 Hz and 5 kHz nonetheless exceeded0.89 in 11 measurements from the opposite edges of the bundles (
Fig. ) and 0.96 in ninemeasurements from the same edges ( Fig. ). Phase-lag estimates showed standard deviationssmaller than 0.14 radians for the opposite bundle edges and 0.08 radians for the same edges.That BAPTA neither lowered the coherence nor increased the phase lag implies that thecoordination of motion among stereocilia does not depend on tip links.To compare our findings with those anticipated for a series-parallel configuration of thehair bundle, we modeled the thermal motions expected for stereocilia interconnected by gatingsprings of various stiffnesses. For the stiffness value estimated in previous experiments on hairbundles from the bullfrog's sacculus , the cross-correlation between the movements ofstereocilia on opposite edges of the modeled bundle was near zero ( Fig. ). The coherence ofthese movements approached the value associated with independent motion at all frequencies;consistent with this result, the phase was essentially random ( Fig. ). Even when the gating-spring stiffness was increased more than fiftyfold in an effort to foster closer coupling betweenstereocilia, the cross-correlation failed to attain the values found in actual experiments ( Fig. ).In this instance, the coherency spectrum displayed coordinated movements of stereocilia on abundle's opposite edges only at low frequencies, but essentially independent motion forfrequencies in excess of 1 kHz ( Fig. ). A simple series-parallel model therefore seemsincapable of reproducing the experimental results with values of gating-spring stiffnesscompatible with the hair bundle's measured stiffness. DISCUSSION
The present results demonstrate that the stereocilia throughout a hair bundle exhibit stronglycorrelated motions up to high frequencies. The coherences measured with the two laser beamsfocused on the same or on opposite edges of hair bundles agree within 2% for frequenciesbetween 100
Hz and 5 kHz. Because quiescent hair bundles display a root-mean-square motionof about 3 nm, these coherence values imply that the average splaying between their successivestereocilia is no more than a few tens of picometers. It follows that stimulus forces exerted on ahair bundle are distributed almost equally among the gating springs. Unlike a simple series-parallel arrangement, the bundle's actual configuration ensures that most of the stimulus energyis delivered to the gating machinery of transduction channels. The results also validate previousestimates of tip-link stiffness and buttress the model of negative hair-bundle stiffness, both ofwhich rely on the assumption that the tip links are arrayed in parallel . Although the presentexperiments involved a receptor organ sensitive to relatively low frequencies, we have observedsimilar behaviors in preliminary experiments performed on auditory hair cells in the gecko'scochlea.What mechanisms might account for the high correlation between the motions ofstereocilia at a bundle's opposite extremes? Viscosity could contribute to the phenomenon, forthe tendency of any stereocilium to separate from a neighbor during high-frequency stimulationwould be opposed by the resultant reduction in the fluid pressure between them . Next, thebasal and lateral links that conjoin stereocilia might resist the separation of these processes. Itshould be noted, however, that the enzymatic digestion used for the present experimentalpreparation largely removed these attachments . The most intriguing possibility is thatstereocilia do not separate because they are forced together, whether at rest or in motion, by thecurvature of the cuticular plate into which they insert . If the stereocilia were prestressedagainst one another, the deflection of any stereocilium during stimulation would allow eachsuccessive stereocilium to relax towards its position of mechanical equilibrium. This movement would compel the hair bundle to move as a unit without imposing the increased stiffnessassociated with stereociliary cross-linking.The negative stiffness of a hair bundle is analogous to the negative resistance of a neuronduring the rising phase of an action potential: both depend upon the collective action of ionchannels that are globally coupled to one another, whether mechanically or through themembrane voltage, and therefore display concerted gating. Such channel interactions resemblethose between protein molecules involved in cooperative phenomena. Molecular cooperativity,such as that in hemoglobin and allosterically regulated enzymes, ordinarily involves molecularcontacts so intimate that changes in the configuration of one subunit or domain are physicallycommunicated to its neighbors. In a hair bundle, however, the stimulus-induced opening andclosing of individual transduction channels—situated micrometers apart on separatestereocilia—alters the balance of forces in the whole structure, in turn affecting the forceexperienced by the entire channel ensemble. METHODS
Experimental preparation.
Sacculi dissected from adult bullfrogs (
Rana catesbeiana )following a protocol approved by the Institutional Animal Care and Use Committee weremaintained in oxygenated saline solution comprising 120 mM NaCl, 2 mM KCl, 1 mM CaCl ,10 mM D -glucose, and 5 mM HEPES at pH min digestion at roomtemperature in 1 mg·ml -1 collagenase (type XI, Sigma Chemical Co.), each sensory epitheliumwas separated from the underlying connective tissue and the otolithic membrane was removed.The epithelium was then folded in its plane of mirror symmetry so that hair bundles protrudedradially from the creased edge and could be imaged in profile. The preparation was secured tothe coverslip bottom of an experimental chamber by placing over it a golden, 100-mesh electron-microscopic grid. Interferometric recording.
The dual-beam laser interferometer incorporated the basic featuresof a differential interferometer . Two independent illumination pathways were established,one with red light from a 1.8-mW, 633-nm helium-neon laser (117A, Spectra-Physics), the otherwith green light from a 10-mW, 532-nm diode-pumped solid-state laser (85
GCA circa nm. A 40 X objective lens of numerical aperture 0.8 acted as a condenser to focus each beam in the specimenplane to a diffraction-limited spot with a full width at half-maximal intensity of about 300 nm.After traversing the specimen, the light from the two colored beams was collectedthrough a second, identical objective lens and separated with a dichroic mirror into twoindependent measurement pathways. Each beam impinged on a polarizing beam splitter, whichresolved the elliptically polarized light into two components whose magnitudes were measuredwith independent photodiodes. The amplified outputs were passed through eight-poleButterworth anti-aliasing filters with a low-pass corner frequency of 20 kHz, then sampled with16-bit resolution at 10- µ s intervals. For each color channel, the ratio of the difference betweenthe signals of orthogonal polarization to their sum was directly related to the phase differencebetween the two closely spaced optical pathways, and thus to the position of the objectintercepted by the beam.To calibrate each beam of the instrument, the lens forming one end of the associatedbeam-steering telescope was displaced through a known distance with a piezoelectricalmanipulator. This procedure deflected the relevant focal spot in the specimen plane by a fixedamount set by the system's optical parameters. Using video imaging, we measured this shift inthe specimen plane and thus determined the relation between the voltage applied to thepiezoelectrical manipulator and the ensuing movement. Finally, during an actual experiment, thesteering lens was displaced through a known distance and the resultant signal from the photodiodes was measured, thus establishing the calibration factor relating specimen movementto photodiode output.In the absence of a specimen, each beam displayed a spectrally flat instrument noise of5·10 -6 nm ·Hz -1 over the frequency range from 100 Hz to 20 kHz, corresponding to a root-mean-square sensitivity of 0.3 nm. When directed at one edge of a hair bundle, the interferometerdetected thermal motion consistent with previous measurements : the power-spectral densitywas approximately 0.03 nm ·Hz -1 up to a cutoff frequency near 180 Hz, implying a root-mean-square displacement of 3 nm. When the red and green laser beams were focused at the samepoint on a hair bundle, there was no detectable cross-talk between the two measurementchannels.
Data analysis.
Each measurement consisted of a set of twenty, 100-ms-long records sampled at10- µ s intervals, between which the detection apparatus was calibrated independently for eachlaser beam. Discrete Fourier analysis of these records was performed after tapering of the datawith a Bartlett window. Correlation functions were estimated from Fourier power-spectralamplitudes for which tapered records had previously been padded with 100-ms-long strings ofzeros to avoid spurious correlations. Outliers were rejected on the basis of unstable root-mean-square amplitudes, abnormal cross-correlation amplitudes, or pronounced drift in the raw data.For each set of data in a given experimental condition, power spectra and correlationfunctions were computed with the remaining independent estimates. The averagedautocorrelation and cross-correlation functions were obtained after normalization of each recordby its root-mean-square value. The coherence and phase spectra associated with thesemeasurements represented the modulus and phase of the coherency spectrum defined at afrequency f by € γ ( f ) = C XY ( f ) / S X ( f ) S Y ( f ) If X ( t ) and Y ( t ) represent the signals from the two laser beams for an individual record, C XY ( f )denotes the associated cross-spectrum estimate at the frequency f , S X ( f ) and S Y ( f ) specify the two power-spectral estimates at that frequency, and ... indicates averaging over several independentrecords. Modeling.
To compare our findings with those expected if the stereocilia were to operate in theseries-parallel configuration depicted in
Fig. , we simulated the outcome of the principalexperiments. The modeled hair bundle consisted of a file of seven stereocilia, each subjected tolinear frictional forces from the surrounding solution that included a deterministic componentwith an effective friction coefficient 20 nN·s·m –1 and a stochastic white noise representing thethermal motion at an ambient temperature of 295 K. Each stereocilium was connected to thecuticular plate by a flexional spring of stiffness 5 µ N·m –1 and to each of its immediate neighborsby a gating spring whose stiffness was adjusted to either 1,000 µ N·m –1 or 53,000 µ N·m –1 . Theformer value corresponds to previous estimates of gating-spring stiffness in the bullfrog'ssaccular hair cells ; the latter value is tenfold that proposed in a series-parallel model . Tomimic the experimental protocol, we generated data in sets of twenty, 100-ms-long recordssampled at 10- µ s intervals. We then analyzed the results by the procedures described above andaveraged the results of ten independent repetitions. ACKNOWLEDGMENTS
The authors thank A. J. Hinterwirth for assistance in constructing the interferometer, B. Fabellafor programming the experimental software, and O. Ahmad, M.O. Magnasco, K. Purpura, and J.Victor for useful discussions. The members of our research group provided helpful commentson the manuscript. The research reported in this paper was funded by the U.S. NationalInstitutes of Health. T.R. was supported by funding from the F.M. Kirby Foundation and fromthe U.S. National Institutes of Health. A.S.K. was supported by Howard Hughes MedicalInstitute, of which A.J.H. is an Investigator. AUTHOR CONTRIBUTIONS
A.J.H. conceived the experiments, A.S.K. performed them, and T.R. conducted the data analysis.A.S.K., T.R., and A.J.H. wrote the paper.
COMPETING INTERESTS STATEMENT
The authors declare that they have no competing financial interests.1. Hudspeth, A.J., Choe, Y., Mehta, A.D. & Martin, P. Putting ion channels to work:mechanoelectrical transduction, adaptation, and amplification by hair cells.
Proc. Natl. Acad.Sci. USA , 11765-11772 (2000).2. Fettiplace, R. Active hair bundle movements in auditory hair cells. J. Physiol. , 29-36(2006).3. Marquis, R.E. & Hudspeth, A.J. Effects of extracellular Ca concentration on hair-bundlestiffness and gating-spring integrity in hair cells. Proc. Natl. Acad. Sci. USA , 11923-11928 (1997).4. Howard, J. & Hudspeth, A.J. Compliance of the hair bundle associated with gating ofmechanoelectrical transduction channels in the bullfrog's saccular hair cell. Neuron , 189-199 (1988).5. Martin, P., Mehta, A.D. & Hudspeth, A.J. Negative hair-bundle stiffness betrays amechanism for mechanical amplification by the hair cell. Proc. Natl. Acad. Sci. USA ,12026-12031 (2000).6. Silber, J., Cotton, J., Nam, J.H., Peterson, E.H. & Grant, W. Computational models of haircell bundle mechanics: III. 3-D utricular bundles. Hear. Res. , 112-130 (2004).
7. Nam, J.H., Cotton, J.R., Peterson, E.H. & Grant, W. Mechanical properties and consequencesof stereocilia and extracellular links in vestibular hair bundles.
Biophys. J. , 2786-2795(2006).8. Hudspeth, A.J. & Corey, D.P. Sensitivity, polarity, and conductance change in the responseof vertebrate hair cells to controlled mechanical stimuli. Proc. Natl. Acad. Sci. USA , 2407-2411 (1977).9. Crawford, A.C., Evans, M.G. & Fettiplace, R. Activation and adaptation of transducercurrents in turtle hair cells. J Physiol , 405-434 (1989).10. Karavitaki, K.D. & Corey, D.P. Hair bundle mechanics at high frequencies: a test of series orparallel transduction. In
Auditory Mechanisms. Processes and Models (ed. Nuttall, A.L., Ren,T., Gillespie, P., Grosh, K. & de
Boer, E) 286-291 (World Scientific Publishing Co. Pte. Ltd.,Singapore, 2006).11. Denk, W., Webb, W.W. & Hudspeth, A.J. Mechanical properties of sensory hair bundles arereflected in their Brownian motion measured with a laser differential interferometer.
Proc.Natl. Acad. Sci. USA , 5371-5375 (1989).12. Martin, P., Bozovic, D., Choe, Y. & Hudspeth, A.J. Spontaneous oscillation by hair bundlesof the bullfrog's sacculus. J Neurosci , 4533-4548 (2003).13. Nowak, M., Vaughan, B.A., Wilms J., Dove J.B. & Begelman, M.C. Rossi X-Ray TimingExplorer observation of Cygnus X-1. II. Timing analysis. Astrophys. J. , 874-891 (1999).14. Assad, J.A., Shepherd, G.M. & Corey, D.P. Tip-link integrity and mechanical transduction invertebrate hair cells.
Neuron , 985-994 (1991).15. Bashtanov, M.E., Goodyear, R.J., Richardson, G.P. & Russell, I.J. The mechanical propertiesof chick (Gallus domesticus) sensory hair bundles: relative contributions of structuressensitive to calcium chelation and subtilisin treatment. J. Physiol. , 287-299 (2004).16. Iwasa, K.H. & Ehrenstein, G. Cooperative interaction as the physical basis of the negativestiffness in hair cell stereocilia.
J. Acoust. Soc. Am. , 2208-2212 (2002).
17. Henderson, S., Mitchell, S. & Bartlett, P. Propagation of hydrodynamic interactions incolloidal suspensions.
Phys. Rev. Lett. , 088302 (2002).18. Hudspeth, A.J. Mechanoelectrical transduction by hair cells in the acousticolateralis sensorysystem. Annu. Rev. Neurosci. , 187-215 (1983).19. Jacobs, R.A. & Hudspeth, A.J. Ultrastructural correlates of mechanoelectrical transduction inhair cells of the bullfrog's internal ear. Cold Spring Harb. Symp. Quant. Biol. , 547-561(1990).20. Denk, W. & Webb, W.W. Optical measurement of picometer displacement of transparentmicroscopic objects. App. Optics , 2382-2391 (1990).21. Denk, W. & Webb, W.W. Forward and reverse transduction at the limit of sensitivity studiedby correlating electrical and mechanical fluctuations in frog saccular hair cells. Hear. Res. , 89-102 (1992).22. Denk, W., Keolian, R.M. & Webb, W.W. Mechanical response of frog saccular hair bundlesto the aminoglycoside block of mechanoelectrical transduction. J. Neurophysiol. , 927-932(1992). Figure 1
The hair bundle and its possible modes of motion. ( a, left ) A scanningelectron micrograph depicts a saccular hair bundle, which comprises about 60cylindrical stereocilia and a single kinocilium at its tall edge. ( a, right ) A schematicdiagram of a slice along the central file of a hair bundle illustrates the geometricalarrangement of the stereocilia. The bundle is about 8 µ m in height and eachstereocilium is roughly 500 nm in diameter; tip links of exaggerated thickness aredepicted between the adjacent stereocilia. Although the stereocilia differ substantially inheight, the extensions of the associated tip links are nearly identical over thephysiological range of stimulation . The spots at the upper edges of the resting hairbundle represent the diffraction-limited laser beams employed in interferometricmeasurements, here positioned on the bundle's opposite edges. The calibration barcorresponds to 5 µ m. ( b, left ) Upon application of a stimulus to the bundle (arrow), theforce would spread decrementally from the longest to the progressively shorterstereocilia if these were arranged in the series-parallel configuration suggested by thebundle's structure. ( b, right ) By contrast, a principally parallel configuration woulddistribute the stimulus force more-or-less equally among the stereocilia. ( c, left ) Underthermal motion, the stereocilia would display relatively independent positionalfluctuations in the series-parallel arrangement. ( c, right ) In the more parallelconfiguration, the stereocilia would be constrained to move together.
Figure Experimental preparation and control measurements ( a ) The laterallyprotruding hair bundles are apparent in a micrograph of a folded sensory epitheliumsecured by an electron-microscopic grid. The hair bundle selected for measurements,designated by a dashed square, is depicted at a higher magnification on the right.Each calibration bar corresponds to 10 µ m. ( b ) When the green and red laser beamsare focused at the same point on the tips of the longest stereocilia, the two records ofthermal motion are highly similar. ( c ) The red dashed line represents the averaged autocorrelation for 19 records acquired with the red laser beam, whereas the continuousblue line corresponds to the cross-correlation computed from the green- and red-laserrecords. The cross-correlation peaks at 0.97 ± Figure Coherency of stereociliary motion for quiescent hair bundles. ( a ) When thegreen and red beams are positioned on the opposite edges of a hair bundle, the twotraces are very similar apart from different amplitudes owing to different stereociliarylengths. ( b ) For this hair bundle, the cross-correlation, averaged over 20 records (blueline) and superimposed on the corresponding autocorrelation (red dashed line), reachesa peak value of 0.95 ± c ) The averaged coherency spectrum for 29measurements from the opposite edges of 18 hair bundles shows uniformly high valuesand a negligible phase lag at frequencies up to 10 kHz. The standard deviations forthese measurements indicate the degree of variability in the different coherence andphase estimates. The phase spectrum shown at a higher magnification demonstrate asystematic deviation of the mean phase lag from zero at frequencies close to analogfilters' cutoff. This deviation occurs in all records and results from non-identical phasedelays introduced by the anti-aliassing filters. ( d ) In 38 measurements from the samecells, the spectrum for the beams focused on the same edge of the hair bundle is highlysimilar to that in ( c ). Figure Coherency of stereociliary motion for oscillating hair bundles. ( a ) Althoughthe interferometric records from the opposite edges of a spontaneously active hairbundle document displacements substantially larger than those observed in a quiescentbundle, the two traces are again very similar. ( b ) Averaged over 20 records, the cross- correlation peaks at 0.99 ± Fig. , owing to the low-frequencyoscillatory movements of the bundles. The negative values of the correlations for timesexceeding ±7 ms reflect the bundle's tendency to move in the opposite direction afterdwelling at one extreme or the other. As shown on a coarser time scale in the insert,the cross-correlation also illustrates the roughly 70-Hz periodicity associated with theoscillations. ( c ) In 29 measurements from nine hair bundles, the averaged spectrumobtained with the beams directed at the bundles' opposite edges displays a highcoherence and negligible phase lag. ( d ) The corresponding spectrum for 17 controlmeasurements from the same hair bundles is essentially identical to the foregoing.
Figure Effect of severing tip links on hair-bundle motion. ( a ) Measurements from theopposite edges of a quiescent hair bundle portray the enhanced thermal motion afterdestruction of the tip links and other interstereociliary filaments by BAPTA. ( b ) Thecross-correlations before and after the treatment document the altered correlation timeof the bundle's motion. Severing the connections between stereocilia softened the hairbundle, thereby increasing the inversely related time constant for mechanical relaxation.The peak cross-correlation in 20 records was 0.97 ± ± c ) The coherencyspectrum averaged across 11 measurements taken from four cells treated with BAPTAconfirms the strong correlation between the movements of the bundles' opposite edges.( d ) The control spectrum for measurements from the same edges of the hair bundlesdiffers negligibly from that in ( c ). Figure Modeling of hair-bundle movements. ( a ) Simulations of the thermalmovements expected for stereocilia in a series-parallel hair-bundle configuration, withthe two indicated values of gating-spring stiffness, yield cross-correlations much lower than those observed experimentally. ( b ) The coherency spectrum obtained for a gating-spring stiffness of 1,000 µ N·m –1 shows a nearly flat coherence near 0.2 and anessentially random phase for nearly all frequencies. This behavior, which ischaracteristic of finite samples of independent random processes, implies negligiblecoupling between stereocilia. ( c ) Even for an unrealistically large value of the gating-spring stiffness, 53,000 µ N·m –1 , the coherency spectrum for a series-parallel modeldisplays results very different from those observed experimentally. The averagedcoherence lies below 0.8 for all frequencies exceeding 500 Hz and reaches the valueexpected for independent random processes around 1 kHz; the phase becomes nearlyrandom above the latter frequency.kHz; the phase becomes nearlyrandom above the latter frequency.