Damping Properties of the Hair Bundle
Johannes Baumgart, Andrei S. Kozlov, Thomas Risler, A. James Hudspeth
DDamping Properties of the Hair Bundle
Johannes Baumgart ∗ , Andrei S. Kozlov † , Thomas Risler ‡ , ∗∗ ,§ , andA. J. Hudspeth † ∗ Institute of Scientific Computing, Department of Mathematics,Technische Universität Dresden, 01062 Dresden, Germany; (current address:
Max Planck Institute for the Physics of Complex Systems,Nöthnitzer Str. 38, 01187 Dresden, Germany ) † Howard Hughes Medical Institute and Laboratory of Sensory Neuroscience,The Rockefeller University, 1230 York Avenue, New York, New York 10065, USA ∗∗ Institut Curie, Centre de Recherche, F-75005 Paris, France ‡ UPMC Univ Paris 06, UMR 168, F-75005 Paris, France § CNRS, UMR 168, F-75005 Paris, France
Abstract.
The viscous liquid surrounding a hair bundle dissipates energy and dampens oscillations, whichposes a fundamental physical challenge to the high sensitivity and sharp frequency selectivity ofhearing. To identify the mechanical forces at play, we constructed a detailed finite-element modelof the hair bundle. Based on data from the hair bundle of the bullfrog’s sacculus, this model treatsthe interaction of stereocilia both with the surrounding liquid and with the liquid in the narrow gapsbetween the individual stereocilia.The investigation revealed that grouping stereocilia in a bundle dramatically reduces the totaldrag. During hair-bundle deflections, the tip links potentially induce drag by causing small but verydissipative relative motions between stereocilia; this effect is offset by the horizontal top connectorsthat restrain such relative movements at low frequencies. For higher frequencies the coupling liquidis sufficient to assure that the hair bundle moves as a unit with a low total drag.This work reveals the mechanical characteristics originating from hair-bundle morphology andshows quantitatively how a hair bundle is adapted for sensitive mechanotransduction.
Keywords: fluid coupling, coherence, fluid-structure interaction
PACS:
INTRODUCTION
The key initial step of hearing [6], the transformation from mechanical motion into anelectrical signal, takes place in the hair bundles. Each hair bundle comprises manyapposed elastic stereocilia that are located in a viscous liquid that dissipates energy(Fig. 1 A–B). In this work we focus on the mechanical environment of the mechan-otransduction process.Simultaneous measurements of the stereociliary motion at the bundle’s opposite edgesreveal a high coherence (Fig. 1 C–D, [11]). This unitary movement of the bundle issignificantly less dissipative than the displacement pattern with relative motion [10]. Westudied in detail the mechanics of the close apposition of the stereocilia to understand theforces at play, which must be sufficiently high to work against the pivotal stiffness of theindividual stereocilia and to couple them. Our findings confirm previous observations byKaravitaki and Corey [9] and provide futher insights into the mechanics. a r X i v : . [ q - b i o . S C ] M a y B
10 ms n m C ohe r en c e Frequency (Hz) CD Tip linkTop connectorKinociliary bulb
FIGURE 1.
Structure and movement of a hair bundle. (A) a scanning electron micrograph of ahair bundle from the bullfrog’s sacculus illustrates an array of closely apposed, cylindrical stereociliaseparated by small gaps. The calibration bar corresponds to 2 µm. (B) a schematic diagram depictsa single file of stereocilia in a hair bundle’s plane of symmetry. (C) Simultaneous records of themovement at the two edges of a thermally excited hair bundle convey the impression of highlycoherent motion, which is confirmed by the calculated coherence spectrum (D).
In the following, we present estimates for the drag between closely apposed stereociliaand of the bundle due to the external liquid. These results are discussed in the contextof a detailed finite-element model of the hair bundle. Finally, a simplified mechanicalrepresentation of the mechanics of the passive hair bundle is proposed.
DRAG FORCES IN THE BUNDLE
At the length scales of the hair bundle and at audible frequencies, the liquids of the innerear are nearly incompressible and have low inertia. Furthermore, non-linearities can beneglected as amplitudes of motion are small compared to the geometrical dimensions.Experimental data show that even the displacement between two adjacent stereocilia issmall compared to their relative separation [10]. This allows us to simplify the governingequations and to find analytical estimates for the drag caused by the viscous liquid.
Relative motion between stereocilia
The squeezing mode of motion induces the highest drag. In the gap between thecylinders the velocity profiles over the gap height are quadratic with a large velocitygradient next to the surfaces. An analytical approximation was derived for small gapswith the minimum gap distance varying linearly over the cylinders’ length. For a relativemotion between the two cylinders this reads c squeeze = π η h ξ ( + ξ ) χ ( + ξ ) with ξ = (cid:114) g t g b and χ = (cid:114) rg t . (1)Here h is the cylinders’ height, r their radius, g t and g b the wall-to-wall distance at thetip and bottom, and η the fluid’s dynamic viscosity. A similar drag coefficient estimationwas conducted by Zetes [14] for parallel cylinders. Further details are given in [1, 10].
10 10010 −1 Gap (nm) D r ag ( n N ⋅ s ⋅ m − ) SqueezeSlide
FIGURE 2.
Drag coefficient of pivoting cylinders moving in the plane of their axes. The cylinders’height measure h = r = .
19 µm, and their wall-to-wall distance at the bottom g b = . g t varies along the abscissa. The fluid’s dynamicviscosity is set to η = · s. The drag of two cylinders moving toward their common center causesa high drag due to the squeezing flow in the gap (Squeeze, Eq. 1). If the two cylinders are moving inthe same direction, the related drag is lower by orders of magnitude (Slide, Eq. 2). In Fig. 2 the drag coefficient as computed from Eq. 1 is given for a typical stereocil-iary geometry. The drag diverges with decreasing wall-to-wall distance at the tip anddecreases by about a factor of five per decade around the typical wall-to-wall distanceof 10 nm.
Sliding motion of stereocilia
For the sliding motion the drag is induced by the shear of the liquid between the twostereocilia. If the common translatory motion is removed, the only remaining motionthat induces drag is their relative motion as both cylinders move along their axis butin opposite directions. Based on the analytical solution by Hunt et al. [7] for the dragof a cylinder moving along its axis parallel to a plane wall and taking into account thelever-arm ratios of the pivotal motion, the drag coefficient reads c slide = π η r h (cid:90) hz = ( g o ( z ) / r + ) dz with g o ( z ) = g b − g b − g t h z . (2)The integral was evaluated numerically for given geometries (Fig. 2). For this slidingmotion the drag values with about 0.2 nN · s · m − are significantly lower than for therelative modes and almost independent of the wall-to-wall distance. DETAILED MODEL
We implemented a realistic physical model of the hair bundle with an appropriaterepresentation of the fluid-structure interactions that is able to identify the relevantphysical effects [1, 2, 10]. To solve the boundary-value problem of the displacementfields of solid and liquid, we employed the finite-element method in three-dimensionalpace. In contrast to previous models [3, 5, 13, 14] this can resolve the liquid motionbetween and around the stereocilia simultaneously, without any constrain on the knowngeometry [8], by discretizing the complex geometry with hexahedral elements (Fig. 3).
FIGURE 3.
Mesh of the finite-elementhair-bundle model with the mesh of the liquidshown in the symmetry plane.
The stiffness of the horizontal top con-nectors can be determined from coherencemeasurements of hair bundles without tiplinks. Furthermore, recent stiffness mea-surements on these bundles without tiplinks [10] yield a pivotal stiffness for the in-dividual rootlet of only 10 aN · m · rad − . Thestiffness values of the tip links and top con-nectors are set to 1 and 20 mN · m − , respec-tively, to match the experimental observa-tions.An external force is applied at thekinociliary bulb of the model in the stim-ulus direction to compute the drag as ratioof force divided by velocity (Fig. 4). Thepurely liquid-coupled bundle has a drag co-efficient of up to 4,400 nN · s · m − at a fre-quency of 1 mHz. The drag coefficient de-cays as the coupling forces of the viscousliquid between the stereocilia overcome thepivotal stiffness of the stereocilia. For fre-quencies of 0.1 kHz and higher the drag isaround 50 nN · s · m − . This value is slightlyhigher than the experimentally measureddrag of 30 ±
13 nN · s · m − [10] of a bundlewithout tip links and the calculated drag of29 nN · s · m − for a hemi-ellipsoid displaced at the tip pivoting around one of the minoraxes [12]. This implies a minor contribution by internal losses. If the external liquid isremoved in the model and the bundle moves coherently, the drag coefficient becomesaround 13 nN · s · m − . The equivalent drag of a liquid-filled hemi-ellipsoid displaced atthe tip and subjected to uniform shear is 2.4 nN · s · m − , which is a lower bound as it isunlikely that the bundle with internal liquid shears uniformly.The top connectors fully block the relative modes and the associated drag. The value isconstant around 55 nN · s · m − . A bundle with only tip links as elastic coupling elementsbetween stereocilia exhibits the highest damping compared to the other three cases.Below the frequency of 20 mHz the drag is around 6,000 nN · s · m − . With increasingfrequency the drag always exceeds that of the purely liquid-coupled bundle. The obliquetip links transfer about the same displacement amplitude from the tall stereocilium to thenext shorter one, but the lever arms of the center of the pivotal motion with respect tothe tip-link connection point differ. The shorter stereocilium displaces at a lower heightand therefore rotates further, causing additional relative motion with associated drag.From around 3 kHz to higher frequencies the viscous coupling of the liquid overcomesthe tip-link stiffness and the drag aligns with the purely liquid-coupled solution. The Frequency (Hz) D r ag ( n N ⋅ s ⋅ m − ) No linksTop con.Tip linksTop & tipDenkKozlovInternal −3 −2 −1 FIGURE 4.
Drag of the hair bundle at the kinociliary bulb. Stereocilia are always coupled by theviscous liquid and elastically coupled only by horizontal top connectors (Top con.), tip links (Tiplinks), both types of elastic links (Top & tip), and no elastic links (No links). As comparison aregiven: the minimal drag for a bundle without surrounding liquid (Internal), and experimental datafrom Kozlov et al. [10] from hair bundles without tip links and from Denk et al. [4] from intact hairbundles. The inset, which has axis labels identical to those of the main panel, displays the behaviorat very low frequencies. drag of a bundle with both types of elastic links between stereocilia follows closely thedrag of a bundle with only top connectors for frequencies above 1 kHz. For frequenciesfrom 0.1 Hz to 0.1 kHz the relative motion induced by the tip links increases the drag to85 nN · s · m − . CONCLUSIONS
FIGURE 5.
Sketchof bundle mechanics.
If two closely apposed stereocilia move to their common centerthe related drag coefficient is larger by orders of magnitudecompared to a unitary motion. A coherent and low-dissipationmotion can be assured by the horizontal top connectors or – ifthe frequency is sufficiently high – by the viscous liquid fillingthe gap.Based on the detailed model the essential mechanics can besimplified as given in Fig. 5. The two rigid stereocilia are in-terconnected by an oblique tip-link stiffness, a horizontal top-connector stiffness, and a damper representing the coupling liq-uid. At the rootlet the pivotal stiffness might be put in serieswith a pivotal damper to mimic the drag by the external liquid.The parameter values of the springs and dampers should be cho-sen such that the coupling forces are higher for the horizontalpart than for the oblique tip links.It is remarkable how the characteristics of the viscous liquidare transformed into an asset in the geometrical arrangement of the hair bundle to reducehe total drag and couple the stereocilia coherently.
ACKNOWLEDGMENTS
We thank A. J. Hinterwirth for assistance in constructing the interferometer and B.Fabella for programming the experimental software; C. P. C. Versteegh for drag mea-surements on a scaled model; M. Fleischer for help with programming the fluid finite-element model; R. Gärtner and A. Voigt for discussions of the finite-element model andstochastic computations; M. Lenz for discussions of stochastic computations and theanalytic derivation of fluid-mediated interactions; and O. Ahmad, D. Andor, and M. O.Magnasco for discussions about data analysis. This research was funded by NationalInstitutes of Health grant DC000241. Computational resources were provided by theCenter for Information Services and High Performance Computing of the TechnischeUniversität Dresden. J.B. was supported by grants Gr 1388/14 and Vo 899/6 from theDeutsche Forschungsgemeinschaft. A.S.K. was supported by the Howard Hughes Med-ical Institute, of which A.J.H. is an Investigator. Fig. 4 is taken from reference [10] andFigs. 1, 2, and 3 from the associated supplemental material with some modifications.