Hydrodynamic forces on hair bundles at high frequencies

Hearing Research, 48 (1990) 31-36 Elsevier

31

HEARES 01406

Hydrodynamic forces on hair bundles at high frequencies Dennis M. Freeman and Thomas F. Weiss Department of Electrical Engineering and Computer Science, and Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts, U.S.A. and Eaton-Peabody Laboratory of Auditory Physiology, Massachusetts Eye and Ear Infirmary, Boston, Massachusetts, U.S.A. (Received 5 September 1990; accepted 25 March 1990)

We have analyzed a model for the motion of hair bundles of hair cells at high frequencies. In the model, hair-cell organs are represented as a system of rigid mechanical structures surrounded by fluid. A rigid body, that represents a hair bundle, is hinged to a vibrating plate that represents the sensory epithelium. These structures are surmounted by a second vibrating plate that represents a tectorial structure. The analysis shows that at high frequencies, fluid forces cause the rigid body to move as though it were attached to the plates with a system of levers. As a result, the angular displacement of the rigid body is proportional to the displacements of the plates even when there are no mechanical attachments of the body to the tectorial plate. This result is independent of both the size and the shape of the rigid body and independent of the presence and proximity of the tectorial plate, although the constant of proportionality depends upon these factors. Therefore, the mechanical stimulation of hair cells may be particularly simple at high frequencies where the structural differences in hair bundles and tectorial attachments - - that have been shown to be important at low frequencies - - play a less important role. Cochlea; Hair cell; Hair bundle; Micromechanics; Cochlear fluids

Introduction I n a c o m p a n i o n paper ( F r e e m a n a n d Weiss, 1990b), we described results of a theoretical study of the m o t i o n of hair b u n d l e s of hair cells at low frequencies. The results suggest that the freq u e n c y - d e p e n d e n c e of h a i r - b u n d l e m o t i o n dep e n d s critically o n the configuration, proximity, a n d mode "of m o t i o n of tectorial structures; a n d that the d i s p l a c e m e n t of a hair b u n d l e c a n be p r o p o r t i o n a l to the displacement, velocity, acceleration, or to n o simple integral of the m o t i o n of the basilar a n d tectorial m e m b r a n e s . I n this paper, we analyze the m o t i o n of hair b u n d l e s at high frequencies * using a b o u n d a r y - l a y e r m e t h o d (Bat-

chelor, 1967; L a n d a u a n d Lifshitz, 1959; Yih, 1979). The results for high frequencies are distinctly different a n d simpler t h a n those for low frequencies. The results suggest that for high frequencies, the f r e q u e n c y - d e p e n d e n c e of h a i r - b u n dle m o t i o n is i n d e p e n d e n t of the presence, proximity a n d m o d e of m o t i o n of tectorial structures; a n d is p r o p o r t i o n a l to either the d i s p l a c e m e n t of the basilar m e m b r a n e (for hair b u n d l e s n o t attached to a tectorial m e m b r a n e ) or to the relative d i s p l a c e m e n t of the tectorial a n d basilar m e m b r a n e s (for hair b u n d l e s attached to a tectorial m e m b r a n e ) . These results suggest that, at high frequencies, all hair b u n d l e s exhibit a ' l e v e r - m o d e ' of m o t i o n , a n d all hair cells act as d i s p l a c e m e n t detectors.

* Preliminary results of this study have appeared elsewhere (Freeman and Weiss, 1986a; Freeman, 1987; Freeman and Weiss, 1988).

Description of the model

Correspondence to: Dennis M. Freeman, Research Laboratory of Electronics, Room 36-865, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A.

W e analyze the same m i c r o m e c h a n i c a l structures described i n previous papers ( F r e e m a n a n d Weiss, 1990a; F r e e m a n a n d Weiss; 1990b). A rigid

0378-5955/90/$03.50 © 1990 Elsevier Science Publishers B.V. (Biomedical Division)

32 body represents a hair bundle, a basal plate represents the s e n s o r y e p i t h e l i u m , a n d a t e c t o r i a l p l a t e r e p r e s e n t s o v e r l y i n g t e c t o r i a l s t r u c t u r e s . T h e entire s t r u c t u r e is s u r r o u n d e d b y a v i s c o u s fluid. W e c o n s i d e r t h r e e c o n f i g u r a t i o n s ( F r e e m a n a n d Weiss, 1990a): the u n a t t a c h e d - t e c t o r i a l configuration s h o w n in Fig. 1, t h e a t t a c h e d - t e c t o r i a l c o n f i g u r a t i o n in w h i c h t h e rigid b o d y is m e c h a n i c a l l y att a c h e d n o t o n l y to the b a s a l p l a t e b u t also to the t e c t o r i a l plate, a n d t h e f r e e - s t a n d i n g c o n f i g u r a t i o n in w h i c h t h e r e is n o t e c t o r i a l plate. W h e n t h e m o t i o n s o f the s t r u c t u r e s a r e s m a l l r e l a t i v e to b o t h t h e i r d i m e n s i o n s and the b o u n d a r y - l a y e r thickness, n o n l i n e a r t e r m s in t h e e q u a t i o n s o f m o t i o n for t h e f l u i d are n e g l i g i b l e ( F r e e m a n a n d Weiss, 1986b, 1990a). B e c a u s e the h y d r o d y n a m i c e q u a t i o n s a r e linear, t h e i r s o l u t i o n c a n b e e x p r e s s e d as a s u m o f c o m p o n e n t s t h a t e a c h result f r o m m o t i o n o f a single m e c h a n i c a l s t r u c t u r e w h i l e all o t h e r s t r u c t u r e s a r e s t a t i o n a r y ( F r e e m a n a n d Weiss, 1990a). F o r s i n g l e - b o d y models, there are three such components: a basal c o m p o n e n t t h a t results f r o m t r a n s l a t i o n o f t h e b a s a l plate, a t e c t o r i a l c o m p o n e n t t h a t results

Ut ~ -->

m

L

Ub~--~ Fig. 1. Geometry of the unattached-tectorial configuration. The basal and tectorial plates, which bound the fluid-filled region, are parallel and separated by a distance G. A rigid body with height L is attached by a hinge to the basal plate and does not contact the tectorial plate. ~, ~, and ~ are unit vectors of an inertial frame of reference with origin at the (undisplaced) location of the hinge. ~ (not illustrated) is perpendicular to both ~ and ~ so that 2 = ~ × ~,. The basal and tectorial plates translate sinusoidally in their planes with radian frequency 60 and complex velocities Ut,~ and Ut~, respectively. The complex amplitude of the angular displacement of the body is O.

Fig. 2. Network model for rotation of the rigid body (Freeman and Weiss, 1990a, Fig. 5); the X indicate elements that contribute negligible torque at high frequencies. The angular velocity j 6 0 0 of the rigid body is determined by a balance between the hydrodynamic and mechanical torques on the rigid body. The hydrodynamic torque Th has three components - - each resulting from motion of one structure with the others stationary (Freeman and Weiss, 1990a, Fig. 4); H B and H r are transfer functions that relate torque on the body to the velocity of the basal and tectorial plate, respectively; Z R is a driving point impedance that relates the torque on the body to its angular velocity. The mechanical torque includes two components. The torque Ta is caused by the attachment of the body to the basal plate, and is represented by an arigular compliance C,,. The torque T, is due to the inertia of the body and depends on both the angular acceleration of the body and the rectilinear acceleration of the hinge (Freeman and Weiss, 1990a, p. 11) 1 represents the moment of inertia of the rigid body about the hinge. The dependent source H I U b represents the inertial torque exerted on the rigid body by translation of the hinge. Both H r U, and Ta are negligible for asymptotically high frequencies. Furthermore, the hydrodynamic functions simplify for asymptotically high frequencies. Both H B a n d Z R are generally complex-valued functions of frequency. However, for asymptotically high frequencies, their values become proportional to j60, i.e. their imaginary parts increase linearly with frequency and their real parts are negligible.

f r o m t r a n s l a t i o n o f t h e t e c t o r i a l p l a t e , ~nd a r o t a tional component that results from body rotation. T h i s analysis, w h i c h is b a s e d o n s u p e r p o s i t i o n , l e a d s to a n e t w o r k d e s c r i p t i o n o f t h e m o t i o n o f t h e b o d y (Fig. 2).

Boundary-layer analysis at high frequencies B o t h v i s c o u s a n d l i n e a r i n e r t i a l f l u i d f o r c e s are g e n e r a l l y i m p o r t a n t in m o d e l s of h a i r - b u n d l e m o t i o n ( F r e e m a n a n d W e i s s , 1988; F r e e m a n a n d W e i s s , 1990a). H o w e v e r , v i s c o u s f o r c e s a r e g e n e r ally more important for fluid located near a movi n g s t r u c t u r e t h a n for f l u i d t h a t is d i s t a n t f r o m a n y s t r u c t u r e . A s a result, b o t h v i s c o u s a n d iner-

33 tial fluid forces must be considered within the boundary layer of fluid adjacent to a rigid structure, but inertial fluid forces predominate outside of these layers (Batchelor, 1967, pages 302-308, 353-358). As frequency increases, boundary-layer thickness decreases and the motion of a viscous fluid approaches that of an inviscid fluid everywhere except in the vanishingly-thin boundary layers. Therefore, the motion of a viscous fluid at asymptotically high frequencies can be characterized by analysis of vanishingly-thin boundary layers near rigid objects plus analysis of motion of an inviscid fluid outside those layers (Batchelor, 1967; Landau and Lifshitz, 1959; Yih, 1979).

Properties of asymptotically-thin boundary layers At high frequencies, we approximate the fluid motion near a rigid structure by that of a thin layer of viscous fluid on the surface of the structure and by inviscid fluid elsewhere. The velocity of the viscous fluid that is adjacent to the rigid structure must equal that of the structure, i.e. viscous fluids are subject to "no-slip" boundary conditions. In contrast, the boundary conditions for an inviscid fluid are "slippery" and only the component of fluid velocity that is normal to the boundary must equal that of the boundary. In fact, the equations of inviscid fluid motion generally require some tangential motion of the fluid relative to its boundary. Fluid motion in the boundary layer is thus determined by the sheafing motion that characterizes the difference between the no-slip condition at the rigid surface and the slip condition required for the nearly inviscid fluid motion outside the boundary layer. Although the boundary-layer thickness decreases with increasing frequency, the difference between the fluid velocities at the inner and outer edges of the boundary layer persists. Thus, the rate of fluid shear must increase as the boundarylayer thickness decreases. The boundary-layer thickness is inversely proportional to the square root of frequency. Therefore, as frequency increases, the rate of the fluid shear increases with the square root of frequency (Batchelor, 1967, page 355). We use these general properties to analyze the high-frequency behavior of models of hair-bundle

motion as follows. First, we compute the fluid motion that would result if the viscosity of the fluid were zero. We use this solution to approximate the nearly inviscid fluid motion outside the boundary layers adjacent to all rigid boundaries. We can then approximate the fluid motion within the boundary layer as uniform fluid shear, with the velocity of the fluid that is adjacent to a rigid boundary set equal to that of the boundary, and the fluid velocity at the edge of the boundary layer set equal to the velocity computed for inviscid fluid.

Properties of fluid motion outside the boundary layers The equations of infinitesimal motion of a viscous, incompressible fluid include a momentum equation jtopU(r, to) = - VP(r, to) + gwEU(r, to)

(1)

and a continuity equation

v,. U(r, to) = o

(2)

where P(r, 60) represents the pressure field, U(r, to) represents the fluid velocity field, /~ and p are the viscosity and density of the fluid, to is radian frequency, and j = x/Z-1. For an inviscid fluid, g is equal to zero, and these equations are simplified (Landau and Lifshitz, 1959, page 18). This is easily seen by taking the curl of Equation 1 with g = 0. Since the curl of the gradient of any scalar function of position is zero (i.e. V x VP(r, 60) = 0), the curl of the fluid velocity field is zero, i.e. the fluid velocity field is irrotational. Therefore, the fluid velocity equals the gradient of a scalar velocity potential q/i(r, to), *

U(r, to) =

to).

(3)

Substitution of Equation 3 into Equation 2 shows that the velocity potential satisfies Laplace's Equation, V2~(r, to) = O.

(4)

* Because the fluid fieldsin single-bodymodelsof hair-bundle motion (Fig. 1) are singly-connected(Batchelor, 1967, page 100), ¢(r, to) is a single-valuedfunction of space.

34 The equations of infinitesimal motion for an incompressible, inviscid fluid reduce to this single equation which depends upon a single scalar function of position. Because the velocity potential satisfies Laplace's Equation, the fluid velocity at every point in an inviscid fluid field is uniquely determined by the normal component of the fluid velocity on closed boundaries (Batchelor, 1967, page 102). Therefore, for single-body models of hair-bundle motion, the fluid velocity distribution that is induced in inviscid fluid is determined uniquely by the instantaneous velocities of the structures; * neither the acceleration nor the past history of the motions of the structures are relevant (Batchelor, 1967, page 104). Fluid motion in single-body models of hair bundle motion results from motions of three structures: translation of the basal plate, translation of the tectorial plate, and rotation of the rigid body. However, because the equations of fluid motion are linear, we can compute the results for arbitrary motions of the three structures by summing the results for three components that each correspond to motion of a single structure while the others are stationary. This application of superposition is important because of a special property of irrotational fluids; when an irrotational fluid is set into motion by a single moving structure, the fluid velocities throughout the field are uniquely determined by the instantaneous velocity of the structure. In combination with linearity, this property implies that the fluid velocity and velocity potential are everywhere proportional to (and in phase with) the velocity of any single moving structure. Substituting the definition of the velocity potential (Equation 3) into the momentum equation for an inviscid fluid, P(r, to)= -jtopO(r, to)

(5)

* Although the rigid structures in these models do not fullyenclose the fluid fields, the fluid velocity can be determined over fully-enclosed boundaries (Freeman and Weiss, 1990a, p. 9, column 2) Because the fluid velocity along the boundaries that are not adjacent to rigid structures approaches zero as frequency increases, those boundaries do not contribute to high-frequency (inviscid) fluid motion.

shows that the pressure field is proportional to j0~ times the velocity potential. Therefore, throughout the fluid field, the pressure field is proportional to the acceleration of the single moving structure. Given only that the fluid is inviscid, it follows that the motion of any of the three structures in the single-body models of hair-bundle motion produces a torque on the rigid body that is proportional to the acceleration of the single moving structure.

Hydrodynamic forces on moving structures We approximate the hydrodynamic force on a rigid body at high frequencies as the sum of two components: an inertial component that results from the inviscid fluid outside the boundary layer that is adjacent to the body, and a viscous component that is generated by the shearing motion of the fluid within the boundary layer. Because the fluid shear in the boundary layer increases with the square root of frequency, the magnitude of the viscous component increases with the square root of frequency. However, the magnitude of the inertial component is proportional to jo~ times the velocity of the body. Therefore, at high frequencies, the inertial component dominates. For asymptotically high frequencies, the total hydrodynamic torque on the rigid body approaches that which would result if the viscosity of the fluid were zero. Fluid forces at high frequencies

The basal, tectorial, and rotational components of hydrodynamic torque on the rigid body in Fig. 1 are each defined as the torque that is produced when just one of the mechanical structures moves and the others are stationary. Therefore, the results of the previous section for single moving structures can be used to characterize these components.

Basal component The basal component corresponds to translation of the basal plate in its plane while the tectorial plate is stationary and the hinge is fixed so that the body cannot rotate. Therefore, the basal plate and rigid body move as a single rigid structure; both translate with velocity Ub(~O)f~. The resulting torque on the rigid body is therefore

35 proportional to the acceleration of the basal plate. The transfer function HB(to), which is equal to the ratio of hydrodynamic torque to basal plate velocity (Freeman and Weiss, 1990a, companion paper, p. 10), has only an inertial component. Therefore, at arbitrarily high frequencies

HB(to ) =jtoN 8

(6)

where NB is a real-valued constant with units of mass times distance.

Tectorial component The tectorial component corresponds to translation of the tectorial plate in its plane when both the basal plate and rigid body are stationary. This mode of motion of the mechanical structures induces no motion of an inviscid fluid. Therefore, at high frequencies, the tectorial component of torque on the rigid body approaches zero.

Rotational component The rotational component of hydrodynamic torque on the rigid body is proportional to the angular acceleration of the body when both plates are stationary. Therefore, the rotational impedance ZR(to), which is equal to the ratio of hydrodynamic torque to angular velocity of the body has only an inertial component. Therefore, at arbitrarily high frequencies

z . ( t o ) = -jtoi

(7)

where I is the rotational inertia of the rigid body about the hinge and NO is the product of the mass of the rigid body and the perpendicular distance between the plate and the center of mass of the body. Thus, fluid forces at asymptotically high frequencies act as added mass (Yih, 1979). The simplifications that occur in the network model at high frequencies are indicated in Fig. 2.

Body motion at high frequencies The angular displacement of the rigid body at high frequencies can be determined by combining the results of the previous section with a model for the attachment of the rigid body to the two plates. In this section, we analyze attachments of the body to the basal plate that can be represented by a rotational spring (as in Fig. 2). The results for m o r e complex attachment models (e.g. that include damping terms) are similar provided the mechanical impedance of these attachments does not grow more rapidly with frequency than an inertial impedance. The magnitude of the impedance of a spring decreases as frequency increases. However, the magnitude of the hydrodynamic impedance ZR(to) and that of the inertial impedance jtoI increase with frequency. Therefore, the spring impedance is negligible at arbitrarily high frequencies, and the relation between the angular velocity of the body and basal plate velocity is

where I R is a real-valued constant with the units of rotational inertia.

jtoO( to )

Superposition of torques

where F is a real-valued constant with the dimensions of length. This relation can be interpreted geometrically; motion of the basal plate causes the body to "pivot" about a point that is a distance F from the hinge. Because the angular displacement of the body is proportional to the displacement of the basal plate, we refer to this as a "lever" mode of motion. This lever mode of motion results for all values of the interplate distance G, including G ~ oo; i.e. it is characteristic of both unattached-tectorial and free-standing configurations. However, the added-mass of the fluid, which is characterized by NB and I R, depends on G. Therefore, the distance F to the pivot point, which

The hydrodynamic torque exerted on the rigid body by arbitrary motions of the body and plates is the sum of the component torques (Freeman and Weiss, 1990a, Equation 29). Therefore, at high frequencies

Th ( to ) =jtoNBVb ( to ) + JIRO(to).

(8)

Notice that the form of this torque is identical to that due to the inertia of the body (Freeman and Weiss, 1990a, Equation 37),

T~( to ) = -jtoNoUb ( to ) - to2IO(to)

(9)

NB + NQ I +I

1 r

(10)

36

is determined by the mass of the body and the added-mass of the fluid, depends on G. Hydrodynamic forces play no role in the attached-tectorial configuration. Because of the mechanical connections between the body and both plates, the angular velocity O(o~) of the rigid body is proportional to the relative velocities of the basal and tectorial plates,

jo O(o ) U,(.,) -

= 1 G

(11)

and a lever-mode of motion is characteristic of the attached-tectorial configuration at all frequencies.

Discussion The effects of fluid forces in models of hairbundle motion are usually difficult to analyze, and few general results have been determined. However, special properties of fluid motion at high frequencies lead to a very simple and general mode of motion. Because at high frequencies the hydrodynamic forces act as added mass, models of hair-bundle motion exhibit a lever mode of highfrequency motion in which the angular displacement of a rigid body is proportional to the displacement of the basal plate to which it is hinged. This mode of motion, which results from forces of fluid origin, is similar to that which results from direct mechanical connections between the rigid body and tectorial plate. These results suggest that the mechanical stimulation of hair cells may be particularly simple at high frequencies. Structural differences in hair bundles and tectorial attachments that have been shown to be important at low frequencies (Free-

man and Weiss, 1990b) play a less important role at high frequencies. Our theoretical work suggests that hair cells with quite different hair-bundle morphologies and quite different relations to tectorial structures all act as displacement detectors at high frequencies.

Acknowledgements This work was supported by NIH grants and by the Sherman Fairchild Foundation.

References Batchelor, G.K. (1967) An Introduction to Fluid Dynamics. Cambridge Univ. Press, London. Freeman, D.M. (1987) Hydrodynamic study of stereociliary tuft motion in hair cell organs, RLE Technical Report 523, Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139. Freeman, D.M. and Weiss, T.F. (1986a) Hydrodynamic study of stereo,ciliary tuft motion. In: Advances in Auditory Neuroscience, The IUPS Satellite Symposium on Hearing, page 19, Internat. Union Physiol.Sci., San Francisco. Freeman, D.M. and Weiss, T.F. (1986b) On the role of fluid inertia and viscosity in stereociliary tuft motion: Analysis of isolated bodies of regular geometry. In: J.B. Allen, J.L. Hall, A. Hubbard, S.T. Neely and A. Tubis, (Eds.), Peripheral Auditory Mechanisms, Springer-Verlag, New York, pp. 147-154. Freeman, D.M. and Weiss, T.F. (1988) The role of fluid inertia in mechanical stimulation of hair cells. Hear. Res., 35, 201-207. Freeman, D.M. and Weiss, T.F. (1990a) Superposition of hydrodynamic forces on a hair bundle. 48, 1-16. Freeman, D.M. and Weiss, T.F. (1990b) Hydrodynamic forces on hair bundles at low frequencies. 48, 17-30. Landau, L.D. and Lifshitz, E.M. (1959) Fluid Mechanics, Pergamon Press, Elmsford, New York. Yih, C. (1979) Fluid Mechanics, West River Press, Ann Arbor, Michigan.