Fluid–structure interaction of the stereocilia bundle in relation
to mechanotransduction
D. E. Zetesa) and C. R. Steele
Division of Applied Mechanics, Stanford University, Stanford, California 94305
~Received 17 October 1996; revised 3 February 1997; accepted 4 February 1997!
Current hypotheses regarding mechanotransduction rely upon motion of the stereocilia relative to
the apical surface of the hair cell. The viscosity of the surrounding endolymphatic fluid will,
however, attenuate stereocilia motion at higher frequencies of excitation. To investigate stereocilia
motion for physiologically reasonable deflections and frequencies of excitation, the fluid–structure
interaction of the stereocilia bundle is considered analytically. Solutions in the frequency domain
are determined for stereocilia bundle dimensions at several locations along the cochlear duct of the
chinchilla. Results indicate that motion of the stereocilia is analogous to that of a low-pass filter.
Comparison of these solutions with Greenwood’s frequency-place map demonstrates that motion of
the stereocilia bundle exists without substantial attenuation at least up to frequencies appropriate for
the location of the corresponding hair cell along the cochlear duct. The variation in stereocilia
morphology within the mammalian cochlea thus appears to provide a collection of low-pass
mechanoreceptors, arranged in order of increasing corner frequency across the auditory spectrum.
© 1997 Acoustical Society of America. @S0001-4966~97!01906-1#
PACS numbers: 43.64.Kc, 43.40.At, 43.80.Gx @RDF#
INTRODUCTION
Deflection of the stereocilia bundle relative to the apical
surface of the hair cell mediates mechanotransduction ~Hudspeth and Corey, 1977; Shotwell et al., 1981!. Current hypotheses regarding gating mechanisms for operation of ion
channels have in common a reliance upon motion of the
stereocilia at auditory frequencies ~Hudspeth, 1989; Hackney
and Furness, 1995, for review!. At present, however, it is not
clear how motion of the fluid and surrounding structures influences displacement of the stereocilia. The viscosity of the
endolymphatic fluid will attenuate the magnitude of stereocilia motion for higher frequencies of excitation ~Steele
et al., 1993!. Phase response and mode shapes of the stereocilia may also appear in the range of excitation frequencies.
The fluid–structure interaction of the stereocilia bundle must
therefore be investigated to determine its dynamic response
for physiologically reasonable deflections and frequencies of
excitation. Results can then be interpreted in relation to proposed models of mechanotransduction.
Previous investigators have considered mechanical
analyses of the stereocilia bundle. Jacobs and Hudspeth
~1990!, Pickles ~1993!, and Geisler ~1993! considered geometric models assuming rigid stereocilia and a specified relationship between the angular deflections of each stereocilium in a column. These models may be valid for
physiologically reasonable deflections of the stereocilia
bundle within the quasi-static range of excitation, although
the nature of geometric constraints prevents these models
from being extended to the dynamic case. Several investigators have also considered the response of the stereocilia
a!
Current address: Medtronic AneuRx, 10231 Bubb Road, Cupertino CA
95014.
3593
J. Acoust. Soc. Am. 101 (6), June 1997
bundle using single degree of freedom lumped parameter
models ~Crawford and Fettiplace, 1985; Strelioff et al.,
1985; Allen, 1990; Freeman and Weiss, 1990; Assad et al.,
1992; Authier and Manley, 1995!. Such single degree of
freedom models consider movement of the stereocilia bundle
as a whole, thereby neglecting hydrodynamic forces from
motion of each stereocilium. These models are therefore restricted to analysis of a single mode shape. A more general
multi-degree of freedom finite element model was developed
by Duncan et al. ~1993!. Although motion of each stereocilium was considered, hydrodynamic forces from the surrounding endolymphatic fluid were not included.
Consideration of hydrodynamic forces is necessary for
analysis of mechanotransduction and phase relationships at
auditory frequencies of excitation. In the following analysis,
linear, multi-degree of freedom equations of motion for the
fluid–structure interaction of the stereocilia bundle including
the extracellular linkages are derived and generalized for any
finite number of stereocilia. Solutions in the frequency domain are considered for measured geometries of inner and
outer hair cell stereocilia bundles from the chinchilla cochlea
~Lim, 1980!. No effort is made within this model to describe
a particular ion channel or gating mechanism. Rather, only
the motion of the stereocilia bundle relative to the apical
surface of the hair cell is modeled to allow future quantitative investigation of hypotheses regarding mechanotransduction. These linear equations are valid until the amplitude of
stereocilia motion causes the net force in the tip links to
become compressive, at which point buckling may contribute a significant nonlinearity. An abbreviated form of this
analysis appears in Zetes and Steele ~1996!, using stereocilia
dimensions from the guinea pig cochlea measured in transmission electron microscopy ~Zetes, 1995!.
0001-4966/97/101(6)/3593/9/$10.00
© 1997 Acoustical Society of America
3593
I. ANALYTICAL DEVELOPMENT
Physiological deflections relevant for mechanotransduction are small ~0.003°–1°, Hudspeth, 1989! and parallel to
the axis of morphological symmetry ~Shotwell et al., 1981!.
The apex of the cell also rotates only for large, physiologically unreasonable deflections of the stereocilia bundle ~Strelioff and Flock, 1984!. In the following analysis, linear deflections of the stereocilia in the excitatory/inhibitory plane
are considered from the normal to the hair cell’s apical surface. As observed by Flock et al. ~1977! and Corey et al.
~1989!, when a stereocilium is loaded at its tip, the majority
of elastic deformation occurs near its tapered base at its insertion into the apex of the hair cell. Each stereocilium shaft
has thus been treated in this analysis as a rigid body with
motion about its basal attachment resisted by an apparent
rotational spring. This equivalent system is derived from
Euler–Bernoulli beam theory for a load applied at the tip of
a beam of varying circular cross section similar to that of a
stereocilium. Finally, the linkages connecting the stereocilia
are treated as linearly elastic elements, with spring constants
derived from one-dimensional elasticity. For clarity, details
of the derivation are described for a single column of three
stereocilia and later generalized for n stereocilia in a given
column.
A. System description and nomenclature
A column of three stereocilia as considered in this analysis is illustrated in Fig. 1. The angles between the long axes
of the tall, middle, and short stereocilia with the normal to
the cuticular plate are described by q 1 , q 2 , and q 3 , respectively. The lengths of the tall, middle, and short stereocilia
are similarly denoted by l i (i51,2,3). The diameter of the
stereocilia is denoted by d, the separation between the stereocilia rootlets by s, the distance from the base of the tall
stereocilium to the upper attachment of the first tip link ~connecting the tall and middle stereocilia! by lt 1 , and the distance from the base of the middle stereocilium to the upper
attachment of the second tip link ~connecting the middle and
short stereocilia! by lt 2 . The angles the first and second tip
links make with the line parallel to the surface of the cuticular plate are described by q 12 and q 23 , respectively. To approximate the resultant force on the stereocilia from the
sound-induced shearing motion of the reticular lamina relative to the tectorial membrane ~Pickles, 1988, for review!, a
force F 1 , is applied at the tip of the tallest stereocilium.
Linear equations of motion are of the general form
~1!
where the vector q and its time derivatives ~denoted by a
superposed dot! are 331 vectors with components representing the angular orientation of each stereocilium in a column of three stereocilia;
HJ HJ HJ
q1
q5 q 2 ,
q3
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q̇ 1
q̇5 q̇ 2 ,
q̇ 3
q̈ 1
q̈5 q̈ 2 .
q̈ 3
J. Acoust. Soc. Am., Vol. 101, No. 6, June 1997
In Eq. ~1!, M is the mass matrix composed of the scalar
components of inertia, C is the damping matrix quantifying
the effects of fluid viscosity, and K is the stiffness matrix
representing the coupled structural stiffness of the stereocilia
bundle. These matrices are 333 for a column of three stereocilia, with their elements referred to in standard subscript
notation
F
F
G
M 11
M 12
M 13
M5 M 21
M 31
M 22
M 23 ,
M 33
M 32
C 11 C 12 C 13
G F
C5 C 21 C 22 C 23 ,
C 31 C 32 C 33
B. Linear equations of motion
Mq̈1Cq̇1Kq5F,
FIG. 1. Side view of a column of three stereocilia with the coordinate
systems and notation used in this analysis. Angles q 1 , q 2 , and q 3 , respectively, describe the angle between the long axis of the tall, middle, and short
stereocilia with the normal to the cuticular plate. Similarly, lengths l i
(i51,2,3) denote the lengths of the tall, middle, and short stereocilia. The
diameter of the stereocilia is denoted by d, the separation between the
stereocilia rootlets by s, and the distance from the base of the stereocilia to
attachment of the first tip link ~connecting the middle and taller stereocilia!,
and the second tip link ~connecting the short and middle stereocilia!, by
lt 1 and lt 2 , respectively. Angles q 12 and q 23 describe the angles the first and
second tip links make with the line parallel to the surface of the cuticular
plate, respectively.
~2!
K 11
K 12
K 13
G
~3!
K5 K 21 K 22 K 23 .
K 31 K 32 K 33
Finally, the forcing function F in Eq. ~1!, is a 331
vector representing the external forces applied to the stereocilia bundle. The loading case of a harmonic force applied to
the tip of the tallest stereocilium is given by
F5
H
F 1 l 1 sin v t
0
0
J
,
~4!
where F 1 is the magnitude of the force, v is the frequency of
excitation, and t represents time.
D. E. Zetes and C. R. Steele: Stereocilia frequency response
3594
1. Stiffness contributions
During deflection of the stereocilia bundle, the stereocilia bend and all extracellular linkages undergo some elongation. However, using a geometric analysis, Geisler ~1993!
showed that the elongation of the lateral links is an order of
magnitude smaller than the elongation of the tip links. Assuming that the lateral links have similar material properties
to the tip links, the contribution of the lateral links to the
structural stiffness of the stereocilia bundle is negligible. The
major contributions to the elements of the stiffness matrix K,
therefore come from bending of the stereocilia and elongation of the tip links. The stiffness matrix can then be written
as a matrix sum K5Kcilia1Ktip, where Kcilia and Ktip are
contributions from the stereocilia and tip links, respectively.
The matrix Kcilia is diagonal and composed of the apparent rotational stiffnesses k 1 , k 2 , and k 3 , for the case of a
load applied at the tip of the tall, middle, and short stereocilia, respectively,
K
F
G
k1
0
0
5 0
0
k2
0 .
k3
cilia
0
~5!
The form of these constants k i (i51,2,3), in relation to the
morphology and material properties of the stereocilia is detailed in Sec. C 1.
The matrix Ktip is symmetric and tridiagonal
Ktip5
k tip
l0
3
F
l 12~ a lt 1 1 b 12r !
2l 12a l 2
2l 12a l 2
h 12a l 2 1l 23X
0
2l 23a l 2
0
G
2l 23a l 2 ,
h 23a l 3
~6!
the effect of a pressure gradient in the long direction is
slowly varying in comparison to the short direction ~Ocvirk,
1952!. The dimensions of the stereocilia are such that the
diameter is much smaller than the stereocilium length d!l i
(i51,2,3). It is therefore reasonable and consistent to assume that the effect of pressure gradient in the long, or axial
direction of the stereocilia is small.
The stereocilia bundle may be treated as having two
fluid bearing surfaces, where the first bearing surface is between stereocilia pairs in a given column, and the second is
between stereocilia pairs in a given row, as shown in Fig. 2.
For a local coordinate system oriented as shown in Fig. 2 and
originating on the excitatory face of a stereocilium surface,
the flow field between stereocilia pairs in a given column is
approximated by the familiar Poiseuille and Couette flows
written in two dimensions
v'
1 ]p 2
~ x 2h ~ y,z ! x ! ,
2m ]y
~7a!
w'
Wx
,
h ~ y,z !
~7b!
where v and w are the velocity profiles between stereocilia
pairs in the y and z directions, respectively, W is the net
velocity of stereocilia membranes in the z direction, h(y,z)
is the separation between the stereocilia shafts, m is the viscosity of the fluid, and ] p/ ] y is the pressure gradient in the
y direction. For clarity, this analysis considers the geometrical simplification of the stereocilia arranged in a cubic array.
The separation between the cylindrical shafts of stereocilia
pairs is approximated by
h ~ y,z ! 5h 0 ~ 11 g y 2 1Dqz ! ,
d
uyu< ,
2
~8!
where k tip is the linear stiffness of the tip link ~also detailed
in Sec. C 1!, l 0 is the natural length of the tip link, s is the
separation between stereocilia rootlets, and r is the radius of
the stereocilia shaft. Geometric parameters are given by
a 5s2r,
b 125lt 1 2l 2 ,
l 125lt 1 cos q122r sin q12 ,
X 5 a lt 2 1 b 23 r , and h 125l 2 cos q12 , and the angular orientation of the tip links is given by q 125tan21„(lt 1 2l 2 )/
(s2r)…. Similar equations can be written for the second stereocilia pair to determine b 23 , l 23 , and h 23 .
2. Viscous contributions
Consideration of characteristic dimensions for physiological deflections and auditory frequencies demonstrates
that flow between the stereocilia is predominantly linear and
viscous ~see the Appendix!. Hydrodynamic contributions to
the equations of motion can therefore be written in matrix
form where in this analysis, the elements of the viscous matrix C in Eq. ~5!, are derived from selected portions of lubrication theory. For the case of the stereocilia bundle, these
approximate solutions assume that the dominant viscous effects occur in the region where the stereocilia are close to
each other and their curvature is slowly varying. For fluid
bearing surfaces of relatively small width and finite length,
FIG. 2. Top view of the stereocilia bundle with the stereocilia rows and
columns geometrically simplified for this analysis in the arrangement of a
regular cubic array. The lines drawn connecting the stereocilia denote tip
links connecting stereocilia columns. The smallest separation between the
stereocilia shafts is denoted by h 0 , and the diameter by d. Relative pressures between stereocilia are denoted by p 12 between the tall and middle
stereocilia, and p 23 between the middle and short stereocilia. Upper right
corner illustrates assumed Poiseuille flow profile and net flow Q ii , between
stereocilia pairs in a given row. Lower right corner illustrates assumed Poiseuille flow profile and net velocity in the excitatory/inhibitory plane DU,
between stereocilia pairs in a given column.
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J. Acoust. Soc. Am., Vol. 101, No. 6, June 1997
3595
where g 52/h 0 d, h 0 is the smallest distance between the
stereocilia shafts, Dq is the relative change in angular position between stereocilia pairs in a given column, and d is the
diameter of the stereocilia. A more general stereocilia arrangement is considered in Zetes ~1995!.
For deflections in the excitatory/inhibitory plane, the stereocilia may also separate with a velocity component U in
the x direction which does not affect Eqs. ~8! and ~9! and is
considered when satisfying continuity. The relevant velocity
components are given by
U i 5q̇ i z,
~9a!
W i 52q̇ i ~ d/2!
~9b!
~ i51,2,3 ! ,
where the subscript i has been introduced to denote the linear
velocities U i and W i , and angular velocities q̇ i of each stereocilium (i51,2,3).
Consideration of characteristic dimensions in a fashion
similar to the Appendix demonstrates that hydrodynamic
forces in the subtectorial space are an order of magnitude
smaller than those between the stereocilia. They are therefore
neglected in this analysis. Edge effects at either end of the
‘‘W’’ arrangement of stereocilia rows are also neglected.
a. Continuity. Continuity must be satisfied to ensure
conservation of mass and compatibility of solutions between
all stereocilia pairs. For small deflections of the stereocilia,
flow in the long direction Eq. ~7b!, is predominantly a shearing of the fluid element with small net flow in comparison to
flow in the short direction Eq. ~7a!. For this analysis, an
additional approximation of planar flow is therefore introduced such that continuity is satisfied only in the plane parallel to the hair cell’s apical surface. A more complete analysis considered in Zetes ~1995! demonstrates that this
approximation does not invalidate the previously described
short bearing analysis nor significantly affect the numerical
solutions described in Sec. II.
Global continuity is satisfied by considering control volumes at the intersection of stereocilia rows and columns
Q 112Q 225 ~ U 1 2U 2 ! d,
Q 222Q 335 ~ U 2 2U 3 ! d,
~10!
where Q 11 , Q 22 , and Q 33 denote the net flow between stereocilia in the tall, middle, and short rows, respectively, and
outward flow is defined as positive.
The net flows between the stereocilia rows Q ii
(i51,2,3), are found from integration of the relevant velocity profile across the channel width and are given by
Q 115
Q 335
2p 12h 30 g 1/2
9mp
22 p 23h 30 g 1/2
9mp
Q 225
,
2 ~ p 232p 12! h 30 g 1/2
9mp
,
~11!
,
p 125
p 235
3d m p ~ 2q̇ 1 2q̇ 2 2q̇ 3 ! z
2 g 1/2h 30
3d m p ~ q̇ 1 1q̇ 2 22q̇ 3 ! z
2 g 1/2h 30
,
~12!
.
b. Tractions and resultant moments. Contributions to
the damping matrix C in Eq. ~3! are given by the resultant
moments of the tractions acting on the stereocilia. The
stresses within the fluid are determined from the constitutive
relationship for an incompressible Newtonian fluid written in
dyadic notation
s52 pI1 m „“ n1 ~ “ n! T …,
~13!
where s is the stress tensor and I is the unity tensor. Also in
Eq. ~13!, the velocity profile n is given by Eqs. ~7!–~9! and
similar equations for velocity profiles between stereocilia
rows. The pressure distribution p is given by Eq. ~12! and
consideration of local continuity for each stereocilia pair.
The relevant tractions can be integrated over the stereocilia surfaces to obtain resultant moments about the bases of
the stereocilia from the viscous and pressure forces. The resultant moment on the tall stereocilium for flow between
stereocilia columns is given by
M column
5
1
E E Ss
EE s
`
l2
2`
0
1
xx u x5h 1 s xy u x5h
l2
0
`
2`
xz u x5h r
D
]h
z dy dz
]y
dy dz,
~14!
where the inner limits of integration are infinite to allow the
slowly varying approximation Eq. ~8! to approach the farfield solution external to the journal bearing Eq. ~12!, and the
outer limits of integration are to the height of the middle
stereocilium, since as discussed previously, contributions
from hydrodynamic forces in the subtectorial space are
small.
Substituting for the necessary quantities and integrating,
5
M column
1
m p l 32 ~ 114 g h 20 !
m p l 2d 2
~ q̇ 1 2q̇ 2 ! 1
~ q̇ 1q̇ 2 ! .
3/2 3
4 g 1/2h 0 1
2g h0
~15!
Similarly, between stereocilia within the same row, the
resultant moment on the tallest stereocilium is given by
M row
1 5
2d m p l 32
3 g 1/2h 20
~ 22q̇ 1 1q̇ 2 1q̇ 3 ! .
~16!
The total viscous moment acting on the tall stereocilium
is found by superposition of the resultant moments in Eqs.
~15! and ~16!. Resultant moments on other stereocilia are
found in a similar fashion.
where p 12 and p 23 are, respectively, the pressures between
the tall and middle stereocilia, and the middle and short stereocilia, relative to ambient pressure external to the stereocilia bundle.
Combining Eqs. ~9a!, ~10!, and ~11! and solving simultaneously, the relative pressures between the stereocilia are
given by
As shown in the Appendix, for physiological deflections
and auditory frequencies, the hydrodynamic mass of the fluid
is small and is therefore not included in the equations of
motion. The mass of the stereocilia themselves may be included by considering a diagonal matrix composed of the
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D. E. Zetes and C. R. Steele: Stereocilia frequency response
J. Acoust. Soc. Am., Vol. 101, No. 6, June 1997
3. Inertia contributions
3596
scalar mass moments of inertia of the tall, middle, and short
stereocilium about their attachments to the cuticular plate.
However, assuming the density of the stereocilia is of the
same order of magnitude as water, the inertia terms of the
stereocilia are also small in comparison to other coefficients
in the equations of motion. They are therefore not described
in detail here, since they do not contribute greatly for physiological frequencies of excitation.
C. Generalization for n stereocilia in a column
The matrix formulation of the equations of motion ~1!
can be generalized for stereocilia bundles containing more
than three rows ~i.e., in the bullfrog saccule; Hudspeth, 1989!
by considering a column of n stereocilia with coefficient
matrices of dimensions n3n. Referring to Eqs. ~1! and ~6!,
the nonzero elements of the upper half of the symmetric
stiffness matrix K, are given by
(
k tip
l
lt 1 b ii11 r ! 1k i ,
~a
l 0 ii11 ii11 i
i5 j51,
ktip
h
a
l 1l ii11 ~ a ii11 lt i 1 b ii11 r ! 1k i ,
l 0 ii11 ii11 i
1,i,n, j5i, ~17!
Ki j5
2
k tip
l
a
,
l 0 ii11 ii11
i,n, j5i11,
ktip
h
a
1k ,
l 0 ii11 ii11 i
l 32 ~ 114 g h 20 !
l 2 d 2 4dl 32
2
1
, i5 j51,
2
4
3h 0
2gh0
l3i11~114gh20! li11d2 2dl3i11
1
2
1
,
4
3h0
2gh20
i,n21,
3597
k rot5
j5i11,
, i,n, j5i12,
3h0
~l3i111l3i !~114gh20! ~li111li!d2 2d~l3i111l3i !
2
1
,
4
3h0
2gh20
1,i,n, j5i,
l3i ~114gh20! lid2 2dl3i
1
1
, i5n21, j5n,
2
4
3h0
2gh20
l 3n ~ 114 g h 20 ! l n d 2 4dl 3n
2
1
,
i5 j5n.
4
3h 0
2 g h 20
~18!
J. Acoust. Soc. Am., Vol. 101, No. 6, June 1997
3E ciliap r 4maxr 3min~ lm1r min!
4l ~ r 4max1lmr 3min1r 4min!
,
~19!
where k rot is defined from moment equilibrium such that
Fl5k rotq, where l is the length of the stereocilium, and q is
the angle of rotation. Also in Eq. ~19!, r max is the radius of
the stereocilia in the cylindrical portion, r min is the radius of
the stereocilia at its insertion into the hair cell apex, l taper is
the length over which the change in radius occurs,
m5(r max2rmin)/ltaper , and E cilia is the Young’s modulus of
the stereocilia. The apparent rotational spring constant for
each stereocilium k i (i51,2,3,...,n) is then given by Eq.
~19! with l replaced by lengths l i (i51,2,3,...,n), respectively.
The linear stiffness of the tip links k tip can be written in
terms of material properties of the linkage constituent for a
uniaxial elastic bar
k tip5
2dl3i11
mp
C i j 5 1/2
g h0
The stiffness constants appearing in Eqs. ~5!, ~6!, and
~17! must be written in terms of material properties to accommodate stereocilia bundles of arbitrary dimensions. The
apparent rotational spring constants k i (i51,2,3,...,n) can
be written for a beam of varying cross section similar to that
of a stereocilium using Euler–Bernoulli beam theory. For the
case of an isotropic fixed-free beam with a circular crosssection, a conical taper at the fixed end, and a load F applied
at the tip, as illustrated in Fig. 3, the apparent rotational
spring constant is given by
i5 j5n,
where the tallest stereocilium is denoted by i51 and the
shortest stereocilium by i5n. The lengths of the stereocilia
are denoted by l i , the distance from the insertion of the ith
stereocilia to the apical surface of the hair cell to the upper
attachment of the tip link to the stereocilia shaft is denoted
by lt i , and geometrical parameters a i j , b i j , l i j , h i j , and
q i j are found by analogy with those in Sec. B 1.
Similarly, referring to Eqs. ~1!, ~15!, and ~16!, the elements of the upper half of the symmetric damping matrix C,
are given by
(
1. Generalization of stiffness constants
E tipp r 2tip
l0
,
~20!
where r tip is the radius of the tip link, E tip is the Young’s
modulus of the tip link, and l 0 is the natural length of the
unstretched tip link.
D. Solution for frequency response
The frequency response of the stereocilia can be considered by a separation of variables solution, where the time
dependence is oscillatory. The vector describing the angular
deflections of the stereocilia is given by
q5Im@ q̃e i v t # ,
~21!
where v is the frequency of the input force, q̃ is a complex
coefficient, and the imaginary part is taken to correspond to
the phase of the input force as assumed in Eq. ~3!.
Substituting Eq. ~21! into the linear equation of motion
~1! gives
@ 2Mv 2 1iCv 1K# q̃52iF̃,
~22!
where F̃ is defined similarly to q̃ in Eq. ~21!, as the imaginary form of the forcing vector given in Eq. ~4!. For a column of n stereocilia, vectors q̃ and F̃ have dimensions
n31, and the coefficient matrix has dimensions n3n.
Quantitative results for the amplitude and phase motion
of each stereocilium as a function of the frequency of the
input force are found from solution of Eq. ~22!. For interpretation of results in relation to mechanotransduction, a specific gating mechanism regarding tension in the tip links for
D. E. Zetes and C. R. Steele: Stereocilia frequency response
3597
FIG. 3. A clamped-free beam with varying circular cross section similar to
that of a stereocilium. A load F is applied at the tip of the stereocilium for
derivation of an equivalent system from Euler–Bernoulli beam theory of a
rigid stereocilium shaft with an apparent rotational spring constant k rot at its
base. The parameter l denotes the length of the stereocilia, r max the radius of
the stereocilia shaft, r min the radius of the stereocilia at insertion into the hair
cell’s apical surface, and l taper the length over which the change in diameter
occurs.
operation of ion channels is considered ~Pickles, 1988, for
review!. Within this analysis, the tension in the tip links
connecting neighboring stereocilia is given by
Ti j5
k tip
@~ l j r2slt i ! q i 1l j ~ s2r ! q j # ,
l0
j5i11,
~23!
where the subscript i j denotes the sequential order of the
stereocilia described previously, i.e., T 12 is the tension in the
tip link connecting the first and second stereocilia.
II. RESULTS
A. Amplitude solutions
FIG. 4. ~a! Frequency response of tension in the tip links normalized by the
magnitude of the applied load at the tip of the tallest stereocilium. The
response of the first tip link is similar to that of a low-pass filter, with
amplitude attenuated for higher-frequency excitation. The tension in the
second tip link demonstrates a modest ‘‘resonance’’ before attenuation.
~——— first tip link, ---- second tip link!. ~b! Phase difference of angular
displacement of stereocilia relative to the phase of the applied load. All
stereocilia move in phase for quasi-static, low-frequency excitation. At
higher frequencies, the stereocilia begin to exhibit phase response, approaching the viscous dominated limit of all stereocilia moving one-quarter
cycle out of phase above the highest audible frequencies of the chinchilla.
~——— tall stereocilium, ---- middle stereocilium, -•- short stereocilium!.
In both figures, measured dimensions are for a stereocilia bundle from
OHC1 in the basal region of the chinchilla cochlea ~Lim, 1980!, viscosity
and density of water, and Young’s modulus of the stereocilia and tip links
E cilia5E tip523107 Pa.
The relationship between the tension in the tip links as a
function of the frequency of the input force is shown in
Fig. 4~a!, where the tension in the tip links has been normalized by the magnitude of the applied load at the tip of the
tallest stereocilium. To generate Fig. 4~a! dimensions for
OHC1 from the basal region of the chinchilla cochlea were
used ~Lim 1980!: l 1 51.6 m m, l 2 51.3 m m, l 3 50.9 m m, d
5200 nm, s5400 nm, lt i 5l i 1d (i51,2), r min5d/4, l taperi
5l i /4 (i51,2,3), and r tip53 nm. The Young’s modulus of
both the tip links and stereocilia were E tip5E cilia
5107 N/m2, and the viscosity and density of water were
used. As shown in Fig. 4~a! when a low-frequency force is
applied to the tallest stereocilium, the normalized amplitude
of the tension in the tip links is relatively large and constant.
At higher frequencies of the input force the tension in the
first tip link is attenuated in a fashion similar to a low-pass
filter, while the tension in the second tip link demonstrates a
modest ‘‘resonance’’ before attenuation. At very high frequencies, the viscous forces become so large that motion of
the stereocilia bundle is prevented and a relatively small tension exists in the tip links. This behavior is consistently demonstrated in the first and second tip link for all stereocilia
bundle geometries from the chinchilla cochlea for both inner
and outer hair cells as reported in Lim ~1980!.
The phase difference between the angular position of
each stereocilium and the input force can also be determined
from solution of Eq. ~22!. As shown in Fig. 4~b!, for the
same geometrical and material properties used in generation
of Fig. 4~a!, the stereocilia demonstrate continuously varying
phase solutions. At low excitation frequencies, the stereocilia
move in phase with each other and the input force. At higher
frequencies of excitation, viscous effects contribute and the
stereocilia begin to separate, such that the short stereocilium
moves out of phase with the input force. At higher frequencies, the middle stereocilium also begins to move out of
phase, and finally the tall stereocilium moves out of phase
with the input force. Above the highest audible frequencies
of the chinchilla, all of the stereocilia move toward the viscous dominated limit of a quarter cycle out of phase with the
input force. As with the amplitude solutions described previously, similar behavior was observed for all stereocilia
bundle geometries reported in Lim ~1980!.
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J. Acoust. Soc. Am., Vol. 101, No. 6, June 1997
B. Phase solutions
3598
C. Dependence on stereocilia morphology
The mammalian cochlea is arranged tonotopically, with
neural sensitivity to frequency varying exponentially along
the cochlear duct. The morphology of the stereocilia bundle
is also arranged tonotopically, with shorter stereocilia
bundles in the higher frequency regions, and longer stereocilia bundles in the lower-frequency regions ~see, e.g., Lim,
1980, and Strelioff and Flock, 1984!. It is therefore interesting to investigate the relationship between stereocilia morphology and frequency response of the stereocilia bundle
while holding other parameters, including the loading case,
constant. In Fig. 5, the corner frequencies, defined as 3 decibels below the quasi-static solution from the current analysis,
are shown using stereocilia lengths reported in Lim ~1980!
for all three rows of outer hair cells and one row of inner hair
cells. These solutions are plotted versus the corresponding
frequency by converting the length of the tallest stereocilium
to the location along the cochlear duct ~Lim, 1980! and this
in turn to a frequency location using the frequency-place
map for the chinchilla from Greenwood ~1990!.
In Fig. 5, it can be seen that the corner frequency of
stereocilia motion is slightly above the Greenwood frequency location of the corresponding hair cell. These results
demonstrate that motion of the stereocilia bundle relative to
the apical surface of the hair cell exists without substantial
attenuation at least up to the frequency appropriate for the
location of the hair cell along the cochlear duct. As may be
expected, for smaller diameter stereocilia with larger rootlet
separations than were assumed in generating Fig. 5, the viscous forces on the stereocilia are decreased, and the corner
frequencies correspondingly increased. Similarly, for larger
diameter stereocilia with closer rootlet separations, the corner frequencies are decreased.
FIG. 5. Corner frequency solutions of tension in the first tip link versus
corresponding Greenwood frequency ~1990! for stereocilia bundles from the
chinchilla. Stereocilia dimensional and cochlear position data from Lim
~1980!. Corner frequencies are defined as 3 dB below the quasi-static solution, and are determined analytically for the frequency response of the first
tip link between stereocilia pairs for all three rows of outer hair cells and
one row of inner hair cells. Solutions indicate that motion of the stereocilia
relative to the apical surface of the hair cell bundle exists without substantial
attenuation at least up to frequencies appropriate for the location of the
corresponding hair cell along the cochlear duct. The variation in stereocilia
morphology along the cochlear duct thus appears to provide a collection of
low-pass mechanoreceptors, arranged in order of increasing corner frequency across the auditory spectrum. Material properties and assumed dimensions are the same as for Fig. 4 ~15OHC1, * 5OHC2, s5OHC3,
35IHC!.
III. DISCUSSION
A. Experimental correlation
The results discussed above are from a Young’s modulus of both the stereocilia and the tip links of E cilia5E tip
5107 N/m2. Although this assumption is reasonable for biological materials ~Wainwright et al., 1976!, these values are
only estimates in the absence of experimental values. For
purpose of comparison, these assumed material properties
provide reasonable agreement with order of magnitude stiffness measurements reported in the literature ~Table III,
Szymko et al., 1992, for review!. In this model it can be seen
that an increased material modulus of the tip links
E tip.E cilia , causes both the corner frequency and the maximum frequency over which all of the stereocilia are in phase
with the input force to increase. For a decreased material
modulus E tip,E cilia , the range over which corner frequency
solutions occur for varying bundle geometries is diminished.
It is also interesting to note that viscous coupling provided
between the stereocilia is sufficient such that amplitude and
phase motion similar to that shown in Fig. 4 are provided
when all linkages are absent, although the corner frequency
is lowered and phase response begins earlier.
Quantitative results shown in Fig. 5 imply a relationship
of corner frequency with Greenwood frequency for dimensions of all stereocilia bundles from the chinchilla cochlea
reported in Lim ~1980!. A similar relationship was also reported for dimensions of stereocilia bundles from the guinea
pig cochlea ~Zetes and Steele, 1996!. Physiological measurements have further shown that the height of the stereocilia
bundle in the bullfrog saccule is a major determinant of the
stimulus frequency at which the hair cell is most sensitive
~Hudspeth, 1989!. Finally, quasi-static solutions from this
analysis are in agreement with experimental measurements
by Duncan et al. ~1995!, for stereocilia bundles from the
chick cochlea. Although Duncan et al. did not observe phase
differences ~or ‘‘splaying’’! between the tallest and shortest
stereocilia, their observations do not contradict the phase solutions reported here, since transition to out-of-phase motion
occurs after viscous attenuation. Phase differences of the stereocilia may have occurred at amplitudes below the visual
threshold for detection of motion ~0.11 mm, Pae and Saunders, 1994!.
It should be noted in interpretation of Fig. 5, that the
solutions range from 0.1 to 5 kHz, while the auditory range
of the chinchilla is approximately 0.1–30 kHz ~Greenwood,
1990!. Incomplete geometrical data for the chinchilla stereocilia and absence of experimental values for the frequencyplace map are likely to have caused this discrepancy in
range. The data reported in Lim ~1980! are for stereocilia
lengths only from the most apical 60% of the cochlea, such
that dimensions from the highest-frequency regions are not
reported. Also, in absence of reported values, it has been
necessary to assume values for the taper of the base of the
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D. Dependence on material properties
J. Acoust. Soc. Am., Vol. 101, No. 6, June 1997
3599
stereocilia, diameter, and rootlet separations which are similar to those of the guinea pig ~Zetes, 1995!. In addition, the
frequency map reported in Greenwood ~1990! is based on
data from only the basal 60% of the cochlea, such that it may
be inaccurate below 0.5 kHz.
In light of these difficulties, inspection of the available
solutions in Fig. 5 indicates that the dimensions of the stereocilia bundle and corresponding filtering properties are
consistent with the tonotopic arrangement of the cochlea.
Motion of the stereocilia bundle relative to the apical surface
of the hair cell exists without substantial attenuation at least
up to frequencies corresponding to the location of the hair
cell along the cochlear duct. The variation in stereocilia morphology along the cochlear duct may thus provide a collection of low-pass mechanoreceptors, arranged in order of increasing corner frequency across the auditory spectrum.
B. Analytical approximations and limitations
The equations derived above are for the linear motion of
the stereocilia bundle. The range of validity of such a model
is dependent on the assumptions of viscous dominated laminar flow and linear elastic stability of the bundle constituents. The hydrodynamic approximations discussed in the Appendix are valid for displacements of the tips of the
stereocilia up to 1° ~approximately 20 nm!, at auditory frequencies of the chinchilla. Perhaps more limiting is the possibility of nonlinear contributions to the equations of motion
from buckling of the tip links. Assad et al. ~1991! and Assad
and Corey ~1992! demonstrate that there is a resting value of
tension in the tip links. This resting tension provides a linear
contribution of the tip links to the equations of motion for
stereocilia deflections in both the tensile and compressive
directions. Assuming that no other nonlinearity is apparent
from ion channels resident in the stereocilia ~Hudspeth,
1989!, these equations of motion are valid for sinusoidal oscillation until the amplitude becomes sufficiently large such
that the smallest net force during the cycle is compressive.
This model is therefore reasonable for the linear response of
the stereocilia bundle, at lower sound-pressure levels.
Solutions from this analysis have primarily addressed
the issue of morphological variation as it affects stereocilia
motion. For consistent comparison, the loading case and
other parameters regarding the stiffness of the stereocilia and
linkages have been held constant and expressed in their simplest form. In particular, the choice of an applied load at the
tip of the tallest stereocilium is most appropriate for outer
hair cell stereocilia since inner hair cell stereocilia are not
attached to the tectorial membrane ~Pickles, 1988, for review!. The apparent rotational stiffness of the stereocilia has
also been written in accordance with Euler–Bernoulli beam
theory. To account for shearing deformation as observed in
preparations for electron microscopy ~Tilney et al., 1983!,
similar equations can be written using Timoshenko theory
~Peterson et al., 1996; Zetes, 1995!. When the material properties of the bundle constituents have been quantified experimentally, an analysis of such rigor coupled with the hydrodynamic solutions described above and alternate loading
cases will provide a more accurate prediction of the stereocilia bundle frequency response. Further morphological stud3600
J. Acoust. Soc. Am., Vol. 101, No. 6, June 1997
ies will also allow a full-scale numerical study to provide
understanding of the variation in stereocilia geometry along
the cochlear duct in relation to Greenwood’s frequency and
hypotheses regarding mechanotransduction.
IV. SUMMARY
This paper introduces an analytical model to mathematically simulate the fluid–structure interaction of the stereocilia bundle for physiologically reasonable deflections and
frequencies of excitation. Results from this model were used
to investigate the frequency response of stereocilia bundles
of the chinchilla cochlea, as quantified by tension in the tip
links for the loading case of a force applied to the tip of the
tallest stereocilium. It was found that the influence of the
surrounding viscous endolymphatic fluid in proportion to the
angular velocity of the stereocilia allows the amplitude of the
tension in the tip links to remain constant for low frequencies
of excitation. For higher excitation frequencies, viscous
forces limit the motion of the stereocilia, thereby reducing
the tension in the tip links in a fashion analogous to that of a
low pass filter. Comparison of solutions for differing stereocilia geometries along the cochlear duct demonstrates that
motion of the stereocilia exists without substantial attenuation at least up to frequencies appropriate for the location of
the corresponding hair cell along the cochlear duct. This behavior is consistent with the cochlear tonotopic map. The
variation in stereocilia morphology along the cochlear duct
thus appears to provide a collection of low-pass mechanoreceptors, arranged in order of increasing corner frequency
across the auditory spectrum. Further consideration with this
model and similar analyses may provide insight into the
functional implication of stereocilia arrangements and more
specific comparison of hypotheses regarding mechanotransduction.
ACKNOWLEDGMENTS
This work was supported by NIH Grant No. R01
DC00108 to Charles R. Steele.
APPENDIX
Consideration of the nondimensional form of the governing incompressible fluid equations demonstrates that the
dominant hydrodynamic forces between the stereocilia are
linear and viscous. Defining nondimensional independent
variables for space x 8 , and time t 1 , and nondimensional dependent variables for speed u 8 , pressure p 1 , at a characteristic point in time, incompressible Navier–Stokes can be
written
2E“p 8 1
] u8
1
1
Du8 2 “h 8 5S
1u8 –“u8 ,
Re
F
]t8
~A1!
where E is the Euler number, Re is the Reynold’s number,
F is the Froude number, S is the Strouhal number, and the
body force has been written as a gravitational force
2g“h, where h is the nondimensional elevation and g is
the gravitational constant. These coefficients are given by
D. E. Zetes and C. R. Steele: Stereocilia frequency response
3600
m
1
5
,
Re r 0 UL
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D. E. Zetes and C. R. Steele: Stereocilia frequency response
E5
P0
,
r 0U 2
1 gL
5
,
F U2
S5
L
, ~A2!
TU
where T, L, U, P 0 , and r 0 are characteristic values for the
time, length, velocity, pressure, and density, respectively,
and m and g are physical values for viscosity and gravity.
For the case of the stereocilia bundle of the chinchilla,
T51025 s,
characteristic values are L51027 m,
23
23
2
P 0 510 N/m ,
r 0 5103 kg/m3,
U510 m/s,
23
2
m 510 kg/m s, and g510 kg m/s , where L is the separation between stereocilia shafts, T is the time for a stereocilium to reach its maximum velocity at the highest audible
frequency ~30 kHz, Greenwood, 1990!, U is the maximum
linear velocity of the tip of the stereocilia ~approximately
1026 m tall; Lim, 1980! for a deflection of 1° at the highest
audible frequency, P is the pressure as estimated from Bernoulli’s equation, and m and r 0 are approximations of the
material constants for the endolymphatic fluid from values
for water.
Evaluating the coefficients gives E51, 1/Re5104 ,
1/F51, and S510. It can thus be seen that the viscous
contributions as quantified by the inverse of Reynold’s number are three orders of magnitude larger than the inertia contributions as quantified by the Strouhal number at auditory
frequencies. The nondimensional density and Froude number
also demonstrate that the contribution of the nonlinear convection terms and gravitational body force are small. Hydrodynamic contributions for motion of the stereocilia within
the endolymphatic fluid are therefore reasonably approximated by consideration of only linear viscous and pressure
forces in the governing incompressible fluid equations.
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3601