Department of Physiology, University of Wisconsin Medical School,
Madison, Wisconsin 53706
 |
INTRODUCTION |
Hair cells of the internal ear detect mechanical
stimuli by gating of mechanosensitive ion channels located in their
stereociliary bundles. The common view of transduction is that force is
delivered to the mechanically sensitive channels by extracellular tip
links connecting the top of one stereocilium with the side wall of its taller neighbor (Pickles et al. 1984
). Deflection of the
bundle toward its taller edge transmits force via the tip links to open transducer channels attached at either end of the link (Denk et al. 1995
). Hair cells, like other sensory receptors, possess an adaptation mechanism to reduce their sensitivity in the face of a
sustained stimulus (Crawford et al. 1989
; Eatock
et al. 1987
). Adaptation shifts the transducer activation
curve, changing the range of displacements to which the channel is
sensitive without diminishing the maximum response.
Transducer adaptation is regulated by changes in stereociliary
Ca2+ concentration that reset the range of bundle
displacements detected by the channel (Assad et al.
1989
; Crawford et al. 1989
; Ricci and
Fettiplace 1997
, 1998
). One proposed mechanism for resetting sensitivity entails a force generator that adjusts the tension in the
tip link by translating the tip link's attachment point along the side
of the stereocilium (Howard and Hudspeth 1987
). The
force generator may be myosin I
linking the transducer channel with
the internal actin cytoskeleton (Hudspeth and Gillespie
1994
). Ca2+ influx through open
transducer channels is posited to inhibit the actomyosin interaction,
causing the channel to slip down the stereocilium and relieve the
stimulus to the channel. A difficulty with this mechanism is that
adaptation can occur on a submillisecond time scale (Ricci and
Fettiplace 1997
), too fast for the kinetics of the full
actomyosin cycle. It is conceivable that fast adaptation relies on
another mechanism with kinetics swifter than achievable with actomyosin interactions.
To assess this hypothesis, we have characterized the time course of
transducer adaptation in turtle auditory hair cells to look for fast
and slow components identifiable with different mechanisms. To support
experimental observations, we devised a model for adaptation in which
Ca2+ entering through open transducer channels
binds at two intracellular sites to trigger separate processes leading
to channel closure. The model, incorporating diffusion of
Ca2+ within the stereocilium in the presence of
intracellular Ca2+ buffers, employs computational
techniques introduced in a previous model of hair-cell calcium dynamics
(Wu et al. 1996
). Our model differs from previous
theoretical schemes (Assad and Corey 1992
; Lumpkin and Hudspeth 1998
) in providing an explicit
formulation of the role of stereociliary
Ca2+ in transducer channel
regulation. It takes for its background prior measurements of the
channel's Ca2+ permeability (Ricci and
Fettiplace 1998
), and experimental data on the effects of
extracellular Ca2+ and intracellular calcium
buffers on adaptation in turtle hair cells (Ricci and Fettiplace
1997
, 1998
; Ricci et al. 1998
). Both experimental and theoretical manipulations provide further information about the properties of fast adaptation.
 |
METHODS |
The preparation and techniques for hair-cell recording and
stimulation in the intact basilar papilla were similar to those previously documented (Crawford and Fettiplace 1985
;
Ricci and Fettiplace 1997
). Turtles (Trachemys
scripta elegans, carapace length 100-125 mm) were decapitated,
and the cochlear duct was dissected out and opened. After digestion in
saline [composed of (in mM) 125 NaCl, 4 KCl, 2.8 CaCl2, 2.2 MgCl2, 2 Na
pyruvate, 8 glucose, and 10 NaHEPES, pH 7.6] containing up to 0.1 mg/ml of protease (Sigma type XXIV), the hair bundles were exposed by removal of the tectorial membrane. The preparation was mounted, hair
bundles uppermost, in a silicone elastomer (Sylgard) well of a
recording chamber mounted on the stage of a Zeiss Axioskop FS
microscope. The preparation was perfused with saline containing (in mM)
128 NaCl, 0.5 KCl, 2.8 CaCl2, 2.2 MgCl2, 2 Na pyruvate, 8 glucose, and 10 NaHEPES,
pH 7.6. The upper surface of the hair-cell epithelium facing the
endolymphatic compartment was separately and continuously perfused by a
large pipette with an internal diameter of 100 µm introduced into the
cochlear duct. Hair bundles were stimulated with a rigid glass pipette,
fire-polished to a tip diameter of ~1 µm and cemented to a
piezo-electric bimorph (Crawford et al. 1989
). The
bimorph was driven differentially with voltage steps, filtered with an
eight-pole Bessel at 3 kHz and amplified through a high-voltage driver
of 20-fold gain, to yield a fast stimulator with a 10-90% rise time
of ~100 µs.
Whole cell currents were measured with a List EPC-7 amplifier attached
to a borosilicate patch electrode. Patch electrodes were filled with an
internal solution of composition (in mM) 125 CsCl, 3 Na2ATP, 2 MgCl2, and 10 CsHEPES, pH 7.2 to which various amounts of the calcium buffers
bis-(o-aminophenoxy)-N,N,N',N'-tetraacetic acid
(BAPTA; Molecular Probes, Eugene, OR) or EGTA (Fluka, NY) were added.
Buffer concentrations of 0.1, 1, and 10 mM were used, and with the
highest concentration, the CsCl was reduced to keep the osmolarity
constant. After application of
50% series-resistance compensation,
the electrode access resistance was 3-10 M
, which gave a recording
time constant of 45-150 µs. Transducer currents were measured at a
holding potential that, after correction for the junction potential,
was
90 mV. To inhibit intracellular myosin ATPases, sodium
metavanadate (Aldrich Chemical Company, Milwaukee, WI) was added to the
patch electrode solution and butanedione monoxime (Sigma Chemical, St.
Louis, MO) was dissolved in the extracellular solution.
 |
RESULTS |
Fast and slow components of adaptation
Transduction in auditory hair cells, evoked by step deflections of
the hair bundle, is characterized by rapid opening of the mechanically
gated channels closely followed by an adaptation that, despite
maintenance of the stimulus, causes the channels to shut again. Figure
1A shows a family of
transducer currents measured experimentally in response to a range of
bundle displacements. Over the entire dynamic range, the adaptive
decline in the transducer current could be well described by two
exponential components, one with a time constant of ~1 ms and the
other an order of magnitude slower (Fig. 1B). The fast time
constant,
fast, dominated for small
displacements, but the slower time constant,
slow, became more conspicuous with increasing
stimulus amplitude. The double-exponential fits in Fig. 1A
were derived by determining the value of the
fast (= 0.7 ms) from small responses and then
holding this value fixed while allowing the contribution of
slow to vary for larger stimulus amplitudes
(Fig. 1). These fits showed that over much of the dynamic range,
slow had a constant value of 11 ms, but for
the largest displacements, it increased to 70 ms.

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Fig. 1.
Two components of adaptation. A: transducer currents
recorded in a high-frequency hair cell for step deflections of the hair
bundle. Inward currents are produced by positive displacements toward
the kinocilium. Whole cell recordings made with 2.8 mM external
Ca2+, 0.1 mM intracellular calcium buffer
bis-(o-aminophenoxy)-N,N,N',N'-tetraacetic
acid (BAPTA), at a holding potential of 90 mV. Each trace is the
average of 5-25 responses. The decays in current for positive stimuli
have been fitted with double exponentials, giving values for the fast
and slow time constants ( f and s,
respectively) and their respective amplitudes
(Af and As).
B: fast and slow time constants and the relative
amplitude of the slow component obtained from the fits in
A are plotted against bundle displacement. Note that
both the magnitude and relative proportion of the slow time constant
increase with displacement amplitude.
|
|
For stimuli that elicit less than half-maximal responses, adaptation is
dominated by
fast, which we have used
previously to assay the calcium sensitivity of the underlying process.
Over a range of ionic conditions, the rate of adaptation
(1/
fast) was proportional to
Ca2+ influx (Ricci and Fettiplace
1998
) and inversely proportional to the concentration of
intracellular calcium buffer, BAPTA (Ricci and Fettiplace
1997
). Application of two-exponential fits to transducer currents recorded with different concentrations of BAPTA showed that
the buffer concentration had parallel effects on the limiting values of
both
fast and
slow.
Mean values for
fast were 0.74 ± 0.14 (SD) ms (n = 7) in 0.1 mM BAPTA, 1.32 ± 0.11 ms (n = 9) in 1 mM BAPTA, and 1.68 ± 0.13 ms (n = 9) in 10 mM BAPTA. The corresponding
values for
slow in 0.1, 1, and 10 mM BAPTA respectively were 9.3 ± 1.4 ms (n = 7), 14.9 ± 2.6 ms (n = 9), and 19.5 ± 4.4 ms
(n = 9). All these measurements were obtained with
2.8 mM external Ca2+. Thus increasing the BAPTA
concentration from 0.1 to 10 mM roughly doubled the values of both fast
and slow time constants. The buffer effects may be due to a reduction
in amplitude and a slowing of the intracellular Ca2+
transient after opening of the transducer channel. The buffer results
indicate that the mechanism underlying
slow also must be
Ca2+ dependent.
Other features of the transducer responses, linked to adaptation, also
depend on the stereociliary Ca2+ dynamics (Ricci and
Fettiplace 1997
). Increasing the concentration of intracellular
calcium buffer diminished the extent of adaptation, defined as the
reduction in current in the steady state relative to the initial peak
(Fig. 2). Thus in 0.1 mM BAPTA, there was no steady-state response for small stimuli; this is equivalent to 100%
adaptation. However, with 10 mM BAPTA, the extent of adaptation never
exceeded 50%. The fraction of transducer current turned on at the hair
bundle's resting position also varied with the concentration of
intracellular calcium buffer (Fig. 2). This difference reflects a
translation of the transducer's activation relationship along the
displacement axis (Fig. 2). Previous experiments have indicated that
two manifestations of adaptation, the fast adaptation time constant and
the fraction of current activated at rest, are differentially sensitive
to various experimental manipulations. These include changing the
nature of the intracellular calcium buffer (Ricci et al.
1998
) or treatment with cyclic AMP (Ricci and Fettiplace
1997
). Such observations support the notion that different
aspects of adaptation may be associated with distinct Ca2+-binding sites.

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Fig. 2.
Effects of intracellular BAPTA concentration on the mechanotransducer
current. Families of currents in 3 hair cells recorded with internal
solutions containing 0.1, 1, or 10 mM BAPTA; 1 mM external
Ca2+, holding potential 90 mV. Ordinates denote
transducer current I, scaled to its maximum values,
Imax, for a large positive displacement.
Each trace is the averages of 5-25 responses. For each cell the peak
current, I, scaled to its maximum is plotted against
bundle displacement (bottom right). BAPTA concentrations
and values of Imax, were 0.1 mM, 1.14 nA;
1.0 mM, 1.06 nA; and 10 mM, 0.45 nA. Increasing internal BAPTA
concentration shifts the relationship to the left. Smooth curves are
double Boltzmanns relating the probability of opening,
P, and displacement, x:
P = {[1 + exp
(axO ax)] · [1 + exp (bxO bx)]} 1 where values of a,
b, and xo are: 0.1 mM BAPTA, 6.5 µm 1, 14 µm 1, 0.15 µm; 1 mM BAPTA, 7.5 µm 1, 14 µm 1, 0.09 µm; and 10 mM
BAPTA, 7.5 µm 1, 14 µm 1, 0.03 µm.
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Effects of myosin ATPase inhibitors on fast adaptation
Previous experimental analysis of adaptation in turtle hair cells
has focused on the fast component that dominates the responses (Ricci and Fettiplace 1997
). The prevailing theory for
the mechanism of adaptation involves operation of a myosin ATPase motor
that adjusts the force delivered by the tip links to the transducer channel (reviewed in Hudspeth and Gillespie 1994
). In
support of this mechanism in frog saccular hair cells, agents that
block the ATPase also inhibit adaptation (Yamoah and Gillespie
1996
). We examined the effects on adaptation of two potential
inhibitors of the myosin ATPase: the phosphate analogue, vanadate, and
the membrane-permeable butanedione monoxime (BDM), an inhibitor of myosin II and myosin V ATPases (Cramer and Mitchison
1995
). Vanadate (1 mM), introduced via the patch electrode
solution, or 10 mM BDM perfused extracellularly had similar effects on
the transducer currents (Fig. 3). Both
agents shifted the current-displacement relationship to the right and
decreased its slope. The positive shifts in the current-displacement
relationship were 203 ± 32 nm (n = 5, vanadate)
and 154 ± 52 nm (n = 3, BDM). Similar shifts produced by other ATPase inhibitors have been previously reported (Yamoah and Gillespie 1996
). However, neither agent
significantly diminished the fast component of adaptation (Fig. 3). The
fast time constant,
fast, had mean values of
1.54 ± 0.12 ms (n = 4, control), 1.45 ± 0.14 ms (n = 3, BDM) and 1.65 ± 0.14 ms
(n = 5, vanadate), all with 1 mM internal BAPTA. The
time constant of the slow component,
slow,
measured in the same cells was 7.5 ± 2.0 ms (control), 12.8 ± 1.6 ms (BDM), and 9.5 ± 1.3 ms (vanadate). Because vanadate
was delivered via the patch electrode solution, it was not possible to
obtain a good control in the same cell due to the "wash-in" of
vanadate occurring over a similar time course to that of BAPTA. The
controls therefore represent measurement on other cells. It should be
noted that the effects of the ATPase inhibitors resemble qualitatively
those produced by application of 8-bromo cyclic AMP (Ricci and
Fettiplace 1997
).

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Fig. 3.
Effects of myosin ATPase inhibitors on fast adaptation.
A: before (control) and 10 min after extracellular
perfusion of 10 mM butanedione monoxime (BDM). Displacement amplitudes
and adaptation time constants, fast, were 140 nm and 0.8 ms (control) and 220 nm and 0.76 ms (BDM). B:
current-displacement relationships before and after perfusing BDM.
Holding potential, 80 mV. C: family of transducer
currents recorded 15 min after attaining whole cell configuration with
the patch electrode solution containing 1 mM sodium vanadate.
D: current-displacement relationships for a control cell
and 1 recorded with 1 mM vanadate in internal solution. Fits in
B and D are single Boltzmanns.
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|
Interpretation of the effects of ATPase inhibitors is complicated by
the fact that they also block the CaATPase responsible for
Ca2+ extrusion from hair cells (Tucker and
Fettiplace 1995
). Consistent with those results, both vanadate
and BDM produced prolonged tail currents at the offset of depolarizing
current steps due to sustained activation of the small-conductance
Ca2+-activated K+ (SK)
channels (see Fig. 6 of Tucker and Fettiplace 1995
).
Effects on the transducer current-displacement relationship therefore could be a combination of an elevation of stereociliary
Ca2+ concentration and block of the slow
component of adaptation. However, the lack of any significant effect on
fast argues that fast adaptation is unlikely
to be mediated by a myosin ATPase.
The fast process of adaptation showed linear behavior for small
displacement steps about the resting position of the bundle (Fig.
4). The linearity was most evident under
conditions where the resting probability had been raised by lowering
the concentration of external Ca2+ or by
increasing the amount of intracellular calcium buffer. Thus in Fig. 4
recorded in 0.35 mM Ca2+, the responses for small
positive and negative steps are mirror images of one another, and
fast has a similar value for stimuli in either
direction. However, the same linearity held under other conditions
provided the amplitude of the negative stimulus was sufficiently small
not to turn off the transducer current during the initial peak of the
response. Collected measurements of
fast for
small positive and negative stimuli, obtained under a range of
conditions, are plotted in Fig. 4B, which shows a good
correlation between the adaptation time constants measured with the two
stimulus polarities. This linearity implies that the reaction involved in generating the fast component of adaptation is a reversible one and
contrasts with the behavior expected for a myosin-based motor in which
the adaptation rate for positive stimuli is faster than for negative
stimuli (Assad and Corey 1992
). The fast positive rate
was attributed to "slipping" of myosin's attachment to the actin
cytoskeleton, whereas the slower negative rate was limited by myosin
ascending on the actin core of the stereocilium.

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Fig. 4.
Linearity of fast adaptation. Left: averaged transducer
currents recorded for small positive and negative displacement steps in
0.35 mM external Ca2+ and 3 mM intracellular BAPTA where a
greater fraction of transducer current is turned on at rest. Note the
symmetry of responses for small positive and negative stimuli. Maximum
current, 1 nA. Right: collected measurements of the
correlation between the fast adaptation time constants for small
positive ( pos) and negative ( neg)
stimuli; to obtain a range of time constants, measurements were made in
a various concentrations of intracellular BAPTA (0.1-10 mM) and
external Ca2+ (0.07-2.8 mM). Straight line has slope of
0.93 and correlation coefficient, r = 0.92.
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Outline of the two-site model
The central tenet of the model is that the transducer channels
respond to the difference (x
Xa) between an external stimulus, x, and an internal "set point"
Xa. As the channels open,
Ca2+ ions enter the stereocilium and bind to an
intracellular site triggering a change in the set point that opposes
the external stimulus. Ca2+ is thus part of a
negative feedback loop. The sequence of events ensuing from an increase
in the external stimulus x
x' can be summarized as follows: the channels open in response to the new stimulus (x'
Xa)
promoting Ca2+ influx and binding to an
intracellular site S; the proportion of S bound catalyzes a change in
the set point, (Xa
X'a), causing the channels to adjust
their probability of opening in response to the new stimulus
(x'
X'a). To implement the
Ca2+ feedback, a three-dimensional model of the
stereocilium was constructed to simulate the diffusion of free
Ca2+ ions and mobile Ca2+
buffers within the cytoplasm (see APPENDIX). Major
components of the model are as follows: each turtle hair-cell
stereocilium contains a small number of mechanoelectrical transducer
channels (Ricci and Fettiplace 1997
) represented as a
diffuse Ca2+ source, 10 nm radius, located at the
stereociliary tip (Jaramillo and Hudspeth 1992
).
Ca2+ influx was estimated from the transducer
current per stereocilium and the proportion of the current carried by
Ca2+ (Ricci and Fettiplace 1998
).
The time course of internal Ca2+ is determined by
diffusion and binding to calcium buffers and by extrusion via a plasma
membrane CaATPase known to occur in turtle hair cells (Tucker
and Fettiplace 1995
).
Ca2+ is assumed to interact with two classes of
intracellular binding site, S1 and S2, associated with the fast and
slow adaptation processes, respectively
|
(1)
|
where KDi is the
Ca2+ dissociation constant for the two sites, and
the subscript, i = 1 or 2, corresponds to the fast and
slow sites respectively. Both sites are diffusely distributed over
cylindrical regions of radius 1.5 nm and of longitudinal extent 20-50
nm from the channel for S1 and either 50-100 nm or 150-200 nm for S2.
S1 was positioned to encompass the "crossing points" of the
Ca2+ gradients in different BAPTA concentrations
(see Fig. 10 of Ricci et al. 1998
). Because S2 is
associated with the slower process, it initially was located further
from the channel than S1; the effects of varying the position of S2
will be described in the following text. Owing to the steep
Ca2+ gradient within the stereocilium dictated by
the concentration of diffusible buffer, S1 must have a
Ca2+ dissociation constant
(KD1 = 20 µM) higher than that of S2
(KD2 = 0.5 µM). Previous studies (Ricci
et al. 1998
) indicate that the Ca2+
concentration may decline from several hundred micromolar near the
channel to a few micromolar at a distance of 100 nm, so the dissociation constants were chosen appropriately for the location of
the two sites within the gradient. The dissociation constants are
within the range of values reported for calmodulin (Linse et al.
1991
).
The fraction, fSCa, of each
calcium-binding site occupied catalyzes a change in
Xa that takes place in two stages.
First, a conformational transition is assumed to occur in a modulator molecule converting it from an inactive form M to an active
form M*
|
(2)
|
Each binding site has its own modulator
(M1 and
M2) with distinct kinetics, the rate
constants for the second site being 10 times slower than those for the
first site.
Second, Xa is scaled linearly
according to the concentration of the active form of each type of
modulator
|
(3)
|
in which
i and
i are constants and the modulator
concentration, M*i,
is integrated over the regions specified for each binding site. The
effects of the modulators are assumed to sum independently to control
the set point Xa. Because the
modulator concentration, Mi,
takes values between 0 and 1, the constants
i
and
i determine the dynamic range of the
feedback. A restricted dynamic range is consistent with the limited
extent of adaptation reported by Shepherd and Corey
(1994)
.
Model transducer responses
The theoretical responses for three different intracellular BAPTA
concentrations are given in Fig. 5. The
simulations expressed as the probability of opening of the transducer
channel have been inverted for easier comparison with the inward
currents recorded experimentally. Comparison of the model with the
experimental records in Fig. 2 shows a number of similarities in terms
of the overall shape of the response and their sensitivity to BAPTA. Thus the extent of adaptation was comparable in the different conditions and was reduced with an increase in BAPTA concentration. The
steady-state responses for small displacements in 0.1 mM BAPTA are all
more closely grouped compared with 10 mM BAPTA, reflecting nearly a
100% adaptation in the low buffer concentration. Only a single
external Ca2+ concentration (1 mM) is illustrated
in Fig. 5, but model responses at other Ca2+
concentrations from 0.07 to 2.8 mM showed comparable agreement with the
experimental transducer currents.

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Fig. 5.
Effects of intracellular BAPTA concentration on the model transducer
responses. Theoretical families of transducer channel open
probabilities in response to 50-ms stimulus steps, for 3 BAPTA
concentrations in 1 mM external Ca2+. To compare with
experimental responses (Fig. 2), increasing probability of opening is
plotted downward. Bundle displacements (in µm) are: 0.5, 0.2,
0.05, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6 0.8, 1.0, and 1.2. Transducer activation curves for the 3 BAPTA concentrations are given
in the bottom right, the peak probability of opening for
each trace being plotted against bundle displacement. More points are
plotted than shown in the theoretical responses. Increasing the
intracellular BAPTA concentration shifts the relationship to the left
as with the experimental results in Fig. 2.
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As expected, the model responses exhibited two components of adaptive
decay, one component,
fast, with time constant
of 1 ms, and a second component,
slow, of 14 ms. Fitting of the decays with double exponentials indicated that the
slow component became more pronounced with an increase in stimulus
amplitude. Both
fast and
slow increased with intracellular BAPTA
concentration in a similar manner to the experimental measurements. The
mean values of
fast are plotted in Fig.
6, and the values for
slow were 11, 14, and 70 ms in 0.1, 1, and 10 mM BAPTA, respectively. It should be noted that the magnitude of
slow in the model responses remained
approximately constant over the dynamic range in contrast to the
experimental results where
slow increased at
large displacements. This discrepancy indicates a nonlinearity in the
slow process. The fraction of current activated at rest in the model,
as in the experiments, also increased with buffer concentration due to
a shift in the current-displacement relationship.

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Fig. 6.
Effects of external Ca2+ concentration on the parameters of
adaptation for the theoretical responses. A: response
onsets for 1 mM intracellular BAPTA and 0.07 and 2.8 mM external
Ca2+ concentration; B: response onsets for
10 mM BAPTA and 0.07 and 2.8 mM external Ca2+
concentration. Note that in both A and B, the adaptation
rate is slower and the fraction of current turned on at rest is higher
in the lower Ca2+ concentration. C: fraction
of total transducer current turned on at rest vs. external
Ca2+ with different concentrations of intracellular BAPTA:
0.1 mM ( ); 1 mM ( ), and 10 mM
( ). D: time constant of fast adaptation
vs. external Ca2+ with different concentrations of
intracellular BAPTA, symbols as in A. Time constants
were derived from the fits to small stimuli evoking less than a
half-maximal open probability, a procedure applied to characterize
experimental currents. Theoretical results show similar trends and
ranges of values to those of the experimental transducer currents (see
Fig. 3 of Ricci et al. 1998 ).
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To complement the calcium buffer results, the effects of varying the
external Ca2+ concentration also were examined.
This was implemented by altering the fraction of current carried by
Ca2+ in line with the values determined
experimentally (Ricci and Fettiplace 1998
). Reducing the
external Ca2+ increased the fraction of current
turned on at rest and slowed the adaptation time constant (Fig. 6),
effects that agree qualitatively with the experimental observations
(see Ricci et al. 1998
). However, the
Ca2+ sensitivity of the parameters, especially
the adaptation time constant, was weaker in the model than in the
experimental results. This defect might be corrected by making the
sites bind multiple Ca2+ ions in a cooperative
fashion as occurs with calmodulin-based receptors.
Properties of the second Ca2+-binding site
The model was useful for distinguishing the relative
contributions of the two sites, a manipulation that is difficult to
perform experimentally. The simulations were repeated in the absence of one or other site by setting the scaling constants,
and
, for that site to 0. The responses are shown in Fig.
7 for the case of 1 mM internal BAPTA and
should be compared with the equivalent simulations with both sites
present in Fig. 5. Removing site 1 produced responses with
slow adaptation with a time constant of ~14 ms that was independent
of level. The removal of site 2 gave responses,
characterized over most of the range solely by a fast time constant of
1 ms similar to that seen in the two-site model. Neither set of
responses in Fig. 7 for a single Ca2+-binding
site provided as good a match to the experimental results as did the
two-site model.

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Fig. 7.
Contribution of the fast and slow components of adaptation to the
theoretical transducer responses. Top: family of traces
computed with only the 2nd, slower, site (S2). Bottom:
family of traces calculated with only the 1st, fast site (S1). Stimulus
monitor is shown above the theoretical transducer
responses. Equivalent responses when both binding sites are present are
shown top right in Fig. 3. All calculations performed
for 1 mM external Ca2+, 1 mM intracellular BAPTA.
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In the majority of simulations, S2 was placed further from the
transducer channel (150-200 nm) than the
Ca2+-binding site for the fast process (20-50
nm). However, if S2 was moved closer to the channels, similar responses
could be achieved provided that the
Ca2+-dissociation constant for the site
(KD2) was increased. With S2 at
50-100 nm from the channel, it was necessary to raise the Ca2+-dissociation constant,
KD2, from 0.5 µM (the standard
value) to 3 µM. In contrast, it was not possible to alter
significantly the range for S1 and still retain fast adaptation.
Damped oscillatory responses
A consistent feature of the model responses for the lower BAPTA
concentrations was an under-damped oscillatory approach to the steady
state (Fig. 5), a manifestation of resonance stemming from negative
feedback control of the transducer channels. Such resonance
theoretically could produce frequency tuning for sinusoidal stimuli
(Crawford and Fettiplace 1981
), with the transducer
current being maximal at the resonant frequency. An expanded version of the smallest theoretical responses in 1 mM BAPTA from Fig. 5 are shown
in Fig. 8A, where the
oscillations are clearly evident at the onset and termination of the
step. This type of resonance has been observed experimentally at
frequencies ranging from 58 to 230 Hz (Ricci et al.
1998
). Figure 8A includes an example of such
experimental transducer currents recorded with 1 mM intracellular BAPTA. These currents exhibit damped oscillations at a similar frequency, 180 Hz, to the model responses.

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Fig. 8.
Damped oscillations in theoretical and experimental transducer
responses. A: theoretical responses shown at
top for small, ±0.05-µm displacement steps. Responses
were calculated for the standard set of theoretical parameters.
Bottom: examples of damped oscillatory behavior in
experimental transducer currents exhibiting a resonant frequency
comparable to that in the theoretical responses. For both model and
experimental responses, the internal Ca2+ buffer was 1 mM
BAPTA. Note that the resonance is more under-damped in the experimental
than in the theoretical responses. B: effects on
theoretical resonance of altering the kinetics of the modulator
transition, M M* for S1 (the fast
component). In the top traces, both forward and backward
rate constants were accelerated threefold, and for the bottom
traces, the rate constants for the same site were slowed
threefold. Oscillation frequencies for the stimulus offset are given
beside the traces. Note that the resonant frequency
increases as the modulator kinetics speed up.
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The main parameter controlling the resonant frequency in the model
responses was the speed of the fast adaptive process. Figure 8B shows the results of altering the rate constants for the
modulator transition. A threefold increase in the rate constants from
the standard value elevated the resonant frequency from 180 to 270 Hz.
Conversely, a threefold decrease in the rate constants slowed the
adaptation to the point where the resonance was not visible. Two
conclusions may be drawn from these results: first, the resonant behavior stems from the operation of the fast adaptation process; second, some of the variability in the appearance of the oscillations may be caused by differences among cells in the kinetics of the fast
adaptive feedback.
Effects of speed of stimulus onset on adaptation
An important experimental variable influencing the appearance of
the fast component is the rate of onset of the displacement step. In
the present experiments, the driving voltage to the piezoelectric stimulator was filtered with an eight-pole Bessel at 3 kHz. This yielded a 10-90% rise-time in the stimulating probe of ~0.1 ms, which is comparable with or less than the rise time of the transducer current (Crawford et al. 1989
). When the driving voltage
was filtered at 100 Hz, equivalent to a rise time of 3 ms, both the
onset and adaptation time constants were slowed (Fig.
9). The example illustrated shows that
the fast adaptation time constant,
fast,
increased from 1.3 to 4.6 ms. The additional filtering also
desensitized transduction (Fig. 9A), such that a larger
stimulus was required to produce the same peak current amplitude. As a
consequence, the current-displacement relationship for the stimulus
filtered at 100 Hz was shifted to more positive displacements relative to that for the 3-kHz filtered stimulus.

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Fig. 9.
Effects of stimulus onset speed on the fast time constant of
adaptation. A: averaged transducer currents for
displacement steps for which the driving voltage to the
piezoelectric element were shaped with an 8-pole Bessel filter set
at 3,000, 500, 100, and 50 Hz. With the slower stimulus onsets, both
the activation and adaptation of the transducer current were slowed,
and the current magnitude was reduced for the same stimulus amplitude.
Maximum current: 0.72 nA, 2.8 mM external Ca2+, 1 mM
intracellular BAPTA. B: experimental transducer currents
for small displacement steps filtered with an 8-pole Bessel filter at
3,000 Hz (left) or 100 Hz (right). To
achieve a similar peak current, the stimulus amplitude was increased
from 0.09 µm (left) to 0.22 µm
(right). Fits to the current decays are smooth lines
superimposed on the noisy experimental traces. Note that the
predominant time constant of adaptation, , increases more than
threefold as the stimulus onset is slowed. C: equivalent
theoretical responses for submaximal stimuli shaped with a single-pole
low-pass filter of time constant 0.1 ms (left) and 2 ms
(right). To achieve a similar peak open probability, the
stimulus amplitude was increased from 0.15 µm (left)
to 0.4 µm (right). As with the experimental responses,
the time course of adaptation is slowed by increased filtering of the
stimulus. 2.8 mM external Ca2+, 1 mM intracellular BAPTA in
both B and C.
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In the computed responses, the stimulus onset was normally
instantaneous, but filtering of the stimulus with a single pole filter
of time constant 0.1 ms had no effect on
fast.
However, when the filter time constant was raised to 2 ms,
fast increased from 0.7 ms to 5.6 ms (Fig.
9C). For theoretical as with the experimental responses, it
was necessary to increase the stimulus amplitude with the more heavily
filtered step to produce the same magnitude of response. An explanation
for these changes is that with slower stimulus onsets, the rate of
change and extent of the Ca2+ excursion at the
first site are both reduced, which slows and diminishes the magnitude
of fast adaptation. Both experimental and theoretical observations
emphasize the importance of using a stimulus with a rapid attack to
reveal the fast adaptive process.
 |
DISCUSSION |
Two components of adaptation
Characterization of the time course of mechanoelectrical
transducer currents showed that adaptation in turtle auditory hair cells proceeds with at least two time constants differing by an order
of magnitude. To account for this observation, and other evidence
summarized in Ricci et al. (1998)
, we constructed a
model of adaptation of the transducer channels that involved two
processes with different kinetics, each governed by stereociliary
Ca2+ levels. The model reproduced several
features of the experimental responses, including the sensitivity to
the concentrations of external Ca2+ and
intracellular calcium buffer, BAPTA, and a dependence on the onset
speed of the stimulus. The model also mimicked the behavior of the
turtle hair cell's transducer in its capacity to generate damped
oscillatory responses. The resonant behavior depended on the kinetics
of the mechanism responsible for the fast component of adaptation.
Models of hair-cell transducer adaptation assume that intracellular
Ca2+ controls the range of bundle displacements
detected by the mechanoelectrical transducer channel. This assumption
is expressed in our model by the notion of the channel's "set
point." One mechanism by which the set point might be altered invokes
a myosin motor connected to both the transducer channels on the
stereocilium's side wall and the internal actin cytoskeleton (reviewed
in Hudspeth and Gillespie 1994
). The speed of a myosin
motor will be limited by the kinetics of myosin ATPase, which for fast
skeletal muscle has a cycle time on the order of 50-100 ms at room
temperature (Hibberd and Trentham 1986
; Pollard
et al. 1991
). Although the most precise kinetic
information is available for the skeletal muscle myosin II, the
adaptation motor may depend on an unconventional myosin-I known to be
present in hair-cell stereocilia (Hasson et al. 1997
).
The cycle time of myosin-I also may approach 50 ms (Pollard et
al. 1991
).
The properties of the fast component of hair-cell adaptation, its
submillisecond kinetics, its symmetry for small positive and negative
displacements, and its insensitivity to the ATPase-inhibitors vanadate
and BDM, all argue that it does not rely on a conventional myosin-based
motor. An alternative hypothesis is that fast adaptation is mediated by
conformational rearrangements in the channel protein itself
(Crawford et al. 1989
) or in molecules directly
connecting it to the cytoskeleton. A specific mechanism would be that
the Ca2+-dependent modulator, M1
(Eq. 2), is an auxiliary subunit of the transducer
channel, the activation of which alters the gating kinetics of the
channel stabilizing it in its closed configuration (Fettiplace
et al. 1992
). Activation of M1 would
result from association with Ca2+ bound to S1, which itself
may be a separate Ca2+-binding protein like calmodulin or
may be an integral part of M1.
Location of the Ca2+-binding sites
Arrangement of the two Ca2+-binding sites
along the stereociliary axis is convenient for a model constructed in
cylindrical coordinates but may be physically unrealistic particularly
with respect to the more distant second site. S1, positioned at 20-50 nm from the center of the transducer channel complex, is of dimensions only slightly greater than ion channels (~10 nm diam), which may be
arranged in a cluster. Furthermore S1 does not need to be located directly on the axis, and its placement anywhere within a hemispherical shell centered on the channel complex would yield similar theoretical results. The spatial extent of S1 might represent the local cytoplasmic distribution of a Ca2+-binding protein like
calmodulin, which has been suggested to mediate calcium's role in
adaptation (Walker and Hudspeth 1996
). The transducer
channels were assumed to be entirely located at the apex of the
stereocilium but channels may be present at both ends of the tip links
(Denk et al. 1995
). In the current model, channels
placed on the side wall of the stereocilium were neglected due to the
added geometric complexity incurred, which would have removed the
radial symmetry and considerably lengthened the computations. Because
S1 is located close to the channels, our model would still provide an
adequate description of fast adaptation for channels on the side wall.
The location of S2 is more problematic because its distance from the
transducer channels (150-200 nm in most calculations) was large
relative to the size of the channel. However, we found that provided
that the Ca2+-affinity of the site was adjusted,
similar theoretical responses could be achieved with S2 positioned
50-100 nm from the transducer channels. Such distances are within the
dimensions of the electron-dense plaques, representing cytoskeletal
linking proteins or arrays myosin head groups, into which the tip links
insert (Hudspeth and Gillespie 1994
).
Nevertheless, we do not feel it is possible from our results to derive
a precise location for S2, and thus the coordinates for S2 in the model
may not impose major limitations on its physical realization. In
particular, the results neither establish nor eliminate a myosin motor
as the mechanism of the slow component of adaptation.
The kinetics of the mechanoelectrical transducer channel in
turtle hair cells can be fit by a scheme (Crawford et al.
1989
) involving two closed states
(C1 and
C2) and one open state
(Om):
A set of ordinary and partial differential equations (ODEs and
PDEs) was integrated to calculate the spread of free
Ca2+. For each compartment, ODEs computed the
open probability of transducer channels (Eq. A3),
Ca2+-dependent modulation processes (Eqs.
A7 and A8) and the reaction of fixed buffers (Eq. A13). All PDEs were related to the diffusion processes. One PDE
(Eq. A17) determined the concentration of
Ca2+-free diffusible buffer and one PDE described
the total rate of change of [Ca2+] and is a
summation of Eqs. A1, A6, A11, A13, and A17:
This work was supported by National Institutes of Deafness and Other
Communications Disorders Grants RO1 DC-01362 to R. Fettiplace and
RO1-DC-03896 to A. J. Ricci and a Deafness Research Foundation grant to A. J. Ricci.
Present address of Y.-C. Wu: SAP Labs, 3475 Deer Creek Road, Palo Alto,
CA 94304.
Address for reprint requests: R. Fettiplace, 185, Medical Sciences
Bldg., 1300 University Ave., Madison, WI 53706.
The costs of publication of this article were defrayed in part by the
payment of page charges. The article must therefore be hereby marked
"advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.