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Channel gating forces govern accuracy of mechanoelectrical transduction in hair cells

Edited by A. James Hudspeth, The Rockefeller University, New York, NY, and approved October 29, 2003 (received for review May 2, 2003)
Abstract
Sensory hair cells are known for the exquisite displacement sensitivity with which they detect the soundevoked vibrations in the inner ear. In this article, we determine a stochastically imposed fundamental lower bound on a hair cell's sensitivity to detect mechanically coded information arriving at its hair bundle. Based on measurements of transducer current and its noise in outer hair cells and the application of estimation theory, we show that a hair cell's transducer current carries information that allows the detection of vibrational amplitudes with an accuracy on the order of nanometers. We identify the transducer channel's molecular gating force as the physical factor controlling this accuracy in proportion to the inverse of its magnitude. Further, we show that the match of stochastic channel noise to gatingspring noise implies that the gating apparatus operates at the threshold of negative stiffness.
The main peripheral structure that constitutes the mammalian sense of hearing is the cochlea, a snailshellshaped cavity in the temporal bone filled with fluid (1). Sound, via the middle ear, excites these cochlear fluids, which transmit their motion via the organ of Corti to the hair bundles of the sensory hair cells. The hair cells perform the actual mechanoelectrical transduction in the ear. They encode the mechanical motion of their hair bundle into a change of the open probability of mechanically gated transducer channels, resulting in a variation of inward transducer current (2, 3). This signal, the first electrically coded stage in the signalprocessing cascade of the ear, is further transformed by the hair cells, most likely also via electromechanical feedback (4, 5), and sent to the brain.
From the brain's point of view, the transducer channels thus serve as a window onto the mechanical events in the cochlea, which carry the soundevoked information. Here, unlike most other approaches in haircell research, we also follow this informationupstream direction and determine the accuracy with which the position X of a hair bundle can be detected given a certain transducer current, I. Because ion channels have intrinsic stochastic properties that derive from their thermal activation, a specific deflection, X, may result in a range of possible transducer currents. Conversely, in the informationupstream direction, this means that a specific transducer current, I, can result from a range of different deflections of the hair bundle, each having a specific probability of evoking that current. The variance of these deflections will be used here as a measure of accuracy of mechanoelectrical transduction.
In addition to this intrinsicchannel accuracy, the elastic elements that have to be tensioned to engage the transducer channels' gates also will convey Brownian noise to the channels and, consequently, will additionally impair the accuracy with which a transducer current can be decoded into the evoking hairbundle deflection X.
The twostate gatingspring model (6) of mechanoelectrical transduction is used to describe this overall accuracy in terms of its gating parameters and thus allows for the identification of the physical limitations of mechanoelectrical transduction in hair cells.
Materials and Methods
Electro and Mechanophysiology. Experiments were performed on apicalcoil outer hair cells obtained from acutely isolated organs of Corti taken from CD1 mice. Animal procedures were regulated under United Kingdom Home Office guidelines.
Transducer currents in response to fluidjet stimulation were measured under wholecell voltage clamp with an EPC8 patch clamp amplifier (HEKA Electronics, Lambrecht/Pfalz, Germany) at a –84 mV holding potential (including a –4 mV liquid junction potential correction). Outer hair cells investigated by using the dynamic stimulus protocol (see Data Analysis) were obtained between postnatal days 5 and 7 (P5–7). Pipettes were pulled from soda glass and thickly coated with wax. Extracellular solution contained 135 mM NaCl, 5.8 mM KCl, 1.3 mM CaCl_{2}, 0.9 mM MgCl_{2}, 0.7 mM NaH_{2} PO_{4}, 2 mM Napyruvate, 5.6 mM dglucose, and 10 mM Hepes, and vitamins and amino acids were added from concentrates (pH 7.5; 306 mOsm/kg). Intracellular solution contained 137 mM CsCl, 2.5 mM MgCl_{2}, 2.5 mM Na_{2} ATP, 10 mM Na_{2}phosphocreatine, 5 mM Hepes, and 1 mM EGTANaOH (pH 7.3; 292 mOsm/kg). The contribution of basolateral channels to the measured currents was eliminated by the holding potential used and by the replacement of potassium by caesium in the intracellular solution. The average series resistance was 2.7 ± 0.7 MΩ (n = 11) after compensation, and the average membrane capacitance was 6.2 ± 0.3 pF (n = 11), resulting in a typical noise value of ≈1.7 pA^{2} integrated over the recording bandwidth, which was corrected for in the noise measurements. The average membrane input resistance was 435 ± 195 MΩ (n = 11). Current signals were lowpass filtered with cutoff frequencies of 2.5 or 5 kHz (f_{–3dB}, eightpole Bessel), which were always below the cutoff frequency imposed on the measurements by the series resistance of the patch.
Hairbundle displacement was measured by using a differential photodiode system, which was lowpass filtered (f_{–3dB} = 5 kHz, eightpole Bessel) and recorded simultaneously with the transducer current (7). The rms noise caused by the displacement measurement technique was ≈0.3 nm. Fluidjet stimuli were generated, and elicited responses were recorded with a Power 1401 data acquisition board in combination with the signal software package (CED, Cambridge, United Kingdom). Generated stimuli were lowpass filtered (f_{–3dB} = 2 kHz, eightpole Bessel). Current and displacement recordings were oversampled at 50 kHz to allow a proper analysis of the response envelope of the highfrequency component (f_{2}, see Data Analysis). Hairbundle stiffness was measured in response to force steps as described (7) in cells held at –84 mV at P7 and P19 (see Discussion). All experiments were performed at room temperature (≈25°C).
Cramér–Rao Bound on Transducer Accuracy. A discrete version of the Cramér–Rao inequality (8, 9) is used to determine a lower bound of the variance of any possible unbiased estimator, X̂, of hairbundle displacement, X, from a conditional probability density function p(IX) = p(n_{o}X). This function expresses the probability of a total transducer current, I, entering the cell through n_{o} open channels, given a position, X, of the hair bundle's tip. Here, I = i·n_{o}, where i is the current flowing through one open channel. The variance, , of the estimator, X̂, of the input parameter, X, is then equal to or greater than the Cramér–Rao bound, , which stems from intrinsicchannel stochastics described by p(n_{o}X) according to: [1] The inverse of is usually referred to as the information, Inf(X), about X contained in I (see ref. 8) and in this case has dimensions equal to m^{–2}. Assuming a total of N_{ch} equal and independent operational channels, we may use the binomial distribution for p(n_{o}X) (see ref. 10). Then, evaluating the righthand side denominator of Eq. 1 yields for the information: [2] Here, p_{o}(X) is the (average) open probability and p′_{o} (X) is the derivative of p_{o}(X) with respect to X. The information as given by Eq. 2 appears to be closely related to an empirically defined integratedband signaltonoise ratio of haircell transduction (11). In that study it was proposed that a transducer current power signal, S_{I}, in response to a small bundledisplacement power, ΔX^{2}, around an operational position, X, is related to the change of the channel's open probability according to S_{I}(X) = [i · N_{ch} · p′_{o}(X) · ΔX]^{2}, whereas the transducer current noise power is given by (see ref. 12): N_{I}(X) = i^{2}·N_{ch}·p_{o}(X)·(1 – p_{o}(X)). Combined with the new identity in Eq. 2, this finding means that the signaltonoise ratio of haircell transduction equals the information times the displacement power of its bundle: S_{I}/N_{I} = Inf·ΔX^{2}. Conversely, our present result also shows that the Cramér–Rao bound on the accuracy, , equals the value of hairbundle displacement power, ΔX^{2}, that results in a signaltonoise ratio of the transducer current, I, of one (S_{I}/N_{I} = 1).
TwoState GatingSpring Model. For each channel, the basic twostate gatingspring model assumes the action of a gating spring, which at one end is driven by the hair bundle and at the other end engages the channel's gate (6, 13). Thermodynamics then shows that the open probability of the channel, p_{o}(X), is a sigmoidal function of hairbundle position X (Fig. 1a) and can be expressed in terms of the gatingspring constant, K_{s}, and the conformational swing of the channel, D, both as measured at the hair bundle's tip: p_{o}(X) = [1 + exp(–K_{s}D·(X – X_{o})/kT)]^{–1}. Here, k is Boltzmann's constant, T is the absolute temperature, and X_{o} defines the displacement at which p_{o}(X) = 0.5. The product, K_{s}·D, has been termed gating force, Z, (6) and defines an effective operational range of hairbundle displacements, Λ_{90} = 6 kT/Z, centered at X_{0} (Fig. 1a) within which the open probability is ≈90% modulated, apart from the effect of adaptation (14). The model also defines an ensemble averaged stiffness, K_{ch}(X) = K_{s} – K_{gc}(X), of each springchannel complex (Fig. 1b). This stiffness thus consists of the gatingspring constant, K_{s}, and in addition includes a displacementdependent reduction, K_{gc}(X) = Z^{2}·p_{o}(X)·(1 – p_{o}(X))/kT, which is related to the extent of gating and therefore termed gating compliance (6).
The twostate model, extended also to describe differential engagement (15) of the two states, was used to generate fits to the measured data and results (e.g., Fig. 2 b–d, solid lines). Differential engagement has been introduced to the description of gating of mechanoelectrical transducer channels to account for the finite minimum open probability observed at hairbundle positions more negative than approximately –50 nm. At these bundle positions, gatingspring tension is possibly decoupled from the channel's gate.
Data Analysis. A dynamic stimulus protocol (doublesine), consisting of a sum of two sine waves (f1 ≈ 49 Hz and f2 ≈ 1,563 Hz), displaced the bundle through a large part of its operational range (Fig. 2a). By using the doublesine protocol, p_{o}(X) relationships (Fig. 2b) were constructed by lowpass filtering of the current response to obtain the lowfrequency (f_{1}) component and its harmonics and displaying them as a function of the associated lowpassfiltered hairbundle displacement. The envelope of the remaining f_{2} current component was calculated and plotted as a function of the associated displacement component at f_{1} to obtain p′_{o}(X) curves (Fig. 2c). The twostate model was used to fit the p_{o}(X) and p′_{o}(X) data simultaneously. To match the associated p_{o}(X) relationships, p′_{o}(X) data were linearly scaled with a factor (average 0.83 ± 0.09; n = 11) most likely reflecting differences in adaptation (14) at the two frequencies (f_{1} and f_{2}).
Current noise was determined over ≈100ms time windows during steps of different bundle position, X. Noise power density spectra were determined by using Bartlett's periodogramaveraging method (5× averaging) with a Taylor–Kaiser window (β = 1.8) (16).
Unless indicated otherwise, data are presented as the mean ± SD with the number of observations in parentheses.
Results
IntrinsicChannel Stochastics. In Materials and Methods it is derived how the intrinsic stochastic properties of N_{ch} independent and equal transducer channels with an open probability, p_{o}(X), limit the accuracy with which a hair cell's bundle position, X, can be optimally detected. This accuracy, or Cramér–Rao lower bound, σ_{min}, is the inverse of the property referred to as information, Inf(X), about X contained in I and depends on N_{ch}, p_{o}(X), and its derivative, p′_{o}(X), according to Eqs. 1 and 2. The interpretation of σ_{min} as the accuracy of mechanoelectrical transduction is illustrated further by the demonstration that equals the bundle displacement power that leads to a transducer current with a signaltonoise ratio of one (see Materials and Methods and ref. 11).
Cramér–Rao bounds on transducer accuracy, σ_{min}(X), were determined in 11 apicalcoil outer hair cells from simultaneous measurements of transducer current and the position of the hair bundle, which was stimulated with a fluid jet. Fig. 2a shows an example of hairbundle displacement (upper trace) and evoked current (lower trace) measured in response to the doublesine stimulus protocol used. This stimulus contained a largeamplitude, lowfrequency component (49 Hz, 1 period shown) and a smallamplitude, highfrequency component (1,563 Hz, 32 periods shown). Filtering of the responses enabled the construction of p_{o}(X) curves from the lowfrequency components (Fig. 2b, data points). The amplitude of the envelope of the highfrequency component in the current response effectively probes the derivative with respect to displacement, X. This method allows the independent determination of p′_{o}(X) curves when plotted against the lowfrequency component of the displacement (Fig. 2c, data points). The number of operational channels per cell, N_{ch}, was determined from the saturation current in each cell in combination with the known unitary current (i = 9.7 pA at –84 mV; ref. 7). The average value found for N_{ch} was 80 ± 26 (n = 11). Fig. 2d shows the resulting lower bound, σ_{min}(X), as obtained by substituting N_{ch} and the measured p_{o}(X) and p′_{o}(X) data from the same two cells as shown in Fig. 2 b and c into Eqs. 1 and 2. The average optimal accuracy, σ_{min}, amounts to 5.9 ± 1.9 nm (n = 11; range, 2.9–9.9 nm) and is reached at a hairbundle position, X, of 44 ± 18 nm (n = 11). This result thus shows that the intrinsic stochastics of the transducer channels prevents a single outer hair cell from reliably detecting mechanical motion of its bundle smaller than on the order of nanometers.
Physical Limits of Transduction. To identify the underlying physics that governs accuracy and related signaltonoise ratio of haircell transduction, we evaluated Inf (Eq. 2) in terms of the gating parameters that define the twostate gatingspring model (refs. 6 and 13 and Materials and Methods). This model was fitted to the measured p_{o}(X) and p′_{o}(X) data to obtain the gating parameters for each cell (e.g., Fig. 2 b and c, solid curves). Averages of the results are given in Table 1. The gatingspring model can be considered the most concise quantitative description to date of observations on the transducer channel's gating machinery and allows a straightforward interpretation in terms of its physical parameters. The most important parameter is the elementary gating force, Z, defined as the product of the gatingspring constant, K_{s}, and the conformational swing of the channel, D.
Combining the properties of the twostate gatingspring model with Eq. 2 reveals that each transducer channel of a hair cell contributes an amount of information, ΔInf, to the total information, Inf = N_{ch}·ΔInf, equal to: [3] The righthand side of Eq. 3 offers a useful but also profound insight into the process of mechanoelectrical transduction: the amount of information, ΔInf, on hairbundle position conveyed by each transducer channel via its gated current is equivalent to the gating compliance, K_{gc}, divided by the thermal noise energy, kT. Nonlinear distortion in hairbundle mechanics, caused by gating and characterized by the gating compliance, K_{gc} (see Materials and Methods), therefore seems to be inevitable for information transfer. The amount of information is fundamentally constrained by the level of thermal noise energy. It is obvious from Eq. 3 that the information is maximal at X = X_{0} (p_{o}(X_{0}) = 0.5), where it amounts to [Z/(2kT)]^{2} so that the optimal accuracy in estimating the hair bundle's position per channel, σ_{min}(X_{0}), is 2kT/Z. This theoretical result, with our average experimental value for Z (174 ± 60 fN; n = 11), leads to ≈47 nm per channel and, for 80 ± 26 operational channels, to 5.3 ± 2.6 nm per cell. This finding is in line with the result for the optimal σ_{min} (5.9 nm) directly obtained from substituting the experimental results obtained for N_{ch}, p_{o}(X), and p′_{o}(X) in Eqs. 1 and 2 (Fig. 2d).
When increased, the gating force, Z, seems to limitlessly improve accuracy (σ_{min} = 2kT/Z) and related signaltonoise ratio of a hair cell. At the same time, though, this increase would reduce the effective operational range, Λ_{90} = 6kT/Z, within which hairbundle vibrations are predominantly transduced (p_{o} is modulated ≈90% within Λ_{90}; see Materials and Methods and Fig. 1a). The requirement for a hair cell to transduce the physiological range of hairbundle vibrations, therefore, undoubtedly puts an upper limit on the value of the gating force. The ratio of operational range, Λ_{90}, and optimal accuracy, σ_{min}(X_{0}), defines an upper bound on the dynamic range of transduction per channel, which can now easily be inferred to amount to a factor of 3 (≈9.5 dB) and is surprisingly independent of any parameter of the model. The operational range, Λ_{90}, of an ensemble of identical but independent channels is the same as that of one channel. However, increasing the number of channels to N_{ch} improves their collective optimal accuracy to , thereby increasing a hair cell's effective dynamic range to 9.5 + 10·log(N_{ch}) dB. Together with the average number of operational channels found (N_{ch} = 80), this finding leads to a theoretical dynamic range of ≈28.5 dB. This dynamic range is consistent with the ratio (28.4 dB) of the average operational range Λ_{90} (156 ± 50 nm; n = 11) and σ_{min}(X_{0}) (5.9 nm), both directly obtained from our experimental data.
GatingSpring Noise. So far we have only considered signaltonoise ratio and accuracy arising from intrinsicchannel stochastics, which, therefore, are solely related to transduction. However, noise attacks the mechanotransducer system on multiple fronts. At its input, formed by the gating springs, the system also is susceptible to Brownian noise fluctuations. When completely transduced, the Brownian noise of a gating spring evokes a current per channel equivalent to the current evoked by a hairbundle stimulus variance of σ^{2}_{B} = kT/K_{s} (see ref. 17). Because we find from our data a value for K_{s} of 7.4 ± 1.0 μN/m (n = 11), σ_{B} equals on average an equivalent stimulus of ≈24 nm for a single channel. Like σ_{min}, this gatingspring noise also scales down in proportion to the inverse of the square root of the total number of channels, yielding σ_{B} = 2.8 ± 0.6 nm (n = 11) per cell. In the stimulus domain, the variance induced by the gatingspring noise, , can be expected to add independently to the lower bound on the variance, , as obtained from the Cramér–Rao inequality.
How do these two noise components compare? Using Eqs. 1–3, together with σ^{2}_{B} = kT/K_{s}, yields: [4] The ratio of the two noise components thus equals the gating compliance, K_{gc}, normalized to the gatingspring constant, K_{s}, and is denoted here as K_{ngc}(X). Given a gatingspring constant with associated , a smaller summed variance could be obtained by adjusting the gating force so as to reduce . Reducing it considerably more than the inescapable level of σ^{2}_{B} would, however, not pay off because σ^{2}_{B} would continue to dominate the summed variance, whereas the operational range would, unfavorably, decrease. On the basis of these considerations, we hypothesize that a hair cell at X = X_{0} may operate at closely matched noise levels () from which it follows that the normalized gating compliance, K_{ngc}(X_{0}), should be on the order of one (Eq. 4). Interestingly, this condition is equivalent to each gatingspringchannel complex operating close to the threshold of negative stiffness (K_{ch}(X_{0}) ≈ 0) as is obvious from the definitions of K_{ch} and K_{ngc} (Materials and Methods and Eq. 4). A characteristic matching parameter, M_{c}, therefore can be defined according to: . If M_{c} exceeds 1, a region of negative stiffness of the springchannel complex exists (Fig. 1b), and gatingspring noise predominates. If, on the other hand, M_{c} is smaller than 1, stiffness is positive, and intrinsicchannel noise dominates, whereas if M_{c} is 1, the two noise components in the stimulus domain exactly match at X_{0}, and K_{ch}(X_{0}) = 0.
To test the noisematching hypothesis (M_{c} ≈ 1), we determined M_{c} in outer hair cells by using K_{s} and D resulting from fits to our data (Table 1). The average value obtained for M_{c} is 0.50 ± 0.16 (n = 11) and shows that the two components of noise at X_{0} are indeed comparable, as also is apparent from the examples in Fig. 2d by comparing the data on σ_{min}(X_{0}) (4.5 and 5.4 nm) to the horizontal lines representing σ_{B} (3.0 and 2.9 nm, respectively).
Discussion
Two noise contributions, intrinsicchannel stochastics and gatingspring noise, both affecting the primary process of mechanoelectrical transduction in hair cells, were quantified. These contributions amount to on the order of nanometers, when interpreted in terms of equivalent hairbundle noise (σ_{min} and σ_{B}). The optimum accuracy arising from intrinsicchannel stochastics is reached at a deflection, X_{0}, ≈45 nm positive of the equilibrium position of the hair bundle, as measured under our experimental conditions (Fig. 2d). Extracellular calcium concentrations, such as found in the cochlear endolymph (≈30 μM; ref. 18) have been shown to bring X_{0} close to the equilibrium position (19) via a calciummediated process called adaptation (14). This process is therefore a likely candidate to maintain optimal detection accuracy under equilibrium conditions.
On the basis of the hypothesis of matched noise variances in the stimulus domain (), we have demonstrated that the ensemble averaged stiffness of a gatingspringchannel complex of a hair cell becomes negative, a property that has been reported previously in saccular hair cells (20). An unequivocal direct experimental observation in the stimulus domain of the degree of matching of the contributions of mechanical gatingspring noise and intrinsicchannel noise is beyond present experimental possibilities. However, the transducer current noise should reflect both components of the variance in the stimulus domain. We therefore analyzed current noise measurements to investigate whether they were consistent with the inferences made from our analysis on intrinsicchannel noise and the noisematching hypothesis.
Fig. 3 shows the variance of the current noise measured at different positions of the operational point, X, of the hair bundle. It is compared with noise variances (Fig. 3, solid lines) predicted on the basis of fits of the gatingspring model to the cell's measured p_{o}(X) and p′_{o}(X) data (e.g., Fig. 2 b and c). The prediction of the total noise variance (Fig. 3, solid line I), consisting of the sum of the channel stochastics [Fig. 3, solid line II, N_{ch}i^{2}p_{o}(1 –p_{o})] and gatingspring noise [Fig. 3, solid line III, N_{ch}i^{2}(p′_{o}) σ^{2}_{B}], adequately follows the shape of the measured noise as a function of hairbundle position, X. However, the predictions were systematically higher than the actually measured noise with a fixed factor per cell (8.8 ± 3.8; n = 4). The most likely explanation for the lower noise variance measured is that the lowpass filter used cuts off the highfrequency part of the spectrum, producing the predicted shape as a function of X but at a fixed fraction. The steepest rolloff part of the spectrum of the measured noise power (Fig. 4) indeed follows the highfrequency rolloff of the lowpass filter used (Fig. 4, solid line), indicating that the intrinsic cutoff frequency of the transducer channels is beyond the cutoff frequency of the lowpass filter. This result is in line with measured current responses to steps having risetime constants that were found to be limited by the time constant of the stimulus device used (≈50 μs; data not shown) corresponding to rateconstants of the channel exceeding 3 kHz. Fig. 3 shows that at negative hairbundle positions (less than approximately –50 nm) the scaled data are in agreement with the differential engaging of the two states, implying a small but constant (≈1%) open probability in that range. The related constant noise in this range excludes the possibility that the noise caused by the gating springs (Fig. 3, solid line III) greatly dominates the stochastic channel noise of the cell (Fig. 3, solid line II). The scaled measurements are consistent with the predicted ratio of the two noise components of M_{c} (= σ_{B}/σ_{min} = 0.55) for this cell. Three other cells were analyzed and gave similar results. Mechanical noise of the whole nonstimulated hair bundle was measured directly and found to be <4 nm^{2}. This finding is consistent with the Brownian noise expected from a hair bundle with stiffness on the order of millinewtons per meter (7, 16). In the stimulus domain, whole hairbundle noise of outer hair cells is therefore approximately an order of magnitude smaller than the average summed noise of the gating springs, (7.6 nm^{2}), and intrinsicchannel noise, (34.2 nm^{2}). Neglecting the whole hairbundle noise, this combination results in an overall optimal accuracy in the detection of hairbundle position amounting to (n = 11).
An alternative explanation for the apparent lack in measured noise variance that cannot be completely ruled out on the basis of the present experiments is that the ensemble of N_{ch} operational channels of a cell may, depending on the stimulus, effectively split up in subensembles with different open probabilities, leading to a lower noise variance than expected from N_{ch} independently fluctuating channels. Such a noisereducing mechanism may be effected by a feedback mechanism per channel, for instance, Ca^{2+}binding to the channel (3, 4), and could thereby improve the cell's signaltonoise ratio. If the observed noise reduction factor (8.8) would completely arise from such a mechanism, σ_{min}(X_{0}) would decrease with a factor , and the best overall accuracy in detecting hairbundle motion (6.5 nm) would be improved by just a factor of 2 leading to 3.4 nm. Channel interaction, like the cooperative scheme proposed in a recent study (21), is not likely to decrease the channel noise but instead can be expected to increase it. This result can be appreciated from considering the channel noise of N_{ch}/2 independent pairs, each consisting of two fully cooperating channels, thus effectively having twice the conductance of a single channel.
Other types of hair cells than the outer hair cells considered here might have different bundle mechanics and gating parameters resulting in alternative relative contributions of the individual noise sources. It has been reported that in frog saccular hair cells, bundle noise may dominate (22), and it has been suggested that this property may serve the phenomenon of stochastic resonance (23). Also, the ratio of gatingspring noise and intrinsicchannel noise (σ_{B}/σ_{min} = M_{c}) may vary, possibly related to the specific detection function of a hair cell. Mammalian vestibular hair cells, for instance, may possess springchannel complexes with an M_{c} exceeding 1 (15). Results reported on turtle hairbundle nonlinearity (24) show that M_{c} in these cells exceeds 0.75, whereas data on fish lateral line hair cells (25) indicate a value of at least 1. A recent report on transduction in frog saccular hair cells (20) translates into an M_{c} of ≈1.5. The associated collective negative stiffness of the gating springs in those hair cells may even dominate the positive passive stiffness of the hair bundle that results from its pivots in the apical plate and stereociliary sidetoside links, rendering the isolated hair bundle as a whole unstable within a substantial region (≈20 nm). Evidence from outer hair cells so far indicates that their hair bundle's stiffness is dominated by positive passive contributions (15).
What limit is imposed on the threshold of hearing by the noise of the gating apparatus of the transducer channel we considered? First we have to consider whether our results from neonatal hair cells are representative for those in the mature hearing organ. It has recently been demonstrated that transducer channel properties of outer hair cells in rats do not obviously change with the onset of hearing (26). We found the steadystate bundle stiffness of transducing mature hair cells at P19 (3.9 ± 1.2 mN/m; n = 7) to be ≈25% less than the stiffness at P7 (5.1 ± 2.0 mN/m; n = 7). Both values are comparable to previous results from neonatal cochlear cultures (7, 15). The absolute reduction in hairbundle stiffness exceeds the overall gatingspring contribution (15) and is therefore most likely related to a reduction in the bundle's passive stiffness resulting from structural changes of the bundle or the disappearance of the kinocilium during this stage of development. The latter possibility is in line with a previous report on a relative kinociliary contribution to the whole bundle stiffness of 10–25% (see ref. 27). Both possibilities would not directly affect the mechanoelectrical transduction process but could lead to displacement noises scaled up with the same percentage. Also, no signs of spontaneous hairbundle oscillations were found during the neonatal (P7) or the hearing (P19) stage. We may thus assume that similar transductionrelated noise variances as reported in this article apply to the hair cells in the mature hearing organ, including the inner hair cells, which most likely possess very similar transduction characteristics (28).
From our results, we arrive at a displacement variance of (6.5)^{2} nm^{2} contained within a bandwidth of at least 5 kHz. This value corresponds to an upper bound of ≈9 × 10^{–2} nm/Hz^{½} as a bandwidthindependent inaccuracy per outer hair cell and could even be a factor of ≈3 lower if the hair cells transduce one order of magnitude faster (50 kHz). Whatever the further signal processing mechanisms, the inaccuracy set by the elastic engagement of haircell transducer channels cannot be circumvented. Obviously, the threshold in terms of a minimal detectable displacement by a single hair cell may be improved at the expense of the effective bandwidth. Lowering it for instance down to the range of tens to a hundred Hertz would enable a single outer hair cell, if no further noise is added in the filtering process, to match the lowest reported threshold of hearing, which in different organs ranges from fractions of a nanometer to several nanometers (29, 30). This same threshold, however, also may be obtained within a larger bandwidth if more than one hair cell contributes to the signal processing. Therefore, a combination of ensemble averaging across several hair cells and a limited effective bandwidth caused by passive electrical and mechanical filtering, as well as active electromechanical feedback, whether mediated by the hair bundle or by prestin (4, 5), would seem to be required to reach the exquisite displacement threshold that some hearing organs possess.
Acknowledgments
We thank A. B. A. Kroese and D. G. Stavenga for their comments on the manuscript. T.D. was supported by the Netherlands Organization for Scientific Research and the School of Behavioral and Cognitive Neurosciences (University of Groningen). W.M. and C.J.K. are supported by the Medical Research Council.
Footnotes

↵† To whom correspondence should be addressed. Email: s.van.netten{at}phys.rug.nl.

This paper was submitted directly (Track II) to the PNAS office.

Abbreviation: Pn, postnatal day n.
 Received May 2, 2003.
 Copyright © 2003, The National Academy of Sciences
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