Coupling active hair bundle mechanics, fast adaptation, and somatic motility in a cochlear model

One of the central questions in the biophysics of the mammalian cochlea is determining the contributions of the two active processes, prestin-based somatic motility and hair bundle (HB) motility, to cochlear amplification. HB force generation is linked to fast adaptation of the transduction current via a calcium-dependent process and somatic force generation is driven by the depolarization caused by the transduction current. In this article, we construct a global mechanical-electrical-acoustical mathematical model of the cochlea based on a three-dimensional fluid representation. The global cochlear model is coupled to linearizations of nonlinear somatic motility and HB activity as well as to the micromechanics of the passive structural and electrical elements of the cochlea. We find that the active HB force alone is not sufficient to power high frequency cochlear amplification. However, somatic motility can overcome resistor-capacitor filtering by the basolateral membrane and deliver sufficient mechanical energy for amplification at basal locations. The results suggest a new theory for high frequency active cochlear mechanics, in which fast adaptation controls the transduction channel sensitivity and thereby the magnitude of the energy delivered by somatic motility.     

Adaptation in auditory hair cells

The narrow stimulus limits of hair cell transduction, equivalent to a total excursion of about 100nm at the tip of the hair bundle, demand tight regulation of the mechanical input to ensure that the mechanoelectrical transducer (MET) channels operate in their linear range. This control is provided by multiple components of Ca(2+)-dependent adaptation. A slow mechanism limits the mechanical stimulus through the action of one or more unconventional myosins. There is also a fast, sub-millisecond, Ca(2+) regulation of the MET channel, which can generate resonance and confer tuning on transduction. Changing the conductance or kinetics of the MET channels can vary their resonant frequency. The tuning information conveyed in transduction may combine with the somatic motility of outer hair cells to produce an active process that supplies amplification and augments frequency selectivity in the mammalian cochlea.     

Two adaptation processes in auditory hair cells together can provide an active amplifier

The hair cells of the vertebrate inner ear convert mechanical stimuli to electrical signals. Two adaptation mechanisms are known to modify the ionic current flowing through the transduction channels of the hair bundles: a rapid process involves Ca(2+) ions binding to the channels; and a slower adaptation is associated with the movement of myosin motors. We present a mathematical model of the hair cell which demonstrates that the combination of these two mechanisms can produce "self-tuned critical oscillations", i.e., maintain the hair bundle at the threshold of an oscillatory instability. The characteristic frequency depends on the geometry of the bundle and on the Ca(2+) dynamics, but is independent of channel kinetics. Poised on the verge of vibrating, the hair bundle acts as an active amplifier. However, if the hair cell is sufficiently perturbed, other dynamical regimes can occur. These include slow relaxation oscillations which resemble the hair bundle motion observed in some experimental preparations.     

Rho GTPases mediate the regulation of cochlear outer hair cell motility by acetylcholine

Outer hair cells are the mechanical effectors of the cochlear amplifier, an active process that improves the sensitivity and frequency discrimination of the mammalian ear. In vivo, the gain of the cochlear amplifier is regulated by the efferent neurotransmitter acetylcholine through the modulation of outer hair cell motility. Little is known, however, regarding the molecular mechanisms activated by acetylcholine. In this study, intracellular signaling pathways involving the small GTPases RhoA, Rac1, and Cdc42 have been identified as regulators of outer hair cell motility. Changes in cell length (slow motility) and in the amplitude of electrically induced movement (fast motility) were measured in isolated outer hair cells patch clamped in whole-cell mode, internally perfused through the patch pipette with different inhibitors and activators of these small GTPases while being externally stimulated with acetylcholine. We found that acetylcholine induces outer hair cell shortening and a simultaneous increase in the amplitude of fast motility through Rac1 and Cdc42 activation. In contrast, a RhoA- and Rac1-mediated signaling pathway induces outer hair cell elongation and decreases fast motility amplitude. These two opposing processes provide the basis for a regulatory mechanism of outer hair cell motility.     

Cochlear amplification, outer hair cells and prestin

Mechanical amplification of acoustic signals is apparently a common feature of vertebrate auditory organs. In non-mammalian vertebrates amplification is produced by stereociliary processes, related to the mechanotransducer channel complex and probably to the phenomenon of fast adaptation. The extended frequency range of the mammalian cochlea has probably co-evolved with a novel hair cell type, the outer hair cell and its constituent membrane protein, prestin. Cylindrical outer hair cells are motile and their somatic length changes are voltage driven and powered by prestin. One of the central outstanding problems in mammalian cochlear neurobiology is the relation between the two amplification processes.     

Viscoelastic relaxation in the membrane of the auditory outer hair cell.

The outer hair cell (OHC) in the mammalian ear has a unique membrane potential-dependent motility, which is considered to be important for frequency discrimination (tuning). The OHC motile mechanism is located at the cell membrane and is strongly influenced by its passive mechanical properties. To study the viscoelastic properties of OHCs, we exposed cells to a hypoosmotic solution for varying durations and then punctured them, to immediately release the osmotic stress. Using video records of the cells, we determined both the imposed strain and the strain after puncturing, when stress was reset to zero. The strain data were described by a simple rheological model consisting of two springs and a dashpot, and the fit to this model gave a time constant of 40 +/- 19 s for the relaxation (reduction) of tension during prolonged strain. For time scales much shorter or longer than this, we would expect essentially elastic behavior. This relaxation process affects the membrane tension of the cell, and because it has been shown that membrane tension has a modulatory role in the OHC's motility, this relaxation process could be part of an adaptation mechanism, with which the motility system of the OHC can adjust to changing conditions and maintain optimum membrane tension.     

Friction from Transduction Channels’ Gating Affects Spontaneous Hair-Bundle Oscillations

Hair cells of the inner ear can power spontaneous oscillations of their mechanosensory hair bundle, resulting in amplification of weak inputs near the characteristic frequency of oscillation. Recently, dynamic force measurements have revealed that delayed gating of the mechanosensitive ion channels responsible for mechanoelectrical transduction produces a friction force on the hair bundle. The significance of this intrinsic source of dissipation for the dynamical process underlying active hair-bundle motility has remained elusive. The aim of this work is to determine the role of friction in spontaneous hair-bundle oscillations. To this end, we characterized key oscillation properties over a large ensemble of individual hair cells and measured how viscosity of the endolymph that bathes the hair bundles affects these properties. We found that hair-bundle movements were too slow to be impeded by viscous drag only. Moreover, the oscillation frequency was only marginally affected by increasing endolymph viscosity by up to 30-fold. Stochastic simulations could capture the observed behaviors by adding a contribution to friction that was 3?8-fold larger than viscous drag. The extra friction could be attributed to delayed changes in tip-link tension as the result of the finite activation kinetics of the transduction channels. We exploited our analysis of hair-bundle dynamics to infer the channel activation time, which was ～1 ms. This timescale was two orders-of-magnitude shorter than the oscillation period. However, because the channel activation time was significantly longer than the timescale of mechanical relaxation of the hair bundle, channel kinetics affected hair-bundle dynamics. Our results suggest that friction from channel gating affects the waveform of oscillation and that the channel activation time can tune the characteristic frequency of the hair cell. We conclude that the kinetics of transduction channels’ gating plays a fundamental role in the dynamic process that shapes spontaneous hair-bundle oscillations.     

Power Dissipation in the Subtectorial Space of the Mammalian Cochlea Is Modulated by Inner Hair Cell Stereocilia

The stereocilia bundle is the mechano-transduction apparatus of the inner ear. In the mammalian cochlea, the stereocilia bundles are situated in the subtectorial space (STS)—a micrometer-thick space between two flat surfaces vibrating relative to each other. Because microstructures vibrating in fluid are subject to high-viscous friction, previous studies considered the STS as the primary place of energy dissipation in the cochlea. Although there have been extensive studies on how metabolic energy is used to compensate the dissipation, much less attention has been paid to the mechanism of energy dissipation. Using a computational model, we investigated the power dissipation in the STS. The model simulates fluid flow around the inner hair cell (IHC) stereocilia bundle. The power dissipation in the STS because of the presence IHC stereocilia increased as the stimulating frequency decreased. Along the axis of the stimulating frequency, there were two asymptotic values of power dissipation. At high frequencies, the power dissipation was determined by the shear friction between the two flat surfaces of the STS. At low frequencies, the power dissipation was dominated by the viscous friction around the IHC stereocilia bundle—the IHC stereocilia increased the STS power dissipation by 50- to 100-fold. There exists a characteristic frequency for STS power dissipation, CF_STS, defined as the frequency where power dissipation drops to one-half of the low frequency value. The IHC stereocilia stiffness and the gap size between the IHC stereocilia and the tectorial membrane determine the characteristic frequency. In addition to the generally assumed shear flow, nonshear STS flow patterns were simulated. Different flow patterns have little effect on the CF_STS. When the mechano-transduction of the IHC was tuned near the vibrating frequency, the active motility of the IHC stereocilia bundle reduced the power dissipation in the STS.     

Fast adaptation in vestibular hair cells requires myosin-1c activity

In sensory hair cells of the inner ear, mechanical amplification of small stimuli requires fast adaptation, the rapid closing of mechanically activated transduction channels. In frog and mouse vestibular hair cells, we found that the rate of fast adaptation depends on both channel opening and stimulus size and that it is modeled well as a release of a mechanical element in series with the transduction apparatus. To determine whether myosin-1c molecules of the adaptation motor are responsible for the release, we introduced the Y61G mutation into the Myo1c locus and generated mice homozygous for this sensitized allele. Measuring transduction and adaptation in the presence of NMB-ADP, an allele-specific inhibitor, we found that the inhibitor not only blocked slow adaptation, as demonstrated previously in transgenic mice, but also inhibited fast adaptation. These results suggest that mechanical activity of myosin-1c is required for fast adaptation in vestibular hair cells.     

Synchronization of Spontaneous Active Motility of Hair Cell Bundles

Hair cells of the inner ear exhibit an active process, believed to be crucial for achieving the sensitivity of auditory and vestibular detection. One of the manifestations of the active process is the occurrence of spontaneous hair bundle oscillations in vitro. Hair bundles are coupled by overlying membranes in vivo; hence, explaining the potential role of innate bundle motility in the generation of otoacoustic emissions requires an understanding of the effects of coupling on the active bundle dynamics. We used microbeads to connect small groups of hair cell bundles, using in vitro preparations that maintain their innate oscillations. Our experiments demonstrate robust synchronization of spontaneous oscillations, with either 1:1 or multi-mode phase-locking. The frequency of synchronized oscillation was found to be near the mean of the innate frequencies of individual bundles. Coupling also led to an improved regularity of entrained oscillations, demonstrated by an increase in the quality factor.     

The actions of calcium on hair bundle mechanics in mammalian cochlear hair cells

Sound stimuli excite cochlear hair cells by vibration of each hair bundle, which opens mechanotransducer (MT) channels. We have measured hair-bundle mechanics in isolated rat cochleas by stimulation with flexible glass fibers and simultaneous recording of the MT current. Both inner and outer hair-cell bundles exhibited force-displacement relationships with a nonlinearity that reflects a time-dependent reduction in stiffness. The nonlinearity was abolished, and hair-bundle stiffness increased, by maneuvers that diminished calcium influx through the MT channels: lowering extracellular calcium, blocking the MT current with dihydrostreptomycin, or depolarizing to positive potentials. To simulate the effects of Ca²⁺, we constructed a finite-element model of the outer hair cell bundle that incorporates the gating-spring hypothesis for MT channel activation. Four calcium ions were assumed to bind to the MT channel, making it harder to open, and, in addition, Ca²⁺ was posited to cause either a channel release or a decrease in the gating-spring stiffness. Both mechanisms produced Ca²⁺ effects on adaptation and bundle mechanics comparable to those measured experimentally. We suggest that fast adaptation and force generation by the hair bundle may stem from the action of Ca²⁺ on the channel complex and do not necessarily require the direct involvement of a myosin motor. The significance of these results for cochlear transduction and amplification are discussed.     

Characterization of adaptation motors in saccular hair cells by fluctuation analysis

The mechanical sensitivity of hair cells, the sensory receptors of the vestibular and auditory systems, is maintained by adaptation, which resets the transducer to cancel the effects of static stimuli. Adaptation motors in hair cells can be experimentally activated by externally applying a transduction channel blocker to the hair bundle, causing the hair bundle to move in the negative direction. We studied the variance in the position of the hair bundle during these displacements and found that it increases as the bundle moves to its new position. Often the variance peaks, and then declines to a steady-state value. We describe both displacement and variance with a model in which a motor acting on the bundle takes approximately 3.6-nm steps whose frequency (approximately 22 s(-1)) declines with the motor's load.     

A virtual hair cell, II: evaluation of mechanoelectric transduction parameters

The virtual hair cell we have proposed utilizes a set of parameters related to its mechanoelectric transduction. In this work, we observed the effect of such channel gating parameters as the gating threshold, critical tension, resting tension, and Ca(2+) concentration. The gating threshold is the difference between the resting and channel opening tension exerted by the tip link assembly on the channel. The critical tension is the tension in the tip link assembly over which the channel cannot close despite Ca(2+) binding. Our results show that 1), the gating threshold dominated the initial sensitivity of the hair cell; 2), the critical tension minimally affects the peak response, (I), but considerably affects the time course of response, I(t), and the force-displacement, F-X, relationship; and 3), higher intracellular [Ca(2+)] resulted in a smaller fast adaptation time constant. Based on the simulation results we suggest a role of the resting tension: to help overcome the viscous drag of the hair bundle during the oscillatory movement of the bundle. Also we observed the three-dimensional bundle effect on the hair cell response by varying the number of cilia forced. These varying forcing conditions affected the hair cell response.     

The calcium-activated potassium channels of turtle hair cells

A major factor determining the electrical resonant frequency of turtle cochlear hair cells is the time course of the Ca-activated K current (Art, J. J., and R. Fettiplace. 1987. Journal of Physiology. 385:207- 242). We have examined the notion that this time course is dictated by the K channel kinetics by recording single Ca-activated K channels in inside-out patches from isolated cells. A hair cell's resonant frequency was estimated from its known correlation with the dimensions of the hair bundle. All cells possess BK channels with a similar unit conductance of approximately 320 pS but with different mean open times of 0.25-12 ms. The time constant of relaxation of the average single- channel current at -50 mV in 4 microM Ca varied between cells from 0.4 to 13 ms and was correlated with the hair bundle height. The magnitude and voltage dependence of the time constant agree with the expected behavior of the macroscopic K(Ca) current, whose speed may thus be limited by the channel kinetics. All BK channels had similar sensitivities to Ca which produced half-maximal activation for a concentration of approximately 2 microM at +50 mV and 12 microM at -50 mV. We estimate from the voltage dependence of the whole-cell K(Ca) current that the BK channels may be fully activated at -35 mV by a rise in intracellular Ca to 50 microM. BK channels were occasionally observed to switch between slow and fast gating modes which raises the possibility that the range of kinetics of BK channels observed in different hair cells reflects a common channel protein whose kinetics are regulated by an unidentified intracellular factor. Membrane patches also contained 30 pS SK channels which were approximately 5 times more Ca-sensitive than BK channels at -50 mV. The SK channels may underlie the inhibitory synaptic potential produced in hair cells by efferent stimulation.     

Ca2+ changes the force sensitivity of the hair-cell transduction channel

The mechanically gated transduction channels of vertebrate hair cells tend to close in approximately 1 ms after their activation by hair bundle deflection. This fast adaptation is correlated with a quick negative movement of the bundle (a "twitch"), which can exert force and may mediate an active mechanical amplification of sound stimuli in hearing organs. We used an optical trap to deflect bullfrog hair bundles and to measure bundle movement while controlling Ca(2+) entry with a voltage clamp. The twitch elicited by repolarization of the cell varied with force applied to the bundle, going to zero where channels were all open or closed. The force dependence is quantitatively consistent with a model in which a Ca(2+)-bound channel requires approximately 3 pN more force to open, and rules out other models for the site of Ca(2+) action. In addition, we characterized a faster, voltage-dependent "flick", which requires intact tip links but not current through transduction channels.     

The Number and Distribution of Calyceal Hair Cells in the Inner Ear Utricular Macula of Some Reptiles

The utricular macula in the inner ear was examined in two turtle, two lacertilian, two snake and one crocodilian species. The orientation of hair cells was found to be similar to the general pattern for this macula described previously from fish to mammals. The calyceal hair cells, characterized by their embracing afferent nerve endings, are distributed in a single belt running parallel with the anterior and lateral borders of the macula in the examined reptiles.
The number of hair cells in a large number of calyces was counted for some of the examined reptile species, and the total number of hair cells calculated in two specimens of a turtle, crocodile and snake utricular macula. In the turtle and snake maculae, the calyceal hair cells form about 10% of the total hair cell number, while the crocodiles, which were very young, only had 2% calyceal hair cells. The total number of hair cells was found to be much higher in the crocodiles than in the other reptiles examined. The presented data, as well as data from the literature of the avian and mammalian inner ear, indicate that the number of this very sensitive hair cell type in the utricular macula is independent of locomotion type.     

A virtual hair cell, I: addition of gating spring theory into a 3-D bundle mechanical model

We have developed a virtual hair cell that simulates hair cell mechanoelectrical transduction in the turtle utricle. This study combines a full three-dimensional hair bundle mechanical model with a gating spring theory. Previous mathematical models represent the hair bundle with a single degree of freedom system which, we have argued, cannot fully explain hair bundle mechanics. In our computer model, the tip link tension and fast adaptation modulator kinetics determine the opening and closing of each channel independently. We observed the response of individual transduction channels with our presented model. The simulated results showed three features of hair cells in vitro. First, a transient rebound of the bundle tip appeared when fast adaptation dominated the dynamics. Second, the dynamic stiffness of the bundle was minimized when the response-displacement (I-X) curve was steepest. Third, the hair cell showed "polarity", i.e., activation decreased from a peak to zero as the forcing direction rotated from the excitatory to the inhibitory direction.     

Multifrequency forcing of a Hopf oscillator model of the inner ear

In response to a sound stimulus, the inner ear emits sounds called otoacoustic emissions. While the exact mechanism for the production of otoacoustic emissions is not known, active motion of individual hair cells is thought to play a role. Two possible sources for otoacoustic emissions, both localized within individual hair cells, include somatic motility and hair bundle motility. Because physiological models of each of these systems are thought to be poised near a Hopf bifurcation, the dynamics of each can be described by the normal form for a system near a Hopf bifurcation. Here we demonstrate that experimental results from three-frequency suppression experiments can be predicted based on the response of an array of noninteracting Hopf oscillators tuned at different frequencies. This supports the idea that active motion of individual hair cells contributes to active processing of sounds in the ear. Interestingly, the model suggests an explanation for differing results recorded in mammals and nonmammals.     

Theoretical conditions for high-frequency hair bundle oscillations in auditory hair cells

Substantial evidence exists for spontaneous oscillations of hair cell stereociliary bundles in the lower vertebrate inner ear. Since the oscillations are larger than expected from Brownian motion, they must result from an active process in the stereociliary bundle suggested to underlie amplification of the sensory input as well as spontaneous otoacoustic emissions. However, their low frequency (<100 Hz) makes them unsuitable for amplification in birds and mammals that hear up to 5 kHz or higher. To examine the possibility of high-frequency oscillations, we used a finite-element model of the outer hair cell bundle incorporating previously measured mechanical parameters. Bundle motion was assumed to activate mechanotransducer channels according to the gating spring hypothesis, and the channels were regulated adaptively by Ca²⁺ binding. The model generated oscillations of freestanding bundles at 4 kHz whose sharpness of tuning depended on the mechanotransducer channel number and location, and the Ca²⁺ concentration. Entrainment of the oscillations by external stimuli was used to demonstrate nonlinear amplification. The oscillation frequency depended on channel parameters and was increased to 23 kHz principally by accelerating Ca²⁺ binding kinetics. Spontaneous oscillations persisted, becoming very narrow-band, when the hair bundle was loaded with a tectorial membrane mass.