Cochlear outer hair cells (OHCs) are fast biological motors that serve to enhance the vibration of the organ of Corti and increase the sensitivity of the inner ear to sound. Exactly how OHCs produce useful mechanical power at auditory frequencies, given their intrinsic biophysical properties, has been a subject of considerable debate. To address this we formulated a mathematical model of the OHC based on first principles and analyzed the power conversion efficiency in the frequency domain. The model includes a mixture-composite constitutive model of the active lateral wall and spatially distributed electro-mechanical fields. The analysis predicts that: 1) the peak power efficiency is likely to be tuned to a specific frequency, dependent upon OHC length, and this tuning may contribute to the place principle and frequency selectivity in the cochlea; 2) the OHC power output can be detuned and attenuated by increasing the basal conductance of the cell, a parameter likely controlled by the brain via the efferent system; and 3) power output efficiency is limited by mechanical properties of the load, thus suggesting that impedance of the organ of Corti may be matched regionally to the OHC. The high power efficiency, tuning, and efferent control of outer hair cells are the direct result of biophysical properties of the cells, thus providing the physical basis for the remarkable sensitivity and selectivity of hearing.
Outer hair cells (OHC) in the mammalian cochlea are essential to the remarkable sensitivity of hearing. These highly specialized cells actively feed mechanical power into the organ of Corti and amplify its mechanical vibrations in response to sound [1]–[5]. How this is achieved at auditory frequencies is a subject of considerable debate. Five biological motor mechanisms have been described in outer hair cells that may contribute [2], [3], [5], [6]. Motors localized to the hair bundles include: actin-myosin motors associated with slow bundle movements and adaptation mechano-electrical transduction (MET) currents [7], [8]; Ca2+ sensitive reclosure or conformational change of the MET molecular apparatus associated with fast bundle movements and adaptation [9]; and electrically-driven bundle displacement that act independent of MET function [10]. Motors localized to the soma include: cytoskeletal remodeling mechanisms [11], [12] and electrically-driven changes in length [13]–[15]. The ability of each of these mechanisms to feed mechanical power into cochlea is limited by their intrinsic thermodynamic properties. As such, some of these motors can be ruled out as key to amplification of mechanical motions in the cochlea simply because they are too slow. The mammalian cochlear amplifier is extremely fast and capable of cycle-by-cycle action, in some species at frequencies exceeding 50 kHz [16], [17]. This rules out mechanisms that require cyclic phosphorylation, transport and/or protein synthesis. In non-mammalian species, that do not have OHCs or the protein prestin, bundle-based motors underlie the active amplification process [18], [19]. In mammals, the evidence indicates OHC somatic motility is a key contributor [20]–[24], and this is the motor we focus on here. OHC somatic electromotility is driven by the MET current entering the cell and likely draws thermodynamic power from the electo-chemical potential between fluid compartments in the cochlea. The apical surfaces of OHCs are bathed in high-potassium endolymph, biased to approximately +50 to +80 mV, and their basal poles bathed in high-sodium perilymph at 0 mV reference. This endocochlear potential is maintained by the stria vacularis and associated cells [25]–[27]. When the hair bundle is displaced and MET channels open at the tips the stereocilia, ionic currents (primarily K+ and Ca2+) are driven into the OHC. A fraction of this MET current enters the apical face of the soma at the base of the stereocilia. In the absence of phosphorylation, it is likely that this current carries the thermodynamic electrical power input that drives the OHC mechanical power output. Here, we analyze how this electrochemical energy is converted into useful mechanical work by somatic electromotility using the model illustrated in Fig. 1. The current model is fundamentally piezoelectric in nature and extends concepts developed by Iwasa [28], [29] to address frequency-dependent power conversion efficiency. New results include the force vs. velocity, and power vs. velocity curves for OHCs (c. f. skeletal muscle cells [30]), and the frequency-dependent power efficiency that arises from intrinsic limitations on impedance matching between the cell and the load. Results indicate that OHCs are broadly tuned to have maximum power efficiency at a best frequency, thus contributing to tuning and the place principle in the cochlea. Furthermore, results provide an interpretation of how efferent activation may directly attenuate and de-tune the power output of OHCs and thereby providing a means for the brain to command exquisite control over the cochlear amplifier in a frequency dependent manner.
Dissipative drag from the cytoplasm and the extracellular space are unavoidable. As a first approximation we modeled the axial component of the drag acting on the plasma membrane using a version of the Navier-Stokes equations. Assuming small displacements from the resting configuration, and ignoring the convective nonlinearity, the Navier-Stokes equations reduce to (20) where is the density of the fluid, r is radial coordinate, is the effective viscosity, and is the axial velocity. To approximate the visco-elastic properties of the materials, we used a complex-valued viscosity of the form, where, is the frequency, is a material constant, and the is a parameter that determines the relative contributions of viscosity vs. elasticity of the material. When this model reduces to the standard Newtonian viscous fluid and when this reduces to the standard shear elastic solid. For biological materials ζ falls between these two extremes – e. g. for the tectorial membrane [55]. These equations account for both the visco-elastic drag and entrained fluid mass. We solved the equations to obtain the velocity field resulting when a cylinder oscillates in the axial direction with displacement. Having the velocity field, we computed the axial shear stress acting on the cylinder wall per unit axial displacement (21) where are Hankel functions, is the non-dimensional Womersley number (complex-valued), and a is the cylinder radius. With this, the damping parameter appearing in the momentum equation (Eq. 8) is. This model is approximate, but matches the viscous analysis of Tolomeo and Steel [43] if the length of the cell is much longer than the diameter, motions are axial, and the viscosity is strictly real valued, i. e. . Model parameters were estimated from known dimensions and physical constants combined with voltage clamp and mechanical data shown in Figs. 2–3 as well as microchamber data in Fig. 4. All other results (Fig. 5–8) and voltage clamp data in Fig. 4 are model predictions and the associated data were not used to estimate parameters. The model uses a reference thickness to describe the multi-component composite lateral wall and it is important to note that some parameters cannot be independently separated from this reference thickness (e. g. , , appear as groups). Coefficients appearing in the cable equation were computed from the physical parameters listed below using: , , and. Coefficients appearing in the wave equation were computed using, and. Dimensions were based on OHCs from the guinea pig cochlea. Data in Fig. 2–3 were used to find the effective stiffness, piezoelectric coefficient, electrical permittivity and conductance of the membrane. These data are for relatively low stimulus frequencies where the intracellular axial resistance has negligible effect on the results. To estimate the axial resistance we used the corner frequency where the capacitance measured at the basal pole of the cell begins to roll off (Fig. 3). The fraction of the membrane occupied by the motor was set to 80% () and the passive component to 20% (). The overall cell compliance was estimated from the slope of the compliance vs. cell length reported by Frank et al. [48], reproduced in Fig. 2C, using as well as the gain reproduced in Fig. 4 (solid, microchamber curve). An iterative optimization routine was run to refine the initial estimates of, and to simultaneously fit data in Fig. 2–4. Specific optimized numerical parameters include: OHC radius a = 4. 5e-6 m; composite mechanical stiffness C* = 1. 4e6 N/m2 (based on, and); plasma membrane conductance; apical face membrane conductance; basal membrane conductance; transduction current gain; composite reference thickness; OHC length; length of the active lateral wall was, and was set by requiring passive basal pole to have a passive capacitance of 7 pF; intracellular axial resistance ri = 5. 76e10 Ohm/m; composite piezoelectric coefficient at rest (C/m2) at rest; plasma membrane area specific capacitance; density; transduction current adaptation time constant; fluid viscosity; and fractional viscosity coefficient. We note that the mixture fraction is not uniquely determined by currently available data and it is possible to find alternative mixture fractions and stiffness parameters that result in the same composite stiffness C*. Nevertheless, it was necessary to use a value of to simultaneously fit all of the data and explain the magnitude of voltage dependent capacitance under unloaded and zero strain conditions. Additional experiments, perhaps involving voltage-dependent capacitance measurements under controlled mechanical loads, have the potential to resolve this ambiguity and reveal more about the lateral wall motor, but are not necessary for the purpose of the present power analysis since the composite parameters would not change. Experimental procedures and animal care were designed to advance animal welfare and were approved by the Baylor College of Medicine animal care and use committee. All physical parameters were deduced from the published literature, with the exception of the intracellular electrical resistance, . To estimate, we isolated OHCs from the guinea pig cochlea [56] and examined the frequency dependence of the input electrical impedance under whole-cell voltage clamp (Axopatch 200 B, Molecular Devices, Sunnyvale, CA). OHCs were harvested from euthanized guinea pigs. Cells were patch-clamped at the base with quartz pipettes covered with Sylgard, and hyperpolarized to minimize the voltage-dependent nonlinear capacitance. K+ and Ca2+ ion channels were blocked with the addition of (C2H5) 4N (Cl), CsCl and CoCl to the bathing and/or pipette solutions [57]. The input admittance was determined with a single sinusoidal voltage (0. 015 V peak to peak, 90–3200 Hz) superimposed on top of a −0. 13 V holding potential after correcting for the inherent phase shifts of the amplifier [58]. 210 measurements were averaged at each frequency. The resistance and capacitance were calculated from the input admittance [57] accounting for the series resistance (∼6 Mohm, remained constant throughout experiment). Experiments were conducted at room temperature.
There are four major observations that can be drawn from the present work. The first addresses how OHCs operate at high frequencies given their electrical capacitance [9], [62], [81]–[83]. Capacitance is thermodynamically conservative and present results confirm that the ability of OHCs to supply mechanical power to the cochlea is not limited by electrical capacitance [84], even at frequencies much higher than the membrane time constant (e. g. Fig. 7). This is true because capacitance is not dissipative. Instead, present results suggest the most serious factor that may limit power output by OHCs is how well the “impedance” of the hair cell is matched to that of the cochlear partition (e. g. Fig. 6). OHCs driving against an excessively stiff cochlear partition, for example, would be inefficient. The second observation is that OHCs may be tuned to maximize their power output at a best frequency, albeit broadly tuned. Although OHC displacement and force are quite flat over a broad range of frequencies when driven by voltage (e. g. Fig. 5, present model and published data [48]), OHC power output is tuned when one considers the mechanical power output relative to the electrical power input. The predicted tuning is dependent upon cell length and correlates with the cochlear place principle [78], thus indicating that tuning of OHCs may contribute to the sharp mechanical and afferent neural tuning in the living cochlea. The third observation addresses how the MET channels would be expected to further tune output of the somatic motor. MET adaptation generates high-pass filtered MET currents [47], [66], [85], [86]. Since the filtering is upstream of the somatic motor it would further sharpen tuning of OHC somatic motor output by attenuating low-frequency amplification. In the context of the organ of Corti, MET adaptation would also be expected to alter the phase of the OHC force possibly to maximize power input to the cochlea near the best frequency [67] and, additionally, might introduce a non-optimal phase that would sharply attenuate cochlear gain at both low and high frequencies. Because of these factors, the influence of tuning in isolated OHCs on tuning curves in the cochlea would be expected to be even more significant than implied by the OHC motor efficiency alone (Fig. 7). The fourth observation is that OHC somatic power output may be controlled by the brain via efferent activated ionic conductance (s). The model predicts that increasing the conductance of the basal pole would reduce OHC power output and tuning, thus providing a plausible explanation for a fast mechanism that may be used by the brain to control both sensitivity and frequency selectivity of hearing (e. g. Fig. 7). Finally, it is important to note that the OHC somatic motor is not present in non-mammals, yet these animals also exhibit many of the properties of the mammalian cochlear amplifier [87], [88]. The MET apparatus itself is clearly a key contributor to hair bundle motility and amplification [9], [89]. In addition, there is an MET-independent component of hair bundle motility driven by voltage [10]. This voltage-dependent component has analogy to the somatic motility addressed here, and may be involved in tuning and the power stroke of hair bundle motility with potential relevance to active bundle amplification in high frequency hearing organs [90]. These hair-bundle features occur upstream of the somatic motor and the two clearly interact with each other via micromechanical environment and electrical fields [91].
The sense of hearing is exquisitely sensitive to quiet sounds due to active mechanical amplification of sound-induced vibrations by hair cells within the inner ear. In mammals, the amplification is due to the motor action of “outer hair cells” that feed mechanical power into the cochlea. How outer hair cells are able to amplify vibrations at auditory frequencies has been somewhat of a paradox given their relatively large size and leaky electrical properties. In the present work, we examined the power conversion efficiency of outer hair cells based on first principles of physics. Results show that the motor is highly efficient over a broad range of auditory frequencies. Results also show that the motor is likely controlled by the brain in a way that allows the listener to focus attention on specific frequencies, thus improving the ability to distinguish sounds of interest in a noisy environment.
physiology/sensory systems
neuroscience/sensory systems
computational biology
Power Efficiency of Outer Hair Cell Somatic Electromotility