Kinetic Membrane Model of Outer Hair Cells.

The effectiveness of outer hair cells (OHCs) in amplifying the motion of the organ of Corti, and thereby contributing to the sensitivity of mammalian hearing, depends on the mechanical power output of these cells. Electromechanical coupling in OHCs, which enables these cells to convert electrical energy into mechanical energy, has been analyzed in detail using isolated cells using primarily static membrane models. The mechanical output of OHCs was previously evaluated by developing a kinetic theory based on a simplified one-dimensional model for OHCs. Here, a kinetic description of OHCs is extended by using the membrane model, which was used for analyzing in vitro experiments. This theory predicts, for systems without inertial load, that elastic load enhances positive shift of voltage dependence of the membrane capacitance because of turgor pressure. The effect of turgor pressure increases with increasing elastic load. For systems with inertia, the magnitude of mechanical power output could be ∼5% higher than the value predicted by the one-dimensional model at the optimal turgor pressure. Published by Elsevier Inc.

Significance

This work is an attempt to develop a physical model to clarify the mechanism of outer hair cells in performing their role as an amplifier in mammalian hearing. Specifically, this work extends a static model of these cells into a dynamic one to evaluate mechanical power production, which is essential for the function of these cells. It clarifies the assumptions essential for an earlier phenomenological theory, a one-dimensional model. In addition, it describes the effect of turgor pressure on mechanical power generation.

Introduction

Outer hair cell (OHC) motility, often referred to as electromotility, is essential for the sensitivity, frequency selectivity, and dynamic range of the mammalian ear (1). This motility is piezoelectric (2, 3, 4, 5), based on a membrane protein SLC26A5 (prestin) (6), and it is driven by the receptor potential generated by the sensory hair bundle of each cell. Even though the biological role of this motility has been confirmed by replacing it with its nonfunctional mutants (7), the mechanism with which OHC motility plays this role has not been fully clarified.

This problem involves two factors. One is the speed of conformational transitions of prestin, the motile molecule, and the other is the magnitude of the receptor potential, which drives the motile mechanism.

First, prestin’s conformational transitions, which are elicited by voltage changes, accompany charge transfers. This charge movement contributes to the membrane current, together with the regular capacitive current of the plasma membrane. Because the conformational transitions depend on the operating point, the resulting capacitance is voltage dependent. For this reason, it is often referred to as nonlinear capacitance. The frequency dependence of nonlinear capacitance gives rather low characteristic frequencies ∼15 kHz (8) and 3 kHz (9).

Conformational transitions are also monitored as the current noise spectrum (10). Sealed patches formed on OHCs show a characteristic frequency of 35 kHz, higher than ∼15 kHz for nonlinear capacitance (11). This discrepancy suggests different modes of motion. These observations indicate that those characteristic frequencies likely reflect the mechanical relaxation processes rather than the intrinsic transition rates of the motile molecules.

This interpretation is reinforced by cell displacements elicited by voltage waveforms applied through a suction pipette. In one mode of experiments, the frequency range observed was 8.8–6 kHz or lower (9,12). In quasi-isometric conditions, however, the response extends up to 80 kHz (12).

Second, the intrinsic electric circuit of the cell attenuates the receptor potential at auditory frequencies. For the membrane potential to change, an electric current needs to charge up or down the capacitor, which is formed by the plasma membrane. This factor increasingly attenuates the receptor potential with increasing stimulation frequency, with the roll-off frequency lower than the operational frequency (13).

Various hypotheses have been proposed to address the issue of this voltage attenuation (14, 15, 16, 17, 18). Mechanical load on OHCs is a possible mechanism for reducing the membrane capacitance (19, 20, 21, 22, 23). Specifically, elastic load reduces nonlinear capacitance (20), and inertial load can eliminate the total membrane capacitance near resonance (21). Thus, attenuation of the receptor potential can be avoided near resonance frequency.

The compatibility of OHC electromotility with their biological role could be tested by comparing the optimal power production of the cell with power dissipation in the cochlea (24,25). Such an examination was attempted by using a simple model, which describes the mechanical property of the cell only by the length and the axial stiffness (one-dimensional (1D) model of OHCs for brevity) (20) and assuming the energy loss due to the shear in the space between the tectorial membrane and the reticular lamina, which is essential for stimulating hair cells. This study showed that electromotility is compatible with their biological role up to 10 kHz (21).

This work develops a kinetic theory based on the membrane model (26) and examines the physical basis of a more phenomenological treatment (20). It also predicts the effect of turgor pressure, which could play a modulatory role. Besides power production, the amplifier gain of OHCs is discussed. This approach could facilitate extension to a more complex theory, in which the motile element has more than two states to account for anion sensitivity and slower relaxation processes (27, 28, 29, 30).

Methods

The system

Here, we consider a system in which an OHC has a mechanical load consisting of viscous, elastic, and inertial components (Fig. 1; Table 1). This model system provides an OHC with the simplest possible environment so that its performance can be examined. The biological role of this cell could be inferred from this examination.

Figure 1

The membrane model and mechanical load. (A) An OHC (red rectangle) is subjected to mechanical load, consisting of viscous η (left), elastic K (middle), and inertial m (top) components. The stiffness of the cell is k. In its lateral membrane, the cell has a membrane protein, which undergoes conformational transitions between two states. (B) The membrane model of OHC is shown. The cell body of an OHC is approximated by a cylinder of radius r and length L. Motile elements (upper right), which undergo area changes a in the axial direction and a in the circumferential direction, are uniformly embedded in the cylindrical plasma membrane at density n. The conformational transitions involve transfer of electric displacement q across the membrane. To see this figure in color, go online.

Table 1

Parameters of the Membrane Model

Notation	Definition	Value Used	References
d1	axial modulus	0.046 N/m	(35)
d2	circumferential modulus	0.068 N/m	(35)
g	cross modulus	0.046 N/m	(35)
az	axial area change	−4.5 nm2	(31)
ac	circumferential area change	0.75 nm2	(31)
q	mobile charge	−0.8 e	(2)
n	density of motile element	9 × 1015/m2	(31)
r	radius	5 μm	(35)
L0	cell length	25 μm
C0	structural capacitance	10 pF	a
η	drag coefficient	160 nNs/m	(21)
Notation	Meaning	Definition
κ	axial modulus	d1 + d2/4 − g
k	axial stiffness	2πrκ/L0
K	elastic load	2πrKe/L0
γ	operating point parameter	β〈C〉(1−〈C〉)
ωr	resonance frequency	(k+K)/m	b
ωη	damping roll-off	(k + K)/η	b
ω¯	reduced frequency	ω/ωr
a1	unitary length change	A/(4πrκ)

The definitions of other parameters are a = −(a + 2a), A = −(2d1 − g)a + (d2 − 2g)a, φ = d1d2 − g2,  = K/(k + K), α2 = 1 + γn(A2 + φa2), and ζ = γNq2/C0. Of these parameters, , α, and ζ are dimensionless. i0 is the ac component of transducer current. The parameter values of the motile element reflect that the extended state E is taken as the reference. e, the electronic charge.

The experimentally obtained regular capacitance of an OHC is close to estimates based on the geometrical surface area with the standard value ∼1 μF/cm2 for specific capacitance.

Angular frequency.

Let us assume that the lateral wall is uniform and that the cell maintains its characteristic cylindrical shape. The latter assumption could be justified for low frequencies, where inertial force is not significant compared with elastic force within the cell. The limit of the validity is examined later in Results and Discussion, after experimental values for material properties are provided.

In addition, we consider movement of this cell in response to small fast changes in the membrane potential, from an equilibrium condition, ignoring metabolic processes that maintained the physiological condition. Changes in pressure, if present, are assumed gradual and therefore have only a modulatory role. The volume of the cell is assumed constant during voltage changes.

Electrical connectivity

Here, we initially assume that the membrane potential of the cell is controlled by an extracellular electrode and an intracellular electrode with low impedance to facilitate the evaluation of the membrane capacitance. Later on, for evaluating the power output of an OHC, the mechanotransducers at the hair bundle will be incorporated. The receptor potential that drives the motile element in the OHCs depends on the membrane capacitance as well as the current source.

Static properties of OHCs

First, consider an OHC in equilibrium. The shape of an OHC is approximated by an elastic cylinder of radius r and length L (Fig. 1 B). Small displacements of the cell can be described by constitutive equations (31),andwhere the first equation is force balance in the axial direction and the second force balance in the circumferential direction. The quantities ɛ′ and ɛ′ are respectively small elastic strains of the membrane in the axial direction and the circumferential direction. The quantities d1, d2, and g are elastic moduli of the membrane in the axial direction, circumferential direction, and cross moduli, respectively; P is the internal pressure, and f is the axial tension due to an external force F, which can be expressed as 2πrf. These elastic moduli assume orthotropy (32,33).

Axial stiffness

The material stiffness k in the axial direction and the axial elastic modulus κ of the cell are expressed as

where L0 is the resting length of the cell. The axial modulus κ is obtained as κ = f/ɛ′ under constant volume condition ɛ′ = ɛ′ + 2ɛ′ = const., eliminating ɛ′ and rP from the constitutive equations ((1a), (1b)). It can be expressed as (see Appendix A)

This axial stiffness of the cell is material stiffness without a motile element, which would reduce the stiffness in a manner similar to “gating compliance” (31,34).

Motile element

Here, we assume that each motile element undergoes transition between two states, compact (C) and extended (E). In the following, the fraction of the state is expressed in italics.

Unlike conventional description of molecular transitions, we do not assume that transitions between these states take place in accordance with “intrinsic” transition rates. Instead, we assume that transitions between these states are determined by the mechanical constraints in a manner similar to piezoelectricity.

To incorporate the motile element, let us assume that the total strain in the axial direction ɛ and the one in the circumferential direction ɛ consist of elastic components (ɛ′ and ɛ′) and the contribution of motile elements. Each motile element undergoes electrical displacement q and mechanical displacements a and a (see Fig. 1 B) during the transition from state E to state C. If the motile elements are uniformly distributed in the lateral membrane at density n, the total strains in the two directions are then respectively expressed by (31)and

The constitutive equations are then rewritten byand

Notice that the internal pressure P now consists of two components, dependent on the activity of the motile element and independent of it. The latter is referred to as turgor pressure.

In the presence of external elastic load K, axial tension f can be expressed bywhere K = KL0/(2πr). (5a), (5b) and 6 lead towhere A = −(2d1 − g)a + 2μa and μ = d2/2 − g (see Appendix A). The quantity ɛ is the volume strain, which can be expressed as ɛ + 2ɛ for the cylindrical cell for small strains.

Because the displacement of the cell of our interest is in the auditory frequency range, we assume the cell volume is constant. For this reason, we can regard ɛ as a parameter representative of turgor pressure, which does not depend on the activities of the motile elements and can only change slowly responding to metabolic activity or osmotic pressure.

The axial displacement Z (= L0ɛ) of the cell and the total charge of the motile elements Q, respectively, can be expressed using C, the fraction of state C, byandwhere N is the total number of motile elements, i.e., N = 2πrL0n.

Boltzmann distribution

The fraction of state C in equilibrium should be given by a Boltzmann functionwithwhere ΔG is the energy difference (of state C from state E), V the membrane potential, and β = 1/kT with Boltzmann’s constant k and the temperature T. The voltage V0 determines the operating point. By substituting f and rP, we obtain (see Appendix A)where shorthand notations areand

The volume ɛ is due to turgor pressure P. At the extreme hyperpolarization, when the motile element is in the extended state (i.e., C = 0), P = ɛφ/(2κr).

Equation of motion

If C = C∞, i.e., the distribution of motor states is in equilibrium, the cell does not move. Suppose the membrane potential V changes abruptly; the cell exerts force kπrAn(C∞ − C) because a change ΔC in C gives rise to length change πrAnΔC of the cell because of the cylindrical geometry if ɛ is kept constant. Here, k is the material stiffness of the cell defined by Eq. 3. With mechanical load including mass m and drag coefficient η (see Fig. 1 A), the equation of motion can be expressed aswhere C∞ is the value of C, which satisfies Boltzmann distribution, for the given condition at time t. With the aid of Eq. 8a, this equation can be rewritten in the formwhich is intuitive, considering that z is proportional to C. This equation determines the rates of transitions between the two states because C is the only variable, given C + E = 1.

This is the same equation derived for the one-dimensional model (20), except that the variable C in this membrane model depends on a larger number of factors, including turgor pressure. The property of the OHC as expressed by the equation cannot be rendered as a simple combination, either series or parallel, of an elastic element and a displacement element.

In the absence of the inertia term, Eq. 14 has a typical relaxation equation for a stochastic process. One may question the inertia term in an equation that describes a stochastic process. It will be shown later that this equation with the inertia term is consistent with the equation of piezoelectric resonance provided that the deviation from equilibrium is small (see Results and Discussion).

Small harmonic perturbation

Because one of the main functions of OHCs is to amplify small signals, the response of OHCs to small harmonic stimulation is of special interest. Assume that the voltage consists of two parts, a constant term and small sinusoidal component with angular frequency ω (= 2πf) and amplitude v:

Then, C and C∞ should also have two corresponding componentsandand the first-order terms of the equation of motion turn intowith  = (k + K)/m and ω = (k + K)/η, andwith γ = β. Thus, quantity c obeys the equationwith α2 = 1 + γn(A2 + φa2)/(2κ).

In response to voltage changes, axial displacement Z likewise consists of two components:and

The second equation is derived from Eq. 8a for constant volume strain ɛ.

Nonlinear capacitance

If we express corresponding changes in a similar manner, the charge variable can be expressed as Q = nq + Nqcexp[iωt], and nonlinear capacitance C is given bywhere Re[…]represents the real part because capacitance is charge transfer synchronous to voltage changes (20). Here, shorthand notations are introduced: . Nonlinear capacitance can turn negative near resonance frequency and is capable of nullifying the membrane capacitance, as in the case of the 1D model (21) (see Fig. 4 A).

Figure 4

The membrane capacitance and the power output of an OHC. (A) The membrane capacitance is plotted against the reduced frequency ω/ω. The set of plots is generated by increasing the external elastic load K. The values of K correspond to 0.1k, 0.5k, k, and 2k (from the left to right). Solid red: membrane model. Dashed blue: 1D model. (B) Power output function is plotted against frequency. The set of plots are generated by increasing K in a manner similar to (A). (C) Frequency of maximal power output against the frequency of zero membrane capacitance is shown. The value of K corresponds to, from left to right, 0.1k, 0.25k, 0.5k, k, and 2k. The broken black line indicates equal frequency. (D) Power output based on the membrane model (W: solid red circles) and on the 1D model (W1: open blue circles) is plotted against (= K/(k + K)). W0 = η(a1Nqi0)2/(2π). (E and F) Ratio of power output of the membrane model to that of the 1D model is shown. (E) Dependence on the elastic load is given. The numbers in the ascending order correspond to ɛ = 0 (labeled 0, black), 0.03 (3, black), 0.05 (5, red), 0.08 (8, blue), and 0.1 (9, blue). (F) Dependence on the volume strain ɛ is shown.  = 0.1 (solid black circles), 0.5 (solid red squares), 0.67 (open green upside-down triangles), and 0.8 (open blue triangles). In (A)–(D), ɛ = 0.065. The drag factor ω/ω is assumed to be 5 at  = 0. To see this figure in color, go online.

Comparison with one-dimensional model

How do the predictions of the membrane model differ from 1D models? In the following, the 1D model (20,21) is briefly restated to facilitate the comparison.

One-dimensional model

A 1D model has a single parameter k for the cell’s elasticity and a single parameter a1 for mechanical changes of the motile elements. For length changes z and charge transfer Q, the equation that corresponds to Eq. 8a can be written down (20) asandwhere C represents the fraction of the compact state, as in the membrane model. The comparison of Eqs. 8a and 22a suggests a1 corresponds to A. The free energy difference ΔG1 for the 1D model can be expressed by (20,21)

This energy difference ΔG1 determines C∞ = exp[−βΔG1]/(1 + exp[−βΔG1]) and contributes to the factor γ1 = β in the equation of motion. This γ1 does not depend on turgor pressure—unlike γ, its membrane model counterpart—because ΔG1 does not have a term that depends on turgor pressure.

The equation for c for the 1D model iswith , which is independent of turgor pressure.

Similar to the derivation of Eq. 21, nonlinear capacitance is given by Re[Nqc/v]. Now that c given by Eq. 24, nonlinear capacitance C1 of the 1D model is expressed bywhich can take negative values near resonance frequency (21).

Correspondence between the two models

The charge transfer q is identical in the two models. The density n of the membrane model is related to N by n = N/(2πrL0). The relationship between the mechanical factors can be obtained by comparing Eq. 22a with Eq. 8a. These two equations, together with Eq. 2, lead to the expression of unitary length change a1

Power output and amplifier gain

The analysis above shows that the membrane model and the 1D model lead to similar equations for c. The relation between z and c is the same for both cases because a1kN/(k + K) = An/(κ + K). The difference between the two models originates only from γ and α2, which, respectively, differ from their counterparts γ1 and (Table 2). For this reason, the expression for the 1D model will be used in the following with separate definitions for γ and α2 for the membrane model and their counterparts γ1 and for the 1D model.

Table 2

Correspondence of Parameters in the Membrane Model and in the 1D Model

1D Model	Membrane Model
a1	A/(4πrκ)
k	2πrκ/L0
N	2πrL0n
ΔG1	ΔG
α12=1+γ_1Na12kKˆ	α2 = 1 + γn(A^2Kˆ+φa^2)/(2κ)

Power output

Under physiological conditions, energy output from an OHC depends on the receptor potential v, which is generated by the transducer current elicited by a relative change in the hair bundle resistance. This potential depends on the intrinsic circuit property of the cell as well as charge movement due to changes in c, which can be expressed by (20)where i0 is the steady state current, σ the conductance of the basolateral membrane, and C0 the structural membrane capacitance of the hair cell.

The combination of Eqs. 19 and 27 can be written down in the form

For relatively high frequency, at which ionic currents are overwhelmed by displacement current ωC0, we obtainwith ζ = γNq2/C0.

Now, recall that axial displacement z is related to c with Eq. 20b. Energy output from an OHC has two components. One is elastic energy, (1/2)kz2 per half cycle, which is recovered at the end of a cycle. The other is dissipative energy, (1/2)ηω|z|2 per half cycle, which results in power output W, which is given by

Here, the factor k/(k + K) is due to impedance matching. The factors α2 and ζ introduce turgor pressure dependence. C0, the structural membrane capacitance, should increase with increasing turgor pressure. This issue will be examined later in Results and Discussion.

Power generation W(ω)for the membrane model is closely related to its counterpart W1(ω)for the 1D model (20,21). The only difference is from the expressions of free energy in the Boltzmann function, resulting in choosing the set γ and α2 for the membrane model or γ1 and for the 1D model. Although the membrane model is sensitive to turgor pressure, the 1D model is not.

Amplifier gain

The power gain G of an amplifier is the ratio of output power W(ω) against input power Win(ω), where Wout is already expressed by Eq. 30 for a given value of , the relative change in the apical membrane resistance due to hair bundle stimulation.

A relative change in resistance can be associated with a displacement x of hair bundle tip with , where is the sensitivity of the hair bundle. Because the tips of hair bundles are embedded in the tectorial membrane, the shear between the reticular lamina and the tectorial membrane can also be represented by x. If we can assume that the shear of the subtectorial space x is proportional to the displacement z of the displacement of OHC cell body, we can put x = λz, where λ is a constant. These relationships lead to x = and

Here, we note that the drag coefficient η consists of two components. One is associated with the movement around the cell that is needed to stimulate the cell. Another is associated with the movement of the hair bundle, for which gating of the transducer channel makes the dominant contribution to drag (36). The drag coefficient η is the same for W(ω)and Win(ω) for a given mode of the movement.

The power gain G(ω) can then be expressed by

Equation 32b shows that power gain G(ω) is a product of two factors. One part depends on the reduced frequency (= ω/ω). The other depends on the mechanical resonance frequency ω. For a given value of ω, the condition G(ω) = 1 may not be satisfied if the mechanical resonance frequency ω is too high.

The limiting frequency , which satisfies G = 1, is of special interest because it shows the frequency range in which OHCs can be effective. It has been shown for the 1D model that the frequency is somewhat higher than the corresponding resonance frequency (21).

Amplifier gain at a given location can be related to the limiting frequency . Let the mechanical resonance frequency at location 2 be ω. Then, the amplifier gain G(ω2) at that location can be expressed by Eq. 32b, with ω replaced by ω and as ω2/ω instead of .

Then, we have G(ω2) = 2 if η2/ω = /ω, where ω2 is the best frequency at location 2. η2 and are, respectively, the drag coefficients of location 2 and at the location of the limiting frequency. If location 2 is more apical, ω < , the drag coefficient should satisfy η2 ≤ because the subtectorial gap, which makes a significant contribution to the drag coefficient, is wider for a more apical location. This condition makes the resonance peak sharper at the location. For this reason, the amplifier gain at best frequency ω2 satisfies G(ω2) > .

Inertia-free condition

In the absence of the inertia term, the power output turns intowhich is a monotonically decreasing function of the frequency ω. As Eq. 32a indicates, the power gain G(ω)is a steeply decreasing function of the frequency (1/ω2 at low frequencies and 1/ω4 at high frequencies).

Near resonance

For the system with inertia, the power output has a peakat . Because α2 > 1, such a peak exists if  < 1. However, power production W is a decreasing function of the frequency ω for overdamped systems, in which is large.

These equations for power production are essentially the same as those for the 1D model that has been studied earlier (20,21). The difference is in the definition of α2 and ζ even though these factors are similar.

Results and Discussion

Numerical examination

The membrane model and the 1D model lead to parallel expressions for mechanical and electrical displacements, which in turn lead to nonlinear capacitance and power output. The difference in the two stems from the difference in ΔG and ΔG1. Because the results of the 1D model have been elaborated (21), our focus is whether or not the membrane model leads to different results, using a set of parameter values that have been experimentally determined.

Nonlinear capacitance and factor γ

The operating point factor γ, which contributes to α2 and ζ, is affected by both turgor pressure and external elastic load through ΔG (see Eq. 10). This sensitivity is reflected in nonlinear capacitance in the low frequency limit (Fig. 2).

Figure 2

Nonlinear capacitance at low frequency. The volume strain ɛ is 0 for (A) and 0.10 for (B). Traces represent elastic load (= K/(k + K)): 0 (dotted, black), 0.01 (dashed, blue), and 0.1 (red). (C) Contour plot of peak voltage shift is shown. The abscissa represents . The ordinate axis is volume strain ɛ, which represents static turgor pressure. Voltage shifts are color coded (blue: negative, red: positive), and the values (in mV) are shown in boldface letters in the plot. To see this figure in color, go online.

Increasing external elastic load broadens the voltage dependence as well as shifts the peak in the positive direction (Fig. 2, A and B). An increase in turgor pressure, represented by ɛ, shifts the peak voltage of nonlinear capacitance in the positive direction. This effect of turgor pressure is consistent with earlier studies, both theoretical (2,31) and experimental (2,37,38). In addition, turgor pressure increases the sensitivity of nonlinear capacitance on the elastic load in both shifting the peak as well as broadening of the dependence (Fig. 2 B). A contour plot of peak voltage shift summarizes the dependence on both turgor pressure and elastic load.

The power output of an OHC has been described using the 1D model (20,21). Here, we focus on the issue as to how the predictions of the membrane model compare with those of the 1D model for the given set of the parameters.

Inertia-free condition. Under the inertia-free condition, power output is a monotonic decreasing function of frequency as described by Eq. 33. The zero-frequency asymptotes are γ/( + ζ) for the membrane model and γ1/( + ζ1) for the 1D model. The ratio of these zero-frequency asymptotes is plotted in Fig. 3 A.

Figure 3

Ratio of coefficients for power output of this membrane model to those of the 1D model. (A) The ratio of power output of the membrane model to that of the 1D model at the zero-frequency asymptote is shown. (B) The ratio at high frequencies is shown. Traces correspond to  = 1/11 (dotted), 1/5 (short dashes), 1/3 (medium dashes), 1/2 (long dashes), and 2/3 (full line). To see this figure in color, go online.

For high frequencies, the power output declines proportionally to (1/ω)2. The coefficients are proportional to γ2. The ratio of the coefficient for the membrane model to that for the 1D model is plotted in Fig. 3 B.

These ratios are very close to unity near K = k (blue traces) and deviate significantly for larger elastic load at both ends of the membrane potentials. However, these deviations are not significant near the resting level of the membrane potential (Fig. 3).

The analysis based on the 1D model indicates that the optimal elastic load K to for counteracting viscous drag is K ≈ k for physiological operating point near −50 mV (20).

The comparison shows that the power output of the membrane model is similar to that of the 1D model in the physiological membrane potential range. Outside of this voltage range, the membrane model predicts smaller power output than the 1D model. This smaller output is also dependent on the elastic load.

Near resonance

For the 1D model, nonlinear capacitance is negative near resonance frequency (21) and can make the total membrane capacitance negative (blue dashed traces in Fig. 4 A). The frequency of maximal power output (blue dashed traces in Fig. 4 B) is close to the frequency of zero capacitance in such cases (blue traces in Fig. 4 C).

The membrane model predicts a similar relationship between the membrane capacitance and power output (red traces in Fig. 4, A–C are for ɛ = 0.065). Power output predicted by the membrane model is slightly smaller than that of the 1D model for small load. However, the membrane model predicts larger power output for large load K (Fig. 4 D).

The difference between the two stems from the functional forms of ΔG (Eq. 10) and ΔG1 (Eq. 23). The second term in ΔG has factors that do not exist in ΔG1. These factors shift the transition point even when is small.

The ratio W/W1 of power output predicted by the membrane model to that of the 1D model depends on both turgor pressure and the elastic load (Fig. 4, E and F). The ratio is larger than unity for ɛ ≈ 0.05. For both small and large values of ɛ, the ratio decreases with larger load (Fig. 4 E).

For every fixed value of the elastic load, the power output ratio has a broad maximum (Fig. 4 F). The dependence on turgor pressure increases as elastic load increases. The ratio W/W1 is unity or higher in the range 0.04 < ɛ < 0.07. It is larger for larger loads (up to 1.07 for  = 0.8). The ratio is lower as load increases outside of this optimal range.

Performance of OHCs

For the 1D model, estimated power output of an OHC is between 0.1 and 10 fW near resonance frequency (21). The effectiveness of OHCs under in vivo conditions was estimated by evaluating a limiting frequency at which the power output of the OHC was equal to the viscous loss, assuming that the major contribution is from the gap between the tectorial membrane and the reticular lamina (25,39,40). This frequency is constrained by two factors, resonance frequency and impedance matching; higher resonance frequency requires stiffer elastic load, which leads to poorer impedance matching for power transfer. This evaluation led to the limit of ∼10 kHz if OHC is directly associated with the motion of the basilar membrane (21). To support higher frequencies, OHCs need to be associated to smaller elastic load and smaller mass, requiring multiple modes of motion in the organ of Corti (21). The slightly larger (∼5%) power output predicted by the membrane model leads to a slightly higher limiting frequency.

The role of turgor pressure

The predicted dependence of power output of OHC on turgor pressure (Fig. 4 D) raises a number of interesting questions. What is the range of turgor pressure in vivo? How much turgor pressure can change? Can it function as a control parameter of the cochlear function?

Power output has a plateau with respect to ɛ at about ɛ = 0.05 (Fig. 4 F), which corresponds to a static axial strain of ɛ = −0.018 under load-free conditions. This strain is ∼36% of the maximal amplitude of electromotility (5% of the cell length). The corresponding turgor pressure is 0.14 kPa. It is probable that the physiological turgor pressure could be lower than this value, as is often the case for biological functions. Such an operating condition allows gain control by the parameters.

However, turgor pressure is not a simple control parameter of power output because changes in turgor pressure accompany shifts of the operating voltage (see, e.g., Fig. 2), even though its effect on the maximal power output could be as large as 50% for large elastic load (Fig. 4 F).

Validity of the cylindrical model

It was assumed in the beginning that the cylindrical shape of the cell is maintained as the cell is driven by changes in the membrane potential and undergoes deformation. The conservation of the cylindrical shape of the cell during motion of the OHC requires that the elastic force of the membrane exceeds the inertial force of the internal fluid. Let x be the amplitude of the end-to-end displacement of a cylindrical cell of radius r and length L. This condition can be expressed bywhere κ is the elastic modulus of the cell (in the axial direction), ρ the density of the internal fluid, and ω the angular frequency. The inequality can be expressed by defining a frequency ωbal, at which these two factors are balanced, as

This expression is intuitive in that a smaller displacement, a decrease in cellular dimension, and an increase in the elastic modulus favor the elastic force over the inertial force.

The experimentally obtained value for 2πrκ is 510 nN per unit strain, and it is reasonable to use the density of water (103 kg/m3) for the density ρ. The radius r is 5 μm and the length L 10 μm for a basal cell. If we assume the amplitude x is 1 nm, an approximate magnitude under in vivo condition, the limit can be expressed by the linear frequency∼40 times higher than ∼100 kHz for high frequency mammals such as bats and dolphins. The condition is even more favorable for more apical cells because fbal decreases with regard to the square root of 1/L, whereas the best frequency decreases much more steeply.

This means we can reasonably assume that relative motion of the internal fluid against the plasma membrane can be ignored and that the main mode of cell deformation is elongation and contraction while keeping the cylindrical shape.

The inertia term

The derivation of the equation of motion (Eq. 17) may not appear legitimate in that it introduces the inertia term to a stochastic equation. However, it turns out to be consistent with a standard expression for the admittance of a piezoelectric system.

The standard expression for the admittance Ype of a piezoelectric resonator can be (41)using an equivalent electric circuit with inductance L and resistance R. That implies the correspondence to mechanical resonance system  = 1/(CL) and ω = 1/(RC), leading to

Equation 39 is equivalent to Eq. 19 because Y = iωNqc, and the zero-frequency limit indicates C = γnq2 and α = 1, which corresponds to K = 0 because the external spring does not exist for the piezoelectric element.

This comparison also illustrates a limit of validity for the equation of motion (Eq. 14). Although Eq. 39 for standard piezoelectricity does not depend on the operating point, Eq. 19 for OHC does through the linearization near the equilibrium condition though the operating point factor γ (= . The equation of motion Eq. 14 is valid only within a small range of the membrane potential, in which linearization can be justified.

Turgor pressure dependence of C0

Even though the ratio W/W1 evaluated is independent of C0, turgor pressure dependence of C0 is important for power production W. To my knowledge, experimental data are unavailable, presumably because of the technical difficulty in subtracting voltage-dependent nonlinear components from the total membrane capacitance.

The cylindrical model (Fig. 1 B) predicts, for our set of values for elastic moduli, the ratio ɛ/ɛ of surface strain to volume strain is ∼0.5 (see Appendix D in the Supporting Materials and Methods). If we assume that membrane thickness decreases with an area increase so that the membrane volume is held constant, the relative increase of C0 is twice the area increase. It is similar to the increase of ɛ. Thus, for a small change of volume strain Δɛ, the resulting change ΔW in power production can be expressed as ΔW/W ≈ −2Δɛ.

It is likely that OHCs have some turgor pressure to maintain their shape as well as their efficiency in power production. If we could assume that the resting value of ɛ is 0.05, power output at ɛ = 0.1 is 10% lower and at ɛ = 0 is 10% higher than the standard value as evaluated in an earlier section.

Drag coefficient

Drag is critical for power balance. It has three sources. One is the shear between the tectorial membrane and the reticular lamina (21,39,42). Another is the rest of the organ of Corti and the interior of the tectorial membrane because of their deformability (43). This treatment ignores this component because it is hard to evaluate. The third source is gating of the transducer channel in the hair bundle (36). The issue of drag within an OHC and at its outside surface is discussed in Appendices B and C in the Supporting Materials and Methods.

The drag coefficient due to the shear between the tectorial membrane and the reticular lamina is 160 nNs/m per OHC (21). The drag coefficient of the frog hair bundle is 340 nNs/m for gating of nearly all channels in the bundle (36). It would be reasonable to assume that for low stimulation intensity, the contribution of channel gating to the drag coefficient is less than 20 nNs/m, smaller than the gap drag, even though experimental values are unavailable for an OHC hair bundle.

Conclusions

The membrane model predicts nonlinear capacitance, cell displacement, and power output of OHCs relevant to in vivo conditions. In addition, these predictions are testable by in vitro experiments.

Nonlinear capacitance is sensitive to both turgor pressure and external elastic load. An increased elastic load reduces the peak height and broadens the voltage dependence of nonlinear capacitance. The peak voltage shifts in the positive direction with increasing turgor pressure and elastic load. That is intuitive because increasing internal pressure positively shifts the capacitance peak.

Power output depends on turgor pressure. The optimal power output is expected at ɛ ∼0.06. Under this condition, for ≥ 0.5, the maximal power output is at least 5% higher than the estimate based on the 1D model. However, power output is lower outside of the optimal range of turgor pressure, particularly with larger elastic load.

The membrane model confirms the main predictions of the 1D model; a single mode of vibration of the organ of Corti can be supported up to ∼10 kHz but, to cover the entire auditory range, cannot be supported without multiple modes of motion in the cochlear partition (21). This prediction appears consistent with recent observations with optical techniques in that the organ of Corti shows multiple modes of motion (44, 45, 46).