INTRODUCTION

TOP ABSTRACT INTRODUCTION PIEZOELECTRICITY TWO-STATE MEMBRANE MOTOR MEMBRANE MOTOR IN A... EXAMINATION OF EXPERIMENTAL... DISCUSSION CONCLUSIONS APPENDIX REFERENCES

The outer hair cell is one of the mechanosensory cells in the cochlea and is indispensable for fine tuning of the ear (Mountain, 1980; Liberman and Dodds, 1984). Besides sensory hairs, this cell has a cell body with a motility (Brownell et al., 1985; Ashmore, 1987) fast enough to follow changes in the membrane potential at auditory frequencies (Dallos and Evans, 1995; Frank et al., 1999). Having transduction mechanisms in both directions makes the cell a key element in the feedback loop in the cochlea that enhances the frequency selectivity and broadens the dynamic range of the ear.

Recent studies have revealed that the motility of this cell is based on a membrane motor that directly uses electrical energy (Ashmore, 1989; Iwasa, 1993; Dallos et al., 1993). This membrane motor has two or more conformations that differ in mechanical and electrical states (Iwasa, 1994). Each of these conformational differences is a familiar feature of membrane transport proteins: Charge transferable across the membrane during conformational changes is similar to gating charges of ion channels (Armstrong and Bezanilla, 1973; Heinemann et al., 1992). Differences in the membrane area in these states are similar to those in mechanosensory channels (Sukharev et al., 1999).

Because the conformations have differences in both properties, conformational transitions accompany charge transfer across the membrane and mechanical displacement of the membrane. Thus, electrical and mechanical changes are coupled, analogous to piezoelectricity. Indeed, attempts have been made to describe the lateral membrane of the outer hair cell as a piezoelectric material (Mountain and Hubbard, 1994; Tolomeo and Steele, 1995). However, these reports assumed that piezoelectric response was primarily linear to the electric field, whereas the response of the cell saturates with respect to the membrane potential. In addition, the cellular structure was not adequately addressed.

The present paper addresses the question of how outer hair cell motility compares with piezoelectricity. It also attempts to clarify how phenomenological parameters that describe the properties of the cell as a whole are related to more microscopic variables that characterize the motor and the cell membrane.

In the first part of the paper, analytical relationships between microscopic and macroscopic quantities are derived. That is followed by an attempt to determine microscopic parameters by sorting out existing experimental data. Then these microscopic parameters are used to obtain macroscopic parameters, which are then compared with experimental values. The last step serves as a consistency test.

The present treatment differs from earlier versions of an "area motor" model (Iwasa, 1994; Iwasa and Adachi, 1997) in providing analytical relationships between the microscopic and macroscopic quantities and thereby clarifying the piezoelectric nature of the motor.

	PIEZOELECTRICITY

TOP ABSTRACT INTRODUCTION PIEZOELECTRICITY TWO-STATE MEMBRANE MOTOR MEMBRANE MOTOR IN A... EXAMINATION OF EXPERIMENTAL... DISCUSSION CONCLUSIONS APPENDIX REFERENCES

Before describing the hair cell system, let us review a standard description of piezoelectricity. In a piezoelectric material, a mechanical displacement and an electric displacement are coupled. Such a property can be described by a coupling term in the free energy. In a simple one-dimensional case, the free energy G of a piezoelectric material can be given by (Ikeda, 1990)

(1)

where E is the electric field and F the force applied. The state E = F = 0 represents equilibrium. The first term is the electrical energy and the last term is the mechanical energy. The middle term represents the coupling energy.

The stress-strain relationships are obtained by taking partial derivatives of Eq. 1. Electrical displacement Q and mechanical displacement L are then represented, respectively, by

The coupling terms in those two equations use the same coefficient c₁₂ because they originate from the same free energy term. This symmetry in the coupling coefficients is the reciprocal relationship, characteristic of piezoelectricity. It distinguishes piezoelectricity from other kinds of electromechanical coupling such as electrostriction and electrokinetic effect.

This system is conveniently characterized by the piezoelectric coefficient c₁₂, which gives the magnitude of mechanoelectric coupling, and the coupling coefficient k (Ikeda, 1990), which describes the fraction of energy that is converted from one form to another,

(2)

and cannot exceed unity. The piezoelectric coefficient c₁₂ of typical piezoelectric substance such as quartz or Rochelle salt is constant in a relatively wide range of electric field E (Ikeda, 1990).

	TWO-STATE MEMBRANE MOTOR

TOP ABSTRACT INTRODUCTION PIEZOELECTRICITY TWO-STATE MEMBRANE MOTOR MEMBRANE MOTOR IN A... EXAMINATION OF EXPERIMENTAL... DISCUSSION CONCLUSIONS APPENDIX REFERENCES

To describe the hair cell system, let us start by examining unit properties of a membrane motor. Because a membrane motor cannot exist on its own and needs to be incorporated into a membrane to function, unit properties practically means the properties of a single motor unit incorporated into an infinitely large isotropic membrane. Under this condition, conformational transitions of the motor do not affect the motor's environment.

The membrane motor in the outer hair cell has been described by a two-state model. In its simplest form, two states of a motor unit differ in their charge by q and their cross-sectional area in the membrane (area motor model) by a. They are subjected to isotropic membrane tension T_m (Fig. 1),

(3)

Here, G_l is the free energy of the state with larger membrane area (extended state) and G_s is that of smaller membrane area (compact state). G₀ is a constant and the membrane potential is V_m. The probability of the extended state P is expressed by

(4)

Here,  = 1/(k_BT), where k_B is the Boltzmann constant and T is the temperature.

View larger version (52K): [in this window] [in a new window]   FIGURE 1   A two-state membrane motor and its displacements. (A) Cross sections of the motor in the two states. Transitions between the states are accompanied by transfer of charge q across the plasma membrane and mechanical displacements. (B) A schematic illustration of the motor's area changes. az and ac are area changes in the axial (z) direction and the circumferential (c) direction, respectively. The rectangular shape of the motor in the illustration is for simpler illustration and does not constitute an assumption.

For describing a membrane system, the membrane potential V_m substitutes for the electric field E because membrane thickness is implicitly assumed constant. The average charge Q and the average area A of the motor may then be represented by,
where the subscript s indicates those quantities that correspond to the compact state.

Due to their P-dependent terms, both Q and A are nonlinear with respect to the membrane potential V_m and to tension T_m. Here, changes in response to small increments in these variables are examined. Increments Q and A that correspond to an increment V_m in the membrane potential and an increment T_m in membrane tension are

(5)

(6)

with

(7)

(8)

(9)

C₀ and  are, respectively, the membrane capacitance and the area compliance of the compact state. For the sake of simplicity, we have assumed that the area compliance is the same in the two states, although the two states may differ in their stiffness. This simplifying assumption is justified at least as an approximation because an experimental examination showed that the effect of different stiffness in the two sates is insignificant for membrane motility (Adachi et al., 2000).

The terms a₁₂ and a₂₁ represent coupling. Because charge transfer and area changes of the motor are coupled, changes in the membrane potential that induce charge movement result in area changes. Reciprocally, changes in tension result in charge movement. The reciprocal relationship a₁₂ = a₂₁ indicates that the coupling is piezoelectric.

The membrane capacitance of the motor consists of the regular (or linear) membrane capacitance C₀ and a nonlinear membrane capacitance a₁₁, which has a bell-shaped membrane potential dependence. Likewise, the area compliance of the motor consists of two parts, the structural area compliance , which is voltage independent, and an area compliance a₂₂, which has also a bell-shaped membrane potential dependence. These voltage-dependent terms are due to the voltage and tension sensitivity of the motor.

The coupling coefficient k of the motor is then given by

(10)

The value for the coupling coefficient k increases up to unity as the relative significance of the linear capacitance C₀ and that of the area compliance  decrease. The coefficient k also depends on the value of the membrane potential and tension through P. The coupling coefficient has a maximum at P = 0.5, where the nonlinear capacitance peaks. Numerical values for the coefficients are examined later.

As will be shown below, the actual motor is in a membrane with anisotropic tension. In such a system, the term a · T_m is replaced by the scalar product of the displacement vector and the stress vector.

	MEMBRANE MOTOR IN A CYLINDRICAL CELL

TOP ABSTRACT INTRODUCTION PIEZOELECTRICITY TWO-STATE MEMBRANE MOTOR MEMBRANE MOTOR IN A... EXAMINATION OF EXPERIMENTAL... DISCUSSION CONCLUSIONS APPENDIX REFERENCES

Now we incorporate the membrane motor into a cylindrical cell at a finite concentration. To make the description of such a composite system simple, a number of assumptions has been made (Iwasa, 1994; Iwasa and Adachi, 1997). They are: 1) end effects can be ignored, 2) the total strains are sums of elastic strains and motor strains, 3) the elastic property of the motor is the same as the rest of the lateral membrane, and 4) the volume of the cell is kept constant.

Basic equations

In the following, the theory is briefly described using constitutive equations. The elastic tension is balanced with tension due to pressure P and tension due to an external axial force F_z. Due to the cylindrical geometry of the cell, the constitutive equations for the lateral membrane are given for tension in the axial direction z and the circumferential direction c:

(11)

(12)

Here, R is the radius of the cylinder and d₁, d₂, and c are elastic moduli. It is assumed that the elastic strains '_z and '_c are small. These equations assume that end effects can be ignored. In addition, it should be noticed that the tension is not isotropic. Not only does the effect of external axial force result in anisotropy, but the effect of pressure is anisotropic as well.

The total strains (_z, _c) and the elastic strains ('_z, '_c) are related by assumptions 2 and 3. Thus,

(13)

(14)

where the second terms on the right-hand side represent motor displacements, with n the number density of the motor in the lateral membrane, a_z and a_c the component of the area difference between the two states (Fig. 1), and P the fraction of the state with larger membrane area. The mechanical part of the free energy difference is understood as the scalar product of the tension vector and displacement vector. If the stiffness of the motor is different from the rest of the cell, additional terms that depend on the membrane fraction of the motor must appear in the above equations.

The fraction P is represented by

(15)

where  = 1/(k_BT) with Boltzmann's constant k_B and the temperature T. The free energy difference G_c in the two motor states depends on the membrane potential and membrane tension and is given by

(16)

Because membrane tension is anisotropic, the free energy difference is more complicated than in the isotropic case (i.e., Eq. 3).

So far, the number of equations is two because P is determined by Eqs. 15 and 16 and goes into Eqs. 13 and 14. Eqs. 13 and 14 are used to eliminate '_z and '_c in Eqs. 11 and 12. Those equations are, for example, to be solved for a given membrane potential V_m and external axial force F_z. Then the undetermined variables are three: the axial strain _z, the circumferential strain _c, and pressure P.

The additional condition is the constant volume condition (assumption 4), which can be expressed by

(17)

Because we assume that the strains are small, the volume strain is expressed by _z + 2_c. The value _v0 is due to static internal pressure of the cell at the resting membrane potential. With the constant volume condition, the number of independent variables is two, because _c is expressed by _z.

Effect of voltage and axial force

Now, let us describe the system for a set of values for V_m and F_z. An effect of the motor incorporated into the cell membrane is that the motor, which changes tension, is reciprocally affected by tension, creating a self-consistency condition. It turns out that the problem is how to determine the motor variable P.

With Eqs. 11-14 and 17, it is possible to eliminate RP and _c to express _z as a function of P,

(18)

with density of motor n and constants,

(19)

(20)

The axial strain _z depends on the membrane potential V_m through P, which describes the motor state. The load-free amplitude is determined by b₁n because the motor state variable P changes between 0 and 1. The cell strain _z and the axial compliance of the cell depends on the first term and the last term of Eq. 18, because the motor state P depends on axial force F_z.

The motor variable P is determined by the free energy difference G_c with Eq. 15, and the free energy difference is, in turn, given by Eq. 16. With Eqs. 12-14 and 17, circumferential tension T_c is expressed by _z and P. Axial tension T_z is a sum of T_c/2 and externally applied axial tension (see Eqs. 11 and 12). These substitutions give rise to

(21)

with

(22)

Here, g is defined by Eq. 20, and b₃ is a constant that involves G₀ and _v0. The factor b₂ determines the effect of the motor on itself because the presence of the term b₂P imposes a self-consistency condition.

By substituting G_c in Eq. 15 with Eq. 21, an equation for P is obtained:

(23)

With this equation, P is obtained for a given set of the membrane potential V_m and axial force F_z. The equation shows that P can be treated as an inverse function of V_m or of F_z. For example, for a fixed value of the axial force F_z, the membrane potential V_m and the motor state P correspond one-to-one, and V_m is readily determined for a given value of P. If we impose a condition b₂ = 0, which excludes the effect of the motor on itself, Eq. 23 turns into a Boltzmann function that expresses P. The voltage dependence of P is illustrated in Fig. 2.

View larger version (22K): [in this window] [in a new window]   FIGURE 2   Effect of b2 on the membrane potential dependence of P and its derivative. (A) Voltage dependence of P and its voltage derivative. The motor variable P is determined by Eq. 23 with Fz = 0. The voltage derivative is qP(1  P), where  is defined by Eq. 29. Solid lines, b2 = 0.9; broken lines, b2 = 0. The condition b2 = 0 leads to the two-state Boltzmann function for P. The two values for b2 are intended to show the extremes. A realistic value for b2 is ~0.3. q = 0.9 e. The scale of P is on the left and the scale of dP/dV is on the right. The unit of dP/dV is V1. (B) Comparison of voltage dependences of P(1  P) for b2 = 0.9 and b2 = 0. Peak heights are normalized. Solid line, b2 = 0.9, q = 0.9 e; broken line, b2 = 0, q = 0.74 e, and translated along the axis of abscissas. These comparisons show that P, which is determined by Eq. 23, can be approximated a two-state Boltzmann function, provided that the charge q is adequately adjusted.

Response to small changes in V_m and F_z

Now let us consider the effect of small changes in the membrane potential and axial force on membrane charge and the cell length. An increment Q in charge and an increment L in the length of the cell due to V_m and F_z can be represented by


Here, the first term in Q is the voltage derivative of motor charge NqP. The second term is due to the regular (or linear) membrane capacitance C_lin of the cell. The equation for L is derived from Eq. 18.

An increment in P due to small changes in the membrane potential and axial force can be obtained by using Eq. 23. The substitution of the resulting expression for P in the above equations leads to

(24)

(25)

where Q is an increment of charge, L is an increment of the axial length of the cell, and F_z is axial force applied to the cell.

The coefficients are given by

(26)

(27)

(28)

where g, b₁, and b₂ are defined by Eqs. 20-22. The factor  is

(29)

The reciprocal relationship c₁₂ = c₂₁ is automatically satisfied. Of the coefficients, c₁₁ is the membrane capacitance and c₂₂ is the axial compliance. The coefficient c₁₁ includes the linear capacitance C_lin and the contribution of motor charge to the capacitance. The coefficient c₂₂ likewise includes both the passive compliance, which is 2g/(R), and the contribution of the motor to the compliance. The coupling coefficient k is given by

(30)

Eqs. 26-28 show that the coefficients c₁₁, c₁₂ (= c₂₁), and c₂₂ consist of constants and terms that are proportional to P(1  P). That means that these coefficients have similar membrane potential dependences if their constant terms are excluded. Ratios of these coefficients, such as c₁₂/c₂₂ and k, however, do not have the same voltage dependences, although they share the peak potential and appear similar (Fig. 3).

View larger version (17K): [in this window] [in a new window]   FIGURE 3   The membrane potential dependence of coefficients c11, c12, and c22. The functional forms of these coefficients are given by Eqs. 26-28 and 30. The values for the parameters are given in Table 1 and Table 2. (A) The membrane capacitance c11 (solid line) and the axial stiffness c22 (broken line). The plots are normalized. The peak value of c11 is 3.7 × 1011 C and the peak value of c22 is 118 m/N. (B) c12 and c12/c22. The plots are normalized. The peak value for c12 is 2.0 × 105 m/V (or equivalently C/N). The peak value for c12/c22 is 1.7 × 107 N/V. (C) The coupling coefficient k.

Effect of motor on itself

The factor  that is given by Eq. 29 does not appear in coefficients a₁₁, a₁₂, and a₂₂ of an isolated motor. This factor arises from a self-consistency condition when the motor is incorporated into a cell.

The partial derivative of Eq. 23 with respect to V_m gives rise to
which contains the factor . The departure of this factor from unity indicates the effect of the motor on itself. It is a result of the motor being incorporated into a cell.

This effect is essentially negative cooperativity. Depolarization decreases P and decreases membrane area, resulting in increased pressure, which in turn increases membrane tension. An increase in membrane tension favors the extended state. These interactions thus reduce the motor's sensitivity to the membrane potential. The effect of this factor is illustrated in Fig. 2. The self-consistency condition reduces the sharpness of the motor's dependence on V_m and F_z, while it keeps the normalized dependence of P on V_m or on F_z relatively unchanged. The partial derivative of P with respect to F_z likewise yields the factor .

	EXAMINATION OF EXPERIMENTAL DATA

TOP ABSTRACT INTRODUCTION PIEZOELECTRICITY TWO-STATE MEMBRANE MOTOR MEMBRANE MOTOR IN A... EXAMINATION OF EXPERIMENTAL... DISCUSSION CONCLUSIONS APPENDIX REFERENCES

To test the validity of the model, experimental data are briefly examined here. First, the values for the parameters are determined from the their directly relevant experiments. Second, the quantities that characterize piezoelectricity are estimated for the membrane motor and for the cell as a whole. Third, the predicted cellular coefficients are compared with experimental data, which are not used for determining the parameters.

Determination of parameters

Elasticity

Motor parameters

2

(31)

2

2

2

View larger version (11K): [in this window] [in a new window]   FIGURE 4   The relationship between b2 and 2 (solid line). Deviations from the broken line indicate the importance of the correction indicated by Eq. A1.

Piezoelectric and coupling coefficients

In the following, to examine how efficiently the cell uses the motor elements in its membrane, the coefficients for the motor in isolation and those for the motor in the cell are compared numerically.

Isolated motor

17

5

2

2

Cell as a whole

Consistency tests

Although experimental values for a number of quantities have been used to determine the theoretical parameters, there remain a number of experimentally determined quantities that are still unused. Comparing experimental values and the predicted values for these quantities can be used to examine the consistency of the model.

In the following, the magnitudes of those quantities are examined for testing consistency. Because the voltage dependences of these quantities, i.e., the sharpness of the dependence and shifts, have been used to determine the parameters, these properties are not useful for testing the model.

Amplitude and c12

Ratio c12/c22

The axial compliance c22

	DISCUSSION

TOP ABSTRACT INTRODUCTION PIEZOELECTRICITY TWO-STATE MEMBRANE MOTOR MEMBRANE MOTOR IN A... EXAMINATION OF EXPERIMENTAL... DISCUSSION CONCLUSIONS APPENDIX REFERENCES

Piezoelectric models for describing the voltage-dependent motility of the outer hair cell have been reported earlier (Mountain and Hubbard, 1994; Tolomeo and Steele, 1995). The present work clarifies a number of issues left out in those earlier reports, which are based on a formal piezoelectric description, by relating a two-state membrane motor model with the formal thermodynamic description. One issue that previous treatments had not addressed concerns the nonlinear characteristics of the hair cell motility, which is a natural consequence of a two-state model. Another issue is to clarify the relationship between the motor mechanism and the effect of incorporating the motor into the cell membrane. However, the most significant feature of the present treatment is in relating the cell function to its underlying molecules, which are likely to undergo discrete conformational transitions common with most functional proteins. Such issues cannot be adequately addressed by simply introducing adjustable parameters to simulate the nonlinearity (Spector et al., 1999).

Physical identity of the motor

A basic assumption of the present model is that the membrane motor is a protein or a cluster of proteins, similar to any other functional membrane proteins, which undergo conformational transitions. These transitions involve transfer of charge across the membrane and changes in its membrane area.

The density of the functional motor units has been obtained from experimental data on the membrane capacitance. It is similar to the density of 10-nm particles (Gulley and Reese, 1977; Kalinec et al., 1992; Frolenkov et al., 1998) in the lateral membrane of the outer hair cell determined by electron microscopy. A detailed comparison seems to suggest that the stoichiometry of the functional unit to those membrane particles is 2:1 rather than 1:1 (Santos-Sacchi et al., 1998). Because it is well established that membrane proteins have subunits and subunits can transfer charge independent of each other, the exact stoichiometry does not challenge the validity of the assumption.

Perhaps the observation most supportive of the idea that the motor is a membrane protein is that prestin, a membrane protein specific to the outer hair cell, confers a prominent nonlinear component to the membrane capacitance and voltage-sensitive motility in kidney cells transfected with the mRNA that encodes the protein (Zheng et al., 2000). The significance of prestin was further confirmed recently by a report that prestin that is expressed in a number of mammalian cells shows tension sensitivity similar to the motor in the hair cell membrane (Ludwig et al., 2001). This observation indicates that the membrane protein constitutes the essential part of the motor, consistent with the model described here.

Properties of the motor

The present model is designed to have a minimal number of parameters, all of which can be determined from experimental data. Experimental data unused for determining the parameters can then be used to test the consistency of the model. The attempt of minimizing the number of parameters may lead to oversimplification, in which the model is unable to explain some experimental observations. In the following, attention will be paid to whether such conflicts, if they exist, are fundamental.

Number of motor states

Electrical properties

Elastic properties

Connectivity with the cortical cytoskeleton

The present model assumes a series connection of the elastic element and the motor element, i.e., Eqs. 13 and 14. It is not immediately clear that the microscopic structure of the lateral wall, in which the cortical cytoskeleton and the motor-containing plasma membrane run parallel, intermittently linked by pillars, supports such a series connection if the stiffness of the wall is primarily determined by the cytoskeleton. Although such a result was obtained by considering membrane bending in the cell axis assisted by the cortical cytoskeleton (Raphael et al., 2000), the approach requires assuming numerous parameter values, which are hard to determine. It turns out that membrane bending (flexoelectricity) also belongs to piezoelectricity because it satisfies the reciprocal relationship (Petrov, 1999). However, the expected significance of the cortical cytoskeleton for motile activity does not appear to be consistent with the experimental observation that the motile machinery remains virtually unaffected by dissolving the cytoskeleton (Adachi and Iwasa, 1999).

Comparison with piezoelectric material

The most striking feature of the electromechanical coupling in the outer hair cell is in its piezoelectric coefficient c₁₂, which is ~25 µC/N. This value is four orders of magnitude greater than the best piezoelectric material, which has 2.5 nC/N (Park and Shrout, 1997). Values for more common piezoelectrics range from 2 to 4 pC/N for quartz to ~550 pC/N for Rochelle salt (Ikeda, 1990).

The coupling coefficients k of 0.31 for the outer hair cell and ~0.4 for of its motor are, however, mid-range among common piezoelectric materials that range from 0.1 for quartz up to 0.76 for Rochelle salt. The main factor that makes the coupling coefficient of the outer hair cell unexceptional despite its enormous piezoelectric coefficient is its mechanical compliance, which is extremely large compared with inorganic materials.

	CONCLUSIONS

TOP ABSTRACT INTRODUCTION PIEZOELECTRICITY TWO-STATE MEMBRANE MOTOR MEMBRANE MOTOR IN A...