INTRODUCTION

TOP ABSTRACT INTRODUCTION THE MODEL MODEL PARAMETERS AND... MODEL PREDICTIONS DISCUSSION CONCLUSION APPENDIX A APPENDIX B REFERENCES

Inside the mammalian cochlea are outer hair cells, which possess the ability to convert electrical energy to mechanical energy and generate force at acoustic frequencies greater than 50 kHz (Frank et al., 1999). These outer hair cells play an indispensable role in hearing, enabling sharp frequency discrimination at low sound intensities by supplying energy to the cochlear partition. The electromechanical coupling in the outer hair cell, referred to as electromotility, was first observed by Brownell and coworkers (Brownell et al., 1985) and was subsequently further investigated and quantified (Ashmore, 1987; Santos-Sacchi, 1991, 1992, 1993). Its paramount importance in auditory physiology has been the subject of many recent reviews (Nobili et al., 1998; Dallos, 1997; Ashmore, 1994; Brownell and Popel, 1998; Lim and Kalinec, 1998). Loss or damage to outer hair cells can cause high-frequency hearing loss, and at least one form of inherited deafness is due to a defective membrane channel in the outer hair cells (Kubisch et al., 1999).

Outer hair cell electromotility is unique among biological motile mechanisms in the speed at which it operates. For most motile mechanisms in biology, we now have at least some knowledge of the molecules involved. However, the molecular mechanism of electromotility remains unknown. Among the different biophysical models that have been proposed for electromotility, the two-state molecular motor model adequately accounts for most experimental observations. In its most general form, this model postulates that a motor molecule exists that changes its conformation between two states when the cell's membrane potential is changed (Dallos et al., 1993). The cell is cylindrically shaped, and the model postulates that the motor molecule gives rise to different strains in the longitudinal and circumferential direction of the cell without specifying any requirements on the cell surface area. In other versions of this model, the conformational change is explicitly specified to be an area change (Iwasa, 1994; Santos-Sacchi, 1993). A change in shape as a result of the application of an electric field is a phenomenon known in solid crystals as piezoelectricity, and formal piezoelectric models of outer hair cell motility have been proposed (Mountain and Hubbard, 1994; Tolomeo and Steele, 1995; Spector, 1999; Spector et al., 1999a,b).

The above models do not explicitly take into account the complex trilamellar structure of the outer hair cell's lateral wall. Electron microscopy reveals that the cochlear outer hair cell has a unique multilayered membrane structure (See Fig. 1). The outermost layer, the plasma membrane, appears corrugated or folded in electron micrographs (Smith, 1968; Furness and Hackney, 1990; Ulfendahl and Slepecky, 1988, Dieler et al., 1991; Holley et al., 1992). Extending from the plasma membrane into the cytosol are structures of unknown composition referred to as "pillars" (Forge, 1991). The folds in the membrane appear to anchor at the pillars. The pillars form a connection between the plasma membrane and the cytoskeleton because they are attached to circumferential actin filaments. The actin filaments are crosslinked by thinner filaments (spectrin molecules), which are oriented preferentially in the longitudinal direction of the cell. Beneath the cytoskeleton is another membraneous layer referred to as the subsurface cisterna (SSC). It is not known if this layer is connected to the cytoskeleton. In contrast to the plasma membrane, the SSC appears smooth in electron micrographs (Dieler et al., 1991; Saito, 1983). A complete model of outer hair cell electromotility should consider the lateral wall's unique ultrastructural composition.

View larger version (47K): [in this window] [in a new window]   FIGURE 1   A schematic of the outer hair cell. These cells are cylindrically shaped with lengths ranging from 20 to 90 µm along the cochlea and with a radius of 4-5 µm. The hair bundle, composed of stereocilia, is located at the apex of the cell. The lateral wall is the source of electromotility and it appears smooth under a light microscope. When examined with electron microscopy, the lateral wall appears corrugated. The folds in the membrane appear to terminate at pillar proteins that extend to the cytoskeleton. The cytoskeleton is composed of actin filaments crosslinked by spectrin molecules.

In this work, we propose a model of electromotility that is based on the structure of the lateral wall and the following considerations:

1.	membranes are thin structures in which bending deformations play a pivotal role in the response to external forces;
2.	membranes are liquid crystals and exhibit flexoelectricity;
3.	the plasma membrane and the cytoskeleton are tightly associated.

The phospholipid and protein molecules that comprise biological membranes give the membrane the material property of having a large resistance to area changes (Bloom et al., 1991). From the perspective of thin films, membranes are in a condensed phase characterized by high surface pressure and tight packing of molecules. However, due to their thinness, membranes have a small resistance to bending, which can barely be measured experimentally. The bending resistance of the membrane is close to the Boltzmann thermal energy (kT), and, as a result, bending of pure giant vesicles (Servuss et al., 1976; Schneider et al., 1984) and red cell flickering (Strey et al., 1995) can be driven by thermal fluctuations in the environment and observed microscopically.

The second important consideration for our model of electromotility is that biological membranes are liquid crystals (de Gennes, 1974; Collings, 1990). Liquid crystals are distinguished from solid crystals in that the molecules of solids display positional and orientational order, whereas the molecules of liquid crystals display only orientational order. In biological membranes, the constituent phospholipid and protein molecules not connected to the cytoskeleton, packed at high density or confined in domains, are free to diffuse laterally so there is no positional order, but the molecules are constrained to remain preferentially aligned with their long axis perpendicular to the membrane, which gives the structure an orientational order.

Liquid crystals exhibit a unique coupling between applied electric fields and mechanical deformation. This effect, termed "flexoelectricity," is caused by the curvature-induced electrical polarization of the material. Though first presented as a peculiar kind of piezoelectricity to explain "unusual effects" observed in liquid crystals (Meyer, 1969), flexoelectricity is fundamentally different from the piezoelectric effect observed in solid crystals. Theoretically, flexoelectricity is based on the realization that out-of-plane bending of a liquid crystal will alter the electric field inside the material. Petrov (1975) hypothesized that phospholipid membranes, being liquid crystals, should exhibit flexoelectricity. In particular, biological membranes are composed of protein and lipid molecules that have large dipole moments parallel to their axes, a requirement for flexoelectricity. Experimentally, it has been confirmed in black lipid membranes that membrane polarization changes when the membrane curvature is changed (the flexoelectric response) (Petrov, 1999; Todorov et al., 1994a). Sun (1997) also demonstrated that flexoelectricity can be induced by the pressure from sound waves and is sensitive to membrane composition. However, flexoelectricity is bi-directional, and so, in addition to this mechanoelectrical coupling, flexoelectric materials should also exhibit an electromechanical coupling. Electromechanical transduction in lipid membranes has been observed experimentally (Todorov et al., 1994b). This direct bending of a membrane as a result of the application of an electric field is sometimes referred to as the "converse flexoelectric effect." This converse flexoelectric effect has been invoked to explain voltage-dependent movements in cells observed with atomic force microscopy (Mosbacher et al., 1998).

The plasma membrane of most eukaryotic cells is connected to an underlying cytoskeleton. The cytoskeleton is responsible for structural support of the cell and contains elements that resist deformation. In the red blood cell and the outer hair cell, the cytoskeleton comprises a network of spectrin and actin molecules. In both these cells, experiments have demonstrated a tight association between the plasma membrane and the cytoskeleton (Hwang and Waugh, 1997; Oghalai et al., 1998). The bending of one layer of a multilayered composite membrane can transmit deformation and force to another component. In a discussion of the mechanics of multilayered membranes, Evans and Skalak (1980) proposed that "if the layers are strongly associated, then couples of force resultants between layers can be produced which create a moment resultant." Evans and Skalak also noted that the moment resultant could be the dominant response to external forces. Because the outer hair cell lateral wall is a strongly associated multilayered system, we will present an analysis of how the deformation in the plasma membrane results in a deformation of the cortical lattice. We will show that an active bending element can function as a cellular motor by transmitting forces to an elastic cytoskeletal element.

These considerations will lead us to a formulation of a "membrane-bending" model of outer hair cell electromotility. This model will show that electrically induced local bending of the membrane (flexoelectricity) can phenomenologically describe experimental observations of outer hair cell electromotility. A flexoelectric mechanism would also enable force generation at high frequencies. In addition, the model will account for the isometric force generated in the cell and explain experimental results, such as the weakening of force production in diamide-treated cells. The model explicitly provides for constant (or nearly constant) cell surface area and gives a pivotal role to localized membrane curvature changes, the importance of which is increasingly being realized in cell biology.

	THE MODEL

TOP ABSTRACT INTRODUCTION THE MODEL MODEL PARAMETERS AND... MODEL PREDICTIONS DISCUSSION CONCLUSION APPENDIX A APPENDIX B REFERENCES

The micromechanical picture

We define the elemental motile unit of electromotility as the plasma membrane and associated cytoskeleton between adjacent pillar proteins (See Fig. 2). In particular, the curvature of the plasma membrane between the pillars coupled with the length of spectrin comprises an elemental structural unit in the cell. The curvature of the plasma membrane between the pillars is able to change, and these curvature changes will be induced by alterations in the transmembrane potential. An increase in the curvature causes the membrane to fold up and will shorten the cell, whereas a decrease in curvature causes the membrane to flatten and will elongate the cell, as illustrated in Fig. 2. Possible molecular mechanisms responsible for these curvature changes are discussed later. The result of decreasing the curvature of the plasma membrane is to increase the spacing between the actin filaments and elongate the extensible spectrin molecules. Figure 3 illustrates the geometry for analyzing this curvature deformation. In Appendix A, we derive the geometric relationship between the curvature change and the longitudinal elongation.

View larger version (9K): [in this window] [in a new window]   FIGURE 2   Micromechanical model for membrane deformation. Curvature changes in the elemental motile unit cause extension of the spectrin molecules attached to the pillars. Three units are shown in the figure. A depolarization (+) leads to a decrease in the radius of curvature and a shortening of the cell and hyperpolarization () leads to an increase in the radius of curvature and cell lengthening.

View larger version (11K): [in this window] [in a new window]   FIGURE 3   Geometry for analyzing membrane deformation. The membrane contour follows a smooth arc with a radius of curvature R. Curvature changes map to an equivalent longitudinal deformation x, which is related to the polar angle. See Appendix A for the relationship between these parameters.

These structural units of motility are spaced along the entire length of the lateral wall. The motile units are connected in series, as required by evidence from microchamber experiments that the magnitude of the voltage-induced cell displacement increases from the point at which the cell in inserted in the microchamber in a cumulative manner (Dallos et al., 1991). This electromechanical system is mechanically characterized as a bending element in parallel with a linear elastic element. The bending modulus of the membrane is denoted k_c and the spring constant of the elastic element is denoted k_s. For simplicity, we only analyze membrane deformation in the longitudinal direction, which experimental evidence shows is the primary mode of action of the motor (Dallos et al., 1991). However, a small change in the radius of the cell accompanies the length change.

Plasma membrane mechanical properties

The membrane is treated as a two-dimensional elastic material. In the analysis that follows, we restrict ourselves to the postulated elemental unit, which is modeled as a thin ring-shaped shell wrapped around the circumference of the cell. It is assumed that the shell undergoes pure bending. The classical theory of thin shells defines the moment produced by bending as (Timoshenko and Woinowsky-Kreiger, 1959)

(1)

where  is the stress, h is the thickness of the shell, and z is the direction normal to the shell (see Fig. 4). The subscript p stands for passive. Note that M_p is a moment resultant: it is the sum of the torques about the neutral surface of the membrane. It has the units of force because it is the moment divided by length (force × length/length). The implicit length specified in this definition of the moment is the arc length of the half shell (see Fig. 3).

View larger version (12K): [in this window] [in a new window]   FIGURE 4   Illustration of parameters used to define the bending moment. An initially undeformed sheet of membrane of thickness h is subjected to a torque about its edges. This force causes out-of-plane bending of the material. The z direction is normal to the membrane. See Eq. 1.

In the analysis of Evans and Skalak (1980), the product of the bending stiffness of the membrane and the total curvature gives the force produced by bending the membrane,

(2)

where k_c is the bending rigidity, c_l is the curvature of the membrane in the longitudinal direction, c is the curvature of the membrane in the circumferential direction and c_o is the spontaneous curvature of the membrane. The spontaneous curvature describes any intrinsic tendency of the membrane to curve in the absence of applied external forces and is related to the molecular composition of the membrane.

The bending rigidity is related to the area expansivity modulus, K, of the membrane by the relationship (Waugh et al., 1992),

(3)

where h is the membrane thickness. In the energetic formalism, the bending energy density is the Helfrich elastic energy for curvature deformation in a liquid crystal (Helfrich, 1973),

(4)

where  = c_l + c is the sum of the membrane curvatures. In bilayer membranes, there will be an additional energy arising from differential expansion and compression in the membrane that occurs during bending (Evans, 1974). At present, we neglect this energetic contribution in our model of outer hair cell electromotility. The magnitude of this nonlocal bending energy is of the same order as the local bending energy (Raphael and Waugh, 1996).

Spectrin mechanical properties

The result of changing the membrane curvature will be to change the interpillar distance and elongate or shorten the spectrin molecules. The spectrin molecules will essentially form an elastic element that is in parallel with the bending element (see Fig. 2). The spectrin molecules are represented as a system of springs and the energy stored in the spectrin network is written as

(5)

where k_s is the spring constant, x_o is the resting (unstressed) length of the molecule and n_sp is the number of springs (spectrin molecules) per unit area.

Passive mechanical energy and equilibrium

The internal energy of the membrane-spring system is equated to the mechanical work done by deformation. In our micromechanical model, this work is due to bending of the plasma membrane and extension of the spectrin molecule s. The mechanical work per unit area due to deformation is written as

(6)

Note that this expression only considers deformation in the longitudinal direction in the membrane. In a complete model of electromotility, the curvature change in the circumferential direction will be linked to the deformation of the actin molecules. Because these changes are small, we neglect them as a first approximation. Because the bending and the extension are related geometrically, the bending and extensional terms can be combined into an effective mechanical term, as described in Appendix A. Effectively, the spring displacement x is linearized about the curvature c_l resulting in a relationship,

(7)

where a and b are constant terms.

The equilibrium of the membrane-spring system can be calculated by setting the partial derivative of the internal energy equal to zero. The resulting relationship between the curvature and the spring length in terms of the ratio of the elastic constants for bending and extension is

(8)

This relationship specifies the equilibrium configuration of the membrane-spring system. Given the elastic constants, the resting spring length, and the spontaneous curvature, the equilibrium curvature and spring length are then determined. These parameters will thus specify the resting (equilibrium) state of the elemental unit. We note that it is possible that the bending of the membrane may be compressing the spectrin molecules, or that the spectrin may be inducing membrane curvature. In terms of the equilibrium curvature, the internal energy can be rewritten as (See Appendix A)

(9)

where k_eff is the effective bending stiffness and G is a constant term defined in Appendix A. For convenience, the internal energy can be redefined, omitting the constant term, as

(10)

Alternatively, the curvature can be expressed in terms of the spring length x in Eq. 7, and the energy can be expressed in terms of an effective spring stiffness.

Force generation during displacement from equilibrium

Because the motile unit is composed of an elastic element in parallel with a bending element, the total passive force per unit length in the longitudinal direction that a single elemental unit exerts will be the sum of the forces from the bending element and the elastic element:

(11)

The effective longitudinal displacement of the bending element is equal to the displacement of the spectrin molecules. We locate an active flexoelectric element in the plasma membrane. Hence, passive and active elements in the plasma membrane are in parallel with the elastic element representing the spectrin molecules. To evaluate the force generated in our micromechanical model, we now consider the properties of an active, voltage-sensitive bending element.

Electrically-induced curvature changes

Biological membranes are composed of protein and lipid molecules that possess dipole moments that establish an electric field within the membrane. Hence, the membrane shell will contain dipoles, as illustrated in Fig. 5. These dipoles give the shell an internal electric field and consequently a polarization. The application of an external electric field changes the curvature of the membrane (the converse flexoelectric response). The relationship between the applied electric field, the orientation of the dipoles in the membrane and the resulting curvature is illustrated in Fig. 5. Note that we have assumed a simple picture of flexoelectricity as originating from the rotation of dipoles in analogy with liquid crystals. In real biological membranes the situation can be more complicated because dipoles, quadrupoles, and surface charges contribute to the polarization (Petrov and Sokolov, 1986; Petrov, 1999). Petrov (1975) first proposed the mathematical relationship between the membrane curvature and the membrane polarization,

(12)

where P_s represents the magnitude of the surface polarization vector per unit area, f is the flexoelectric coefficient with units of Coulombs (C), c_l represents the membrane curvature in the longitudinal direction, and c is the curvature in the circumferential direction. The sign of the flexoelectric coefficient f is positive when P_s points outward from the center of curvature of the membrane and negative when it points inward (Petrov, 1999).

View larger version (19K): [in this window] [in a new window]   FIGURE 5   Orientation of dipoles in the membrane. This figure displays the effect of altering the external electric field on the molecular dipoles and, consequently, the curvature and polarization of the membrane. The magnitude and direction of the polarization vector (P) and electric field vector (E) are illustrated as arrows at the left and the right of a curved element of membrane. The top panel illustrates a hyperpolarized membrane in which the molecular dipoles have rotated to minimize the polarization opposed to the applied field. This corresponds to the largest radius of curvature. The middle panel represents a slightly depolarized membrane, at which more of the dipoles are aligned with the field and the polarization is increased. The bottom panel represents a maximally depolarized membrane, in which all the dipoles are aligned with the applied field. This corresponds to the largest curvature and the shortest length of the motile unit.

Below, we derive the fundamental equations of flexoelectricity in an analogous manner to that done for piezoelectric materials. We then apply these equations to our model of outer hair cell electromotility. In the treatment of piezoelectricity, a thermodynamic potential called the electric enthalpy per unit area is defined, which is written as (Cady, 1946; Meyer, 1969; Lines and Glass, 1977),

(13)

where is the total internal energy per unit area, E is the electric field, and D_s is the electric displacement. The internal energy contains contributions from both the mechanical energy described earlier and the electrical energy,

(14)

where _o is the permittivity of free space. The electric displacement per unit length is written as

(15)

where h is the thickness of the membrane. Combining the above equations, we define the electric enthalpy per unit area to be

(16)

The derivative of the electric enthalpy with respect to curvature defines the moment resultant in the membrane,

(17)

Note the moment resultant has units of force (N). The total moment resultant is thus the sum of passive bending resistance and active flexoelectric properties. The derivative of the electric enthalpy, with respect to the electric field, defines the charge displacement for the two dimensional membrane,

(18)

The charge displacement has units of C/m. Eq. 18 predicts that alterations in membrane curvature will produce charge displacement in the membrane.

Eqs. 17 and 18 represent the fundamental electromechanical coupling equations that will be used in our micromechanical model of outer hair cell electromotility. They are analogous to those derived for linear piezoelectric materials (Cady, 1946). They are also analogous to the equations presented by Tolomeo and Steele (1995) in their piezoelectric model of outer hair cell motility. However, in our treatment, the bending stiffness replaces the area elastic modulus, the curvature replaces the areal strains and the flexoelectric coefficient replaces the piezoelectric coefficient. In addition, our electrical energy (D_sE) is defined per unit area and not per unit volume. This corresponds to defining the electric displacement per unit length and not per unit area, as is the case in Tolomeo and Steele (1995). The treatments are equivalent when Eq. 18 is divided by the membrane thickness (h). Note that, if we express the curvature c_l in terms of the spectrin length using Eq. 7, the model appears phenomenologically piezoelectric with the equations equivalent to the previous piezoelectric models. However, the biophysical meaning of the terms in the present model is different.

Nonlinear flexoelectric model

The above analysis assumed a linear relationship between membrane polarization and membrane deformation, as originally proposed by Petrov (1975). However, in our physical model illustrated in Fig. 5, the flexoelectric response saturates when all the membrane dipoles are aligned in the direction of the applied field. Hence, we extend Petrov's original formulation to include nonlinear effects. For a collection of dipoles, the fraction of the dipoles oriented in the direction of the applied field is given by the Langevin function (Cady, 1946). The Langevin function is defined as

(19)

A derivation of the Langevin function is presented in Appendix B. This function is illustrated graphically in Fig. 6. The argument of the function is  = p_oE_p/kT, where p_o is the permanent dipole moment and E_p is the polarizing or local field. The term _o is introduced to account for the spontaneous polarization. The membrane will exhibit a polarization in the absence of an applied field because the molecular dipoles are sterically constrained to remain oriented about one direction in the membrane. The existence of this spontaneous electric field will account for the offset in the electromotility function from zero. Note that the Langevin function enters a nonlinear region when the argument becomes larger than 1.5 and then approaches saturation slowly. For molecules with small dipole moments, the nonlinear region is never reached at low fields and the linear relations presented earlier are valid. However, if molecules have a large dipole moment, or local fields become very large, the membrane polarization will no longer be a linear function of the electric field.

View larger version (7K): [in this window] [in a new window]   FIGURE 6   The Langevin function. This function describes the distribution of dipoles aligned with a field as a function of the polarizing field. At low fields, the function is linear, but becomes nonlinear as the field intensity increases. The function saturates when all of the dipoles are aligned in the direction of the field.

When the membrane polarization is a nonlinear function of the electric field, the polarization can be expressed as a power series:

(20)

where ₁ and ₂ are constants. A nonlinear relationship between the polarization and the electric field will translate into a nonlinear dependence of the flexoelectric coefficient on membrane potential. It has been mentioned in the flexoelectric literature that the flexoelectric coefficient can be voltage dependent (Hristova et al., 1991), yet a formal relationship has not been presented.

The thermodynamic potential in the nonlinear dielectric will take the form,

(21)

Note that the energy will now include a nonquadratic term that arises from the product of the polarization and the electric field. The existence of a nonquadratic term in the thermodynamic potential was postulated by Spector (1999) in his electroelastic model of outer hair cell motility. The nonlinear charge enthalpy per unit area is written as

(22)

The product f(E)E represents the force that orients the dipoles when a field is applied. This flexoelectric force acts on the membrane to change its curvature. This force is proportional to the fraction of dipoles oriented with the applied field at a given value of the field, so the product f(E)E is written as a constant multiplied by the Langevin function,

(23)

Using this relationship, the moment resultant will be written in the nonlinear case as

(24)

The relationship for the charge displacement can be derived by noting that the product of the polarization and the electric field is given by a constant times the Langevin function,

(25)

where P_max = Np_o, with N being the number of dipoles per unit area. From this, it follows that the charge displacement is

(26)

We note that the charge displacement will contain both linear and nonlinear terms. Eq. 24 will be used to describe the nonlinear features of outer hair cell motility, and Eq. 26 can be used to describe the nonlinear features of outer hair cell charge movement.

	MODEL PARAMETERS AND FEASIBILITY

TOP ABSTRACT INTRODUCTION THE MODEL MODEL PARAMETERS AND... MODEL PREDICTIONS DISCUSSION CONCLUSION APPENDIX A APPENDIX B REFERENCES

At the nanoscale level, we estimate that the plasma membrane of the outer hair cell follows an arc with a radius of curvature of about 30 nm between pillar proteins attached to adjacent circumferential (actin) filaments. The interactin (or interpillar) spacing is based on electron microscopic images of Forge (1991). The spacing between the circumferential filaments varies considerably and depends on the fixation buffer and method (Holley et al., 1992). Forge (1991) reported this interactin distance to be 36 nm, whereas Holley et al. (1992) reported that it can vary from 20 to 80 nm. The spectrin molecules crosslink the actin filaments at various angles, yet the average spacing between spectrin molecules is on the order of 10-30 nm and they are not always coincident with the pillars (Holley et al., 1992); the lengths of spectrin molecules range between 30 and 80 nm.

We investigate the feasibility of the membrane-bending model by determining whether the forces generated are sufficient to cause the experimentally measured deformations of the outer hair cell. We first discuss experimentally determined values of the parameters in the model and then determine whether these values make our micromechanical model feasible.

Values of electro-mechanical parameters

Flexoelectric coefficient

20

18

20

18

Bending stiffness

19

19

19

18

19

18

Spectrin elasticity

3

5

5

5

3

2

19

3

19

Experimental voltage-to-length parameters

The feasibility of the model is evaluated for an outer hair cell 80 µm long. For such a cell, there will be approximately 2000 motile units (membrane folds as shown in Fig. 2) along the length. Here we have chosen the spacing between the actin filaments (the interpillar distance) to be 40 nm. A 10-mV hyperpolarization of the membrane potential would increase the cell length by a maximum 0.2 µm in accord with the maximum voltage-to-length function of 20 nM/mV (Santos-Sacchi, 1992). From this, we estimate the increase in the distance between the pillars to be 0.1 nm and calculate the increase in the radius of curvature geometrically via Eq. A2 in Appendix A. The result is that, for a 0.1-nm extension of a spectrin molecule, the radius of curvature will change from 30 to 30.4 nm. The angle  defined in Fig. 3 will increase from 41.2 to 41.8 degrees.

Feasibility estimates

(27)

(28)

20

18

20

18

(29)

(30)

6

5

5

7

	MODEL PREDICTIONS

TOP ABSTRACT INTRODUCTION THE MODEL MODEL PARAMETERS AND... MODEL PREDICTIONS DISCUSSION CONCLUSION APPENDIX A APPENDIX B REFERENCES

Length changes with voltage changes in the linear model

From the linear constitutive equation for the flexoelectric membrane, we can predict the length change in the outer hair cell as a function of the change in the transmembrane potential. For an unloaded motile unit that generates no force, Eq. 28 is rearranged to give

(31)

where V = V₂  V₁. For depolarization, V is positive, whereas for hyperpolarization, V is negative. Note that V represents the difference in membrane potential between two states, not the absolute value of the membrane potential.

When we substitute the geometric relationship between the change in membrane curvature and the longitudinal displacement derived in Appendix 1 (Eq. A4), we obtain

(32)

where b is the conversion factor derived in Appendix A. The total length change in the cell, L_cell, will be the above equation multiplied by the number of folds, designated N_f. A prediction of L_cell versus the change in transmembrane potential is plotted in Fig. 7. As shown, the model predicts maximal cell deformations of the order of a few microns over membrane potential changes consistent with voltage clamp measurements (Santos-Sacchi, 1991). The model also predicts that the slope of the voltage-to-length curve is dependent on the ratio of the effective bending stiffness to the flexoelectric coefficient. We illustrate this in Fig. 7 by showing the predictions for different values of k_eff/f, which effectively alters the gain of the electromotility function. Note that these predictions are symmetric with respect to the origin of both the length and voltage axes. The electromotility data, in addition to being nonlinear, is also skew symmetric about both axes. We deal with these issues in the nonlinear model discussed below.

View larger version (10K): [in this window] [in a new window]   FIGURE 7   Length changes in the outer hair cell. The linear flexoelectric model predicts the length change as a function of transmembrane potential changes as shown here. The magnitude of the response depends on the ratio of the bending rigidity to the flexoelectric coefficient. We illustrate this for three different ratios: , keff = 4 × 1019 J, f = 1 × 1019 C; · · · ·, keff = 4 × 1019 J, f = 2 × 1019 C; - - -, keff = 4 × 1019 J, f = 4 × 1019 C. Cell length increases with hyperpolarization and decreases with depolarization.

Nonlinear outer hair cell length changes with voltage

The nonlinear flexoelectric model allows us to predict for the length change in the cell as a function of the polarizing field. For an unloaded cell that generates no force, Eq. 24 is written as

(33)

The longitudinal displacement of the motile unit is related to the curvature change from the initial state by the geometric relationship derived in Appendix A (Eq. A4). The initial curvature is expressed in terms of the equilibrium curvature via Eq. A7. Combining these equations with Eq. 33, we obtain the relationship,

(34)

This equation follows because a is the initial length of the motile unit at the initial curvature state. The prediction for the length change in the cell can be written as

(35)

where
The parameter L_V0 is a bias term that accounts for the skew symmetry with respect to the length axis of the electromotility function. Its magnitude depends on the passive mechanical properties of the membrane, and its sign depends on the displacement of the spectrin spring from its force-free length at the initial curvature. If the spring is extended, then x₀ < a and the electromotility curve is shifted downward. The saturating value of the length change when the Langevin saturates in the positive direction is equal to L_m + L_V0. Let us assume that our initial state is the force-free state of spectrin (a = x₀). The prediction for the electromotility curve is then given by Eq. 35 with L_V0 = 0. The curve will follow the Langevin function, which is a function of the dipole moment and the polarizing field. A prediction of L_cell versus the change in polarizing field is plotted in Fig. 8 for different values of the dipole moment p_o. This figure illustrates that, when the dipole moment is reduced, a transition from nonlinear to linear behavior is predicted in the experimental region of voltage clamp measurements. The predicted electromotility curves are centered at zero because we have, for now, assumed no spontaneous polarization. Thus, from Eq. 35, the parameter f_o = k_effL_m/N_fb.

View larger version (12K): [in this window] [in a new window]   FIGURE 8   Nonlinear length changes in the outer hair cell as a function of molecular dipole moment. The nonlinear flexoelectric model predicts the slope of the length change as a function of polarizing field will depend on the value of the dipole moment. These curves were generated from Eq. 35 with the parameters Nf = 2000, keff = 4 × 1019 J, fo = 1 × 1012 C/m, LV0 = 0. The value of the molecular moment was po = 125 D (top), po = 50 D (middle) and po = 10 D (bottom). The polarizing field 1 × 108 V/m corresponds to a membrane potential of 180 mV for po = 125 D, 372 mV for po = 50 D, and 474 mV for po = 10 D. (See Appendix B). Cell length increases as the polarizing field is increased and decreases as the polarizing field is decreased. Note that the direction of the polarizing field is opposite from the direction of the transmembrane potential, i.e., a positive polarizing field corresponds to a negative transmembrane potential.

The spontaneous polarization (E_p,o) is accounted for by expressing the argument of the Langevin function in terms of the membrane potential as

(36)

where 1   is the conversion factor between the polarizing field and the applied field (See Appendix B).

The predictions of the membrane-bending model can now be compared to experimental observations to determine values of the parameters. In Fig. 9, we illustrate the result of a nonlinear least squares regression fit to the experimental data of Santos-Sacchi (1992). The data were fit to the equation

(37)

where L_m, , V₀, and L_V0 were the free parameters of the fit. The nonlinear least squares regression yielded the results L_m = 1.27 µm,  = 0.09 mV¹, V₀ = 29 mV, and L_V0 = 0.78 µm (² = 0.00274). We now see that L_V0 accounts for the experimental observation that, at V = V₀, L_cell is not equal to zero.

View larger version (9K): [in this window] [in a new window]   FIGURE 9   Nonlinear regression fit to experimental data. The solid points are the experimental data on electromotility reported by Santos-Sacchi (1992). The solid line is the nonlinear least squares regression fit to the experimental prediction. The nonlinear least squares regression was carried out according to the Levenberg-Marquardt algorithm using the software package Origin (Microcal, Inc.). The parameters are described in the text.

Isometric Force

The isometric force is the force generated when no deformation of the membrane occurs. The magnitude of the isometric force can be estimated from the linear flexoelectric model applying the condition c_l = c_e in Eq. 17. (Note that it is also possible to apply the same condition to Eq. 24 to calculate the isometric force in the nonlinear case). In this case, the active force is the flexoelectric coefficient multiplied by the electric field, fE. The active force per unit length of the membrane equals fE divided by the arc length of the motile unit (s = 2R). This expression takes the form,

(38)

The coefficient of the active force is the derivative of the active force with respect to the membrane potential,

(39)

Assuming f to be 10¹⁸ C, the coefficient of the active force evaluates to 4.5 × 10³ N/Vm, a value similar to that reported by Spector et al. (1999a,b) and Spector (1999) for the longitudinal component of the force in the orthotropic electroelastic model. This coefficient can be converted to a coefficient in terms of force per voltage change by multiplying by