The Local Forces Acting on the Mechanotransduction Channel in Hair Cell Stereocilia

Abstract

In hair cells, mechanotransduction channels are located in the membrane of stereocilia tips, where the base of the tip link is attached. The tip-link force determines the system of other forces in the immediate channel environment, which change the channel open probability. This system of forces includes components that are out of plane and in plane relative to the membrane; the magnitude and direction of these components depend on the channel environment and arrangement. Using a computational model, we obtained the major forces involved as functions of the force applied via the tip link at the center of the membrane. We simulated factors related to channels and the membrane, including finite-sized channels located centrally or acentrally, stiffness of the hypothesized channel-cytoskeleton tether, and bending modulus of the membrane. Membrane forces are perpendicular to the directions of the principal curvatures of the deformed membrane. Our approach allows for a fine vectorial picture of the local forces gating the channel; membrane forces change with the membrane curvature and are themselves sufficient to affect the open probability of the channel.

Introduction

In organs of hearing and balance, hair bundles (Fig. 1, A and B) provide a mechanical-to-electrical signal conversion, which alters the membrane potential of hair cells. Each hair bundle consists of ∼100 actin-filled stereocilia, which are arranged in rows of increasing height and interconnected by various links. Tip links reach between pairs of stereocilia in adjacent rows and are thought to gate cation-selective mechanotransduction channels. The channel is located near the base of the tip link, in the tip membrane of the shorter stereocilia (1). Bundle movement toward the tallest stereocilia (the excitatory direction) increases force on tip links and opens transduction channels, depolarizing hair cells; movement in the opposite (inhibitory) direction decreases tip-link force, closes channels, and hyperpolarizes hair cells (2). The conductance of the stereocilia mechanotransduction channels varies tonotopically, being larger in high-frequency hair cells (3,4); channels are also regulated by several forms of adaptation, which control the time course of current flow into hair cells (5).

Figure 1.

Membrane deformation and forces acting on tip links and transduction channels in hair cells. (A) Cartoon illustrating the location of the tip link and its components PCDH15 and CDH23. In addition, a pair of channels centrally located at the tip-link insertion is illustrated; this location is hypothetical. The channel complex is anchored to the underlying actin core by a hypothetical tether. Finally, bonds attaching the actin core to the membrane are also illustrated. (B) Scanning electron micrograph of a stereocilia pair connected by a single tip link. This view is orthogonal to that in the cartoon of A. Scale bar, 100 nm. (C–F) At any point of the deformed membrane, the circumferential (around the axis of the stereocilia) and meridional (along the surface of the membrane) directions are the orthogonal principal directions of the local curvature, and F₁ and F₂ are the corresponding components of the membrane resultant. In the case of the centrally located channel (C and D), a dotted line around the channel (gray curvilinear rectangle), defines a representative piece of the membrane around the channel (C), and the vertical force, F_t, is the force associated with a hypothetical tether connecting the protein to the actin core of the stereocilia (dashed line). The membrane forces (resultants) act on the edges of the membrane cut around the protein (D). Each edge of the cut is perpendicular to the corresponding meridional direction, and therefore, the membrane forces are equal to the local resultant component F₂. In the case of the acentrally located channel (E and F), the channel is located away from the center of the membrane (E). Due to the acentral position of the channel, one pair of parallel edges of the membrane cut is perpendicular to the circumferential direction, and the other pair is perpendicular to the meridional direction (F), and the membrane forces are associated with the resultants, F₁ and F₂, respectively. To see this figure in color, go online.

An influential model for mechanotransduction suggested that a gating spring connects stereocilia displacement to changes in force applied to the transduction channel (6). The stiffness inferred for the gating spring is ∼10⁻³ N/m (7–10). Although the tip link was initially proposed to be the gating spring (11), the molecular composition of the tip link revealed later suggested that the tip link must be too stiff to match the elasticity of the gating spring (12,13). Electron micrographs show tenting, where the membrane at the base of the tip link, close to the channel, is pulled away from the underlying cytoskeleton (14), suggesting that this membrane configuration adds the necessary elasticity to the transduction apparatus.

To investigate the role of the membrane in transduction mechanics, we used a biophysical model describing the deformation of the membrane in response to force exerted through the tip link (15). We found that hair bundles with a relatively low gating-spring stiffness, such as those found in frog hair cells, could be described by gating springs consisting entirely of the membrane mechanical properties. By contrast, bundles with stiffer gating springs, like those in rat hair cells, required the membrane to be reinforced by an intracellular tether running from the membrane to the stereocilia core. Although direct experimental confirmation of this concept is still absent, our modeling study showed that the tip membrane by itself—or strengthened by a tether connecting it to the stereocilia actin core—can serve as the compliant part of the gating spring (15).

Although the focus of the previous analysis (15) was the role of the tip membrane in the gating spring, here we concentrate on the local forces affecting the channel. The forces that govern the gating of the transduction channel are caused by the movement of the hair bundle and are applied via the tip link. However, the system of forces acting on the channel protein is not known. In principle, the tip link could be directly connected to the channel or attached to the membrane distant from the channel. The channel could be free-floating within the membrane, or it could be connected to the stereocilia actin core by a tether. These options can result in different systems of forces governing the gating of the channel; such systems include both in-plane (membrane) and out-of-plane (tip-link- and tether-associated) components. The near-channel forces are obtained for different biophysical parameters of the membrane and tether, and they are presented as functions of the tip-link force. We found that the membrane forces (resultants) can be significantly different from membrane tension due to the high curvature of the tip membrane. Moreover, two components of the membrane force can differ between each other; for example, one can be tensile whereas the other is compressive. The membrane forces are large enough to change the open probability of the channel. The out-of-plane tether forces are also important for the channel. Overall, the obtained results provide a fine vectorial picture of the three-dimensional system of forces acting on the stereocilia mechanosensitive channel.

Model and Numerical Method

We describe the environment around the mechanotransduction channel, including the membrane in the tip of the shorter stereocilia, bonds between the membrane and the stereocilia actin core, channel (protein), and hypothetical channel tether in terms of the system’s potential energy. The potential energy of the system, Π, is given by the equation

$$\Pi =E_m+E_b+E_p+E_t-A=\underset{S}{\int }\{\frac{1}{2}\kappa {\left(c_1+c_2\right)}^2+\overline{\kappa }{c}_1{c}_2+\gamma +{\delta }_1\left(k_b{z}^2-G_0\right)+\frac{1}{2}{k}_p\left[{\left(\Delta N\right)}^2+{\left(\Delta L\right)}^2\right]\}dS+\frac{1}{2}{\delta }^{2}K{h}_1^2-F{h}_2.$$ (1)

Here, the total potential energy, Π, is equal to the sum of the internal energies of the elastic membrane, E_m, bonds, E_b, channel protein, E_p, tether, E_t, and negative work, −A, produced by the tip-link force. The internal energy of the membrane is described by the first three terms under the integral (over the surface area, S, occupied by the membrane) sign in Eq. 1, and it is characterized by the principal curvatures, c₁ and c₂, and the parameters, κ (bending modulus), $\overline{\kappa }$ (Gaussian curvature modulus), and γ (membrane tension). Note that membrane tension equilibrates the membrane interaction with the large lipid area along the side surface of the stereocilia. As a result, the membrane surface area is not preserved and can change quite significantly as it is shown below. The bond energy is given by the fourth term under the integral sign in Eq. 1. It is characterized by two parameters, G₀ (density of the adhesion energy) and k_b (bond stiffness). It is assumed that the bond can be stretched up to a certain threshold, $z=z_1$, beyond which it breaks. The bond can also be compressed up to a limit, $z=z_0$, above which it cannot be deformed. Thus, the coefficient ${\delta }_1$ in Eq. 1 can be described by the equation

$${\delta }_1=\{\begin{array}{l}1\text{if}\text{the}\text{bond}\text{is}\text{deformed}\text{between}\text{two}\text{thresholds}\left(z_0\le z\le z_1\right)\hfill \\ 0\text{if}\text{the}\text{bond}\text{is}\text{streched}\text{beyond}\text{the}\text{stretching}\text{threshold}\left(z>z_1\right)\hfill \\ \infty \text{if}\text{the}\text{bond}\text{is}\text{compressed}\text{beyond}\text{the}\text{compression}\text{limit}\left(z<z_0\right)\hfill \end{array}.$$ (2)

The last term under the integral sign in Eq. 1 represents the elastic energy of the protein where the ΔN- and ΔL-functions are determined by the inclination of the normal vector and the in-plane deformation of the membrane due to the application of the tip-link force, respectively. We give more explicit equations for these protein-associated terms when we consider the numerical method below. The stiffness of the protein is characterized by the parameter, k_p. The first term outside the integral sign in Eq. 1 represents the elastic energy of the tether if the channel is connected to the stereocilia actin core. Here, K is the stiffness of the tether, h₁ is the height of the deformed membrane at the center of the protein, and the coefficient δ₂ is given by the equation

$${\delta }_2=\{\begin{array}{l}1\text{if}\text{the}\text{channel}\text{protein}\text{is}\text{tethered}\text{to}\text{the}\text{actin}\text{filaments}\hfill \\ 0\text{if}\text{the}\text{channel}\text{protein}\text{is}\text{not}\text{tethered}\text{to}\text{the}\text{actin}\text{filaments}\hfill \end{array}.$$ (3)

Finally, the last term on the righthand side of Eq. 1 (second term outside the integral sign) represents the work done by the tip-link force, F, on the displacement, h₂, equal to the height of the membrane at the point of the force. After the computation of the membrane shape, we determine the forces that affect the gating of the channel that have in-plane and out-of-plane components. We present the in-plane forces in terms of two components of the membrane resultant parallel to the directions of the local principal curvatures. The out-of-plane forces consist of the tip-link force and possibly the tether force if the channel is connected to the stereocilia actin core. As in our previous studies (15–17), we use a two-term description of the membrane resultants. The first term in the membrane resultants is equal to membrane tension, and the second term, which becomes significant in highly curved membranes like the one in the tip of the stereocilia, is determined by the local membrane curvatures. Thus, the membrane resultants can be described by the equations

$$F_1=\gamma +\frac{1}{2}\kappa \left(c_1^2-c_2^2\right)\text{and}{F}_2=\gamma +\frac{1}{2}\kappa \left(c_2^2-c_1^2\right).$$ (4)

These equations can be derived from a refined shell theory (18) as necessary conditions of a minimum of the potential energy, Π (19). Note that this model describes the membrane as a fluid with bending resistance resulting in anisotropic forces in highly curved membranes. The membrane forces (resultants) reduce to isotropic tension in the case of relatively small curvatures. We estimate the membrane forces gating the channel (in its central and acentral locations) by considering a membrane cut around the protein and computing the membrane resultants that act on the cut edges (Fig. 1, C–F).

To compute the deformed state of the membrane and membrane forces, we use Monte Carlo simulation; similar to Atilgan and Sun (20), we discretize the membrane area using a triangular mesh where each vertex, n_m, that belongs to the membrane is surrounded by a hexagonal area (Fig. 2 A). The protein is associated with a single node, n_p, of the mesh at its center, and it is assumed to have cylindrical shape with a hexagonal cross section (Fig. 2 A). Fig. 2 B presents this mesh for the deformed state of the membrane. Then we compute the discrete analogs of energy terms associated with the membrane, bonds, and tether. Regarding the protein, we describe the ΔN- and ΔL-related terms in the energy functional as

$${\left(\Delta N\right)}^2={\sum }_i{\left(\Delta N_i\right)}^2\text{and}{\left(\Delta L\right)}^2={\sum }_{ij}{\left(\Delta L_{ij}\right)}^2.$$ (5)

Here, the first sum in Eq. 5 is over changes, ΔN_i, in six angles, N_i, between the vector normal to the membrane in the reference state and the edge, e_i, associated with the central node of the protein (Fig. 2 A). The second sum in Eq. 5 is over changes, ΔL_ij, in the angles L_ij between six pairs of the neighboring edges, e_i and e_j, associated with the same central node of the protein (Fig. 2 A). Note that in the reference state, $N_i=\pi /2\text{and}{L}_{ij}=\pi /6$.

Figure 2.

Computational method. (A) Triangular mesh with six edges associated with a representative membrane node, n_m. The protein with the central node, n_p, is shown as a gray hexagon, e_i are the six edges of mesh associated with the protein central node, and N is the normal to the membrane in the undeformed state. After deformation, the angles between N are e_i (originally equal to π/2) become equal to N_i and the angles between neighboring e_i and e_j (originally equal to π/6) become equal to L_ij. (B) The triangular mesh after deformation of the membrane. (C) The finer square mesh used to estimate the membrane resultants along the edges of the cut around the protein after deformation of the membrane. The applied force and membrane parameters are the same as those used in B.

Our computational model takes into account a grid of 1755 nodes, with an initial link length of 6 nm between each node. We consider two cases of a single channel protein of radius 3 nm. In one case, the protein is centered at the same node where the tip-link force is applied (the central case); in the other, the protein is centered at a node 12 nm away from the node where the tip-link force is applied (the acentral case). To evaluate the forces that affect channel gating, we consider a representative piece of the membrane bounded by two pairs of edges, where each pair is perpendicular to one of the principal directions of the curvature. Here, we consider a circular initial state of the membrane and an external force applied at the center of the circle. Thus, the shape of the deformed membrane with no protein in it would be axisymmetric. In principle, a finite-sized protein of a hexagonal cross section results in some asymmetry of the solution. However, since the radius of the protein is relatively small (3 nm vs. 50 nm for the radius of the membrane projection area), its effect on the shape of the membrane was found to be insignificant. Thus, we interpret the principal directions involved in the derivation of the membrane resultants as the meridional and circumferential directions.

For an acentrally located channel, one pair of edges of the membrane region around the protein is perpendicular to the circumferential direction, whereas the other is perpendicular to the meridional direction. In contrast, all edges of the membrane piece around a central protein are perpendicular to the local meridional direction. To compute the resultants, we chose the size of the membrane region to include only the protein; we use the edges to be at a distance of 4 nm along the principal directions from the center of the protein.

After each Monte Carlo simulation of the membrane shape, the mesh is deformed and scattered (Fig. 2 B) and does not necessarily contain a data point precisely along the edges of the membrane piece around the protein. Thus, to calculate forces along the principal directions, a finer square grid is required that contains data at those points of interest. This finer mesh is shown in Fig. 2 C for the same deformed state of the membrane as shown by the original triangular mesh in Fig. 2 B. Such a finer mesh is approximated via a standard natural neighbor interpolation method. With this finer mesh, we can similarly produce a functional approximation of F₁ and F₂ via standard linear interpolation and then use that functional approximation and apply it along the square mesh to get values for the resultants at the points of interest.

Results

To expand on our previous study (15), we characterize here multiple realistic configurations of channel, tip link, and tether, all of which will influence membrane structure and transduction-channel gating in the stereocilia tip (Fig. 1). We consider several possible scenarios of channel arrangement, including location of the channel at the point of the tip-link force or away from it, as well as tethering of the channel to the actin core of the stereocilia. For simplicity, the tip-link force is assumed to be applied at the center of the membrane.

Mechanotransduction channels of sensory hair cells are located within the membranes at stereocilia tips, specifically near tip-link bases (1). Although the overall composition of the stereocilia membrane is known (21), its mechanical properties are not available at present, and so we use these parameter values gleaned from the literature. The membrane surrounding the channel is connected to the membrane covering the stereocilia shafts, which provides a large lipid reservoir. We assume that the channel membrane environment equilibrates quickly and can be characterized by constant membrane tension. In our previous analysis of the stereocilia membrane (15), we found that the bending modulus has a more significant effect on membrane deformation and forces than does membrane tension. Membrane tension in typical cell membranes has been estimated to be 1–3 × 10⁻⁴, 3 × 10⁻⁵, and 5 × 10⁻⁵ N/m (22–24); we use an intermediate value of 10⁻⁴ N/m here.

To compute the main forces generated in the stereocilia tip membrane around the mechanotransduction channel, we modeled the channel as a cylindrical inclusion of a hexagonal cross section (Fig. 2 A), with the length of the base side 3 nm (diameter ∼6 nm); it is considered as relatively rigid (but deformable), with a stiffness parameter of k_p = 200 k_BT (20). Although the size of the transduction channel is not known definitively (but see Forge et al. (25)) and could be larger than this, our model channel diameter is greater than is the acid-sensing ion channel (26), a member of the same channel family as the mechanosensitive degenerins of Caenorhabditis elegans (27). The near-channel forces are computed as functions of the tip-link force, membrane properties, and channel arrangement (position/tethering to the actin core). In the case of tethered channels, we considered a range of values for the tether stiffness that are comparable with the available estimates of the gating-spring stiffness (15).

As described previously (15), our model includes a system of bonds that connect the membrane to the actin core; the membrane becomes partially debonded depending on the applied force and membrane parameters. In our computational method, we used values for adhesion energy density (G₀), rupture threshold (z₁), and compression limit (z₀) of 0.37 k_BT, 4 nm, and −0.5 nm, respectively. These values are physiologically realistic. The adhesion energy density (per computational node) (G₀), was estimated from measurements of membrane adhesion energy (per surface area) in red blood cells and fibroblasts (28,29). The bond strength and associated adhesion energy depend on the strain (force) rate, which in our case is determined by the frequency of hair cell excitation. Thus, the bond adhesion energy may be different under high-frequency conditions. Experimental data for the bond adhesion energy at high strain rates that correspond to tens of kHz of hair cell excitation are presently unavailable. The rupture and compression limits are comparable to the thickness of the membrane bilayer and an actin monomer, respectively. In NIH 3T3 cells, the membrane phospholipid PIP₂ plays the role of connecting the membrane to the cytoskeleton (28). Indeed, PIP₂ is prominent in stereocilia membranes, particularly near stereocilia tips (21,29); PIP2 may thus play a similar role in stereocilia adhering the membrane to the cytoskeleton.

Channels located centrally relative to the tip link

The transduction channel may be located immediately adjacent to the base of the tip link. We therefore first consider the cases of tethered and untethered channels that are centrally located at the tip link; under these conditions, the tip link directly gates the channel, although membrane forces are important as well. Fig. 3 shows representative membrane shapes that result from the application of a 50 pN tip-link force for the cases of an untethered channel (Fig. 3 A) or a channel with a tether stiffness of 0.6 pN/nm (Fig. 3 B). Debonded and bonded parts of the membrane are shown in black and gray, respectively; in both tethered and untethered cases, the bonds break in the central part of the membrane area, closer to the applied force, whereas the more peripheral points remain bonded. Because the membrane is reinforced when the channel is tethered, the debonded area is smaller and the vertical extension of the membrane is reduced.

Figure 3.

Computational images of the deformed stereocilia membrane with a central finite-sized channel. Black and gray areas correspond to the debonded and bonded parts, respectively, of the deformed membrane. (A) Untethered channel. Applied force and membrane bending modulus are 50 pN and 40 k_BT, respectively. (B) Tethered channel. Applied force, membrane bending modulus, and tether stiffness are 50 pN, 40 k_BT, and 0.6 pN/nm, respectively.

As illustrated in Fig. 1 D, when the channel is located at the tip-link position, the local membrane forces (resultants) it senses are strictly meridional (F₂). For an untethered channel, no significant change in F₂ is seen until 20–30 pN of force is applied (Fig. 4 A); at that point, the force generated by bending the membrane overcomes the constant membrane tension. If the membrane tension is lower than the value we used here (10⁻⁴ N/m), the transition point where F₂ begins to increase will occur at a lower applied force. Membrane forces are larger for smaller values of the bending modulus; indeed, with the highest bending modulus we used (80 k_BT), there is little effect of tip-link force on F₂. Thus when the channel is untethered, in-plane gating forces are not large unless the bending modulus is small.

Figure 4.

Membrane forces (resultants) and tether force for a central channel. The membrane tension in all cases is equal to 10⁻⁴ N/m. (A and B) Membrane resultants perpendicular to the local meridional direction, one of the principal directions of the membrane curvature, are shown for a central untethered channel (A) with the membrane bending modulus varied (20 (solid line), 40 (dashed line), and 80 k_BT (dotted line)) and for a central tethered protein (B)with tether stiffness varied (0.4 (solid line), 0.6 (dashed line), and 0.8 pN/nm (dotted line)) and the membrane bending modulus equal to 60 k_BT. (C and D) Tether forces for the tethered central channel when the membrane bending moduli are varied (20 (solid line), 40 (dashed line), and 80 k_BT (dotted line)) and the tether stiffness is equal to 0.6 pN/nm (C) and when the tether stiffness is varied (0.4 (solid line), 0.6 (dashed line), and 0.8 pN/nm (dotted line)) and the membrane bending modulus is equal to 60 k_BT (D).

The effects of tethering on F₂ for a centrally located channel are shown in Fig. 4 B. As the tether stiffness increases, F₂ decreases. As with an untethered channel, tip-link force has a relatively small effect on F₂.

When the channel is tethered to the actin core, the tether force itself is part of the system of forces acting on the channel. Although F_t is larger when the bending modulus is smaller, substantial force still must be applied to overcome the membrane's bending modulus (Fig. 4 C). The potential energy (Eq. 1) allows for tether force to be a nonlinear function of the tether stiffness, K, because the vertical coordinate, h₁, in Eq. 1 depends on K. Nevertheless even for large tip-link forces, the tether force F_t varies nearly linearly with tether stiffness, nearly doubling when the tether stiffness doubles (Fig. 4 D).

Channels located acentrally relative to the tip link

Alternatively, the transduction channel may be located distant from the tip link, which would require it to be gated either by in-plane membrane forces or by tether forces, if the tether is present. We therefore consider the case of an acentral channel, located 12 nm away from the center of the circular membrane area, whose radius is 50 nm (Fig. 1 E). Fig. 5 shows representative images of the deformed membrane in the case of untethered (Fig. 5 A) or tethered (Fig. 5 B) acentral channels. With a tethered acentrally located channel, the deformed membrane is not axisymmetric; the membrane shape in the vicinity of the protein is distorted by the presence of the channel. However, we found that this effect was insignificant when we used realistic membrane parameter values.

Figure 5.

Computational images of the deformed stereocilia membrane with an acentral finite-sized channel. The channel is located on the x axis at a distance of 12 nm from the center. Black and gray areas correspond to the debonded and bonded parts, respectively, of the deformed membrane. Membrane tension is equal to 10⁻⁴ N/m. (A) Untethered channel. Applied force and membrane bending modulus are 50 pN and 40 k_BT, respectively. (B) Tethered channel. Applied force, membrane bending modulus, and tether stiffness are 50 pN, 40 k_BT, and 0.6 pN/nm, respectively.

If untethered, an acentrally located transduction channel would be gated exclusively by in-plane forces. In this case, the membrane forces near the channel are characterized by two components that correspond to the principal radii of the curvature, the circumferential (F₁) and meridional (F₂) directions (Fig. 1 F). Due to opposite signs in the curvature-related terms describing these components of the membrane resultant (Eq. 4), F₁ and F₂ membrane forces respond very differently to membrane deformation (Fig. 6, A and B). As seen above for the centrally located channel, F₂ increases as applied force is increased; moreover, as with the central-channel case, F₂ is highest when the bending modulus is smallest (Fig. 6 B). By contrast, F₁ decreases as applied force is increased, eventually reaching negative values (Fig. 6 A). The decrease in F₁ is greatest with the smallest values of bending modulus.

Figure 6.

Membrane forces (resultants) and tether force for an acentral untethered or tethered channel. In all cases, membrane tension is equal to 10⁻⁴ N/m (A and B) The circumferential component, F₁ (A). and the meridional component, F₂ (B), for the untethered channel, where the membrane bending moduli are 20 (solid line), 40 (dashed line), and 80 k_BT (dotted line). (C and D) The circumferential component, F₁ (C). and the meridional component, F₂ (D), for the tethered channel, where the tether stiffnesses are 0.4 (solid line), 0.6 (dashed line), and 0.8 pN/nm (dotted line) and the bending modulus is equal to 60 k_BT. (E and F) Tether force as a function of bending modulus (E), where membrane bending moduli are 20 (solid line), 40 (dashed line), and 80 k_BT (dotted line) and tether stiffness is 0.6 pN/nm, and as a function of tether stiffness (F), where stiffnesses are 0.4 (solid line), 0.6 (dashed line), and 0.8 pN/nm (dotted line), and bending modulus is 60 k_BT.

Finally, the transduction channel could be located away from the tip link yet still be tethered to the cytoskeleton. Under these conditions, the decrease in F₁ is substantially dampened compared to the untethered condition (Fig. 6, A and B), so F₁ never reaches negative values. The tether force, which could gate the transduction channel, depends strongly on bending modulus but less so on tether stiffness. As in the case of a central channel, variations in bending modulus have a far greater impact on F_t (Fig. 6 E) than does the value of tether stiffness (Fig. 6 F). Moreover, values for F_t for the acentral tethered channel are close to those for the central tethered channel.

Discussion

Hair bundle mechanics are traditionally presented in terms of hair-bundle deflection (7,31), a parameter that can be directly controlled in an experiment. However, forces applied to the channel and channel open probability are determined by properties of the gating spring, a phenomenological entity whose nature has yet to be clarified (15). Because the parts of the tip link outside the membrane are thought to be too stiff to serve as the gating spring, the gating-spring compliance must in some way be associated with the membrane, tethered or untethered to the stereocilia actin core. Thus, to estimate forces acting on the channel with the gating-spring approach, we have to know the deformed state of the membrane, which is more easily addressed by modeling than by experiments.

In the traditional analysis of stereocilia mechanics, where hair bundle deflection is used as the main variable, channel opening effectively lengthens the gating spring and produces an apparent decrease of bundle and gating-spring stiffness; by contrast, in our model, we effectively clamp the tip-link force at a constant value. We express all near-channel forces, as well as membrane deformation and height, as a function of a single external force, the tip-link force. For a given deflection of the bundle, the tip-link force can be larger in organs of greater bundle stiffness (e.g., the cochlear outer hair cell) or smaller in organs of lower bundle stiffness (e.g., the bullfrog sacculus). Here, we consider a broad range of tip-link forces to reveal the main features of the resulting membrane forces.

The interrelationship between the tip-link force, open probability of the channel, and bundle stiffness will be affected by properties of the membrane (tethered or untethered) and the system of forces in the stereocilia tip. We can hypothesize that a change in channel conformational state results in some change in the tip-link force, reducing it upon channel opening and increasing it upon channel closing. Although the change in the tip-link force will be sensed by all components of the force involved and will depend on the channel arrangement, it is reasonable to assume that the internal energy of the system will be smaller for an open channel (lower tip-link force), similar to the traditional analysis. Note that the effect of channel opening and closing on near-channel forces is probably associated with the change in the tip-link force and not with the direct effect of conformational changes in the protein on the surrounding membrane, because a 2- to 3-nm throw effect on a 50-nm-radius membrane will be too small. Finally, changes in the tip-link force for a given displacement of the bundle explain how opening and closing of the channels affects hair bundle stiffness; the quantitative assessment of this relationship will be a subject of future analysis.

For the simplicity, we neglect here the inclination of the tip-link force and variation of the membrane projection area and locate tip-link force centrally, which reduces the problem to an axisymmetric one. However, the proposed approach is general, and it has been effective in the analysis of three-dimensional deformation of the stereocilia membrane (15).

Out-of-plane gating of transduction channels

Transduction channels could be gated by forces in or out of the plane of the membrane (32). For example, out-of-plane forces along the tip link and tether, when present, could displace a gating element perpendicular to the membrane, much as voltage translocates the S4 helix in voltage-gated superfamily channels (33). For the channel to be gated out of plane, the tip link must be directly connected to the channel, and the relevant force is that felt at the gating element as the tip link pulls against the channel immobilized in the membrane, with or without a tether. In the absence of a tether, the out-of-plane force displaces the channel protein relative to the membrane bilayer force profile, changing the relationship with the lipids and allowing channel dilation (32). When a tether is present, the impact of the membrane environment on gating is reduced; the meridional resultant, F₂, is much smaller when even modest tether stiffnesses are considered (compare Fig. 4 B with Fig. 4 A).

When an acentrally located channel is tethered, the out-of-plane tether force, F_t, is similar to that seen with the centrally located channel (compare Fig. 6, E and F, with Fig. 4, C and D), and so in this case, forces could gate the channel with a perpendicular domain displacement or a modification of the membrane bilayer force profile. However, the overall out-of-plane force is different for the two locations due to the direct application of the tip-link force to the centrally located channel. Taken together, these results suggest that a channel gated by out-of-plane forces along a tether can be located at or away from the tip-link location.

In-plane gating of transduction channels

Alternatively, in-plane membrane forces could expand the channel's pore, much as is thought to happen with bacterial osmosensor channels (34) and eukaryotic mechanosensitive K⁺ channels (35). Application of the tip-link force to the membrane distorts it and results in the generation of in-plane membrane forces, which are described in terms of membrane resultants. Experiments on force-gated channels from animal cells have demonstrated that in-plane membrane forces alone can change the probability of channel opening (36–38). Under those experimental conditions, such as inside a micropipette, the difference between tension and membrane resultants was probably small due to a curvature much larger than in our case. Thus, we can use the available experimental data that estimate the dependence of mechanosensitive channel gating on membrane tension and compare them with the resultants in our case. These data suggest that the tension required to half open mechanosensitive channels in animal cells is between 10⁻⁴ and 10⁻³ N/m (36–38).

Because channel opening is minimal in the absence of tip-link force (39,40), if our assumed resting membrane tension is appropriate (10⁻⁴ N/m) and channels are gated by in-plane forces, the force required to half-open transduction channels is likely closer to 10⁻³ N/m. The membrane resultants we calculate are above the lower limit (10⁻⁴ N/m) of tension significant to channel opening and do approach 10⁻³ N/m. The largest resultant was observed in the case of a central untethered protein, in particular for a membrane with a smaller bending modulus (Fig. 4 A). For membranes with a bending modulus of 20–40 k_BT that are subjected to large tip-link forces, the resultant component, F₂, can exceed the upper end (10⁻³ N/m) of the experimentally estimated interval of membrane tension significant to channel opening. The applied force required to reach this level (>80 pN) is substantially greater than forces realistically reached in animal hair cells (41). However, this conclusion is dependent on assumptions in our modeling; and gating transduction channels with in-plane membrane forces certainly is plausible.

With an acentral channel, the circumferential resultant, F₁, is relevant. Strikingly, F₁ decreases with an increase of tip-link force, in some cases becoming compressive. This result suggests that if the channel is acentrally located and is gated by in-plane forces, the mode of gating is likely to be complex—expansive meridional forces would be countered by compressive circumferential forces in the membrane.

By contrast, in comparison to the acentral untethered channel, membrane resultants are decreased substantially in the case of an acentral tethered channel (compare Fig. 6, C and D, to Fig. 6, A and B). The tether force is significantly smaller than the external tip-link force for membranes with bending moduli >40 k_BT. However, the tether force becomes much closer to the tip-link force when the bending modulus is 20 k_BT and applied forces are large.

As the channel is moved away from the point in the membrane where the tip link applies force, however, the resultants for the same force and bending modulus are smaller than those for a central protein. For a channel located 12 nm from the tip link, with a bending modulus of 20 k_BT and large tip-link forces (Fig. 6 B), the resultant F₂ is only about two-thirds as large as in the case of the central channel (compare Fig. 6 B with Fig. 4 A). For a channel located 24 nm away from the applied force, the membrane forces decrease even further when compared to those for a centrally located channel (not shown). Together these results show that the efficiency of gating the transduction channel by membrane forces decreases as the channel is moved from the position where force is applied to the membrane, the tip-link base. If the channel was not immediately adjacent to the tip link, the forces felt by the channel would be submaximal and would vary with channel distance from the tip link. Instead, interaction of the channel with the tip link, with or without tethering, would position the channel in the optimal orientation for application of both in-plane and out-of-plane forces.

Membrane resting tension and tip-link resting force

For smaller tip-link forces, there is a minimal change in the computed resultants F₁ and F₂ with applied force (a plateau region); for larger tip-link forces, the reactive forces change nonlinearly with an increasing dependence on force. The deflection point (the point of deviation from the plateau) depends on the bending modulus of the membrane and corresponds to a smaller force if the bending modulus is smaller. This pattern stems from the two terms describing membrane force, where the first term is equal to membrane tension and the second term is determined by membrane curvature. For larger tip-link forces, the membrane forces become dominated by the curvature-related term. Interestingly, the increase in the resultants with tip-link force is greater for smaller bending moduli, despite the fact that the curvature-associated term in the resultants is proportional to the bending modulus of the membrane. The obtained results mean that in softer membranes, the larger overall deformation and changes in the local curvatures are more significant to the resultants than the decrease in the bending modulus.

The sensitivity of the system to applied forces in the plateau region (i.e., <15 pN applied force) is very low, which suggests that if the transduction channel is gated by in-plane forces, the hair cell would need to position the transduction apparatus so that the resting force is outside of the plateau region (Fig. 6, A–D). Indeed, the resting force generated by the slow adaptation mechanism (42) is ideally suited to tension the system so that it is optimally responsive, i.e., at a steep part of the applied force-resultant relationship. This interpretation suggests that one role of the adaptation motor is to position the transduction apparatus at the point where external force leads to significant membrane bending.

On the other hand, the resting tip-link tension in frog hair cells is estimated to be ∼10 pN (43), which would not be sufficiently large to increase membrane resultants sufficiently to readily gate the transduction channel (Fig. 6, A–D). Although some calculations indicate that resting tension in mammalian hair cells might be substantially greater (44), these predictions have not been confirmed. Thus, experimentally confirmed estimates of resting tension (31) suggest that if the values for the bending modulus used here (20–80 k_BT) are accurate, channel gating through membrane resultants is less likely than gating by direct displacement of channel transmembrane domains.

Tether force

The tether force is a result of the equilibrium of the curved membrane under the action of two out-of-plane forces from the tip link and the tether. Although the dependence of this force on the tip-link force does not have a plateau region, it also shows a nonlinear increase for larger applied forces (Figs. 4 and 6). Similar to the resultants, the tether force is larger for softer membranes (smaller bending modulus), which corresponds to a greater height of the membrane and a more extended tether. On the other hand, tether forces increase as the stiffness of the tether increases, despite smaller membrane heights and a less extended tether. In this case, resting force in the tip link and tether could serve to supply sufficient energy to overcome the closed-state energy of the transduction channel (6).

Tenting

Modeling also allows us to predict the three-dimensional shape of the stereocilia tip membrane, which gives insight into the phenomenon of tenting, as seen in electron microscopy images of tip links. Although the membrane tenting seen at the base of the tip link could represent membrane compliance (14), the nature of membrane tenting is not fully clear; experimental observation of tenting required fixing and dehydrating the sample, which could have produced artifactual membrane displacement. In this study, we make no assumptions besides considering the stereocilia tip membrane in the undeformed state as being flat and bonded to the actin core. The deformed state then appears naturally as a result of the application of tip-link forces to the membrane with biophysically reasonable properties, and it displays membrane tenting very similar to that seen in electron micrographs (see also Kim et al. (45)). Modeling of the response of the membrane to force applied through the tip link thus provides a view of membrane tenting that is distinct from that seen in electron microscopy experiments and suggests that it is a feature of living hair cells.

Conclusions

Here, we describe modeling of the stereocilia tip membrane under the influence of an external force applied through the tip link, with explicit consideration of a channel protein and its membrane environment. Out-of-plane forces in the tip link depend on the physical properties of the bilayer, with the membrane bending modulus having the greatest effect, although tethering the channel intracellularly reduces the impact of the membrane properties. In-plane membrane forces are complex and can include both expansive and compressive forces. They can be significantly affected by the membrane curvature and deviate from the membrane tension. Our results show that forces large enough for transduction channel gating could be generated by out-of-plane forces in the tip link and tether, by in-plane forces in the membrane, or by both. However, if the channel is gated by in-plane forces, location of the channel immediately adjacent to or directly attached to the tip link leads to the steepest relationship between tip link and gating forces, which corresponds to the greatest sensitivity of the channel to external force. These modeling results will guide future studies that identify and characterize the mechanotransduction channel in hair cells.