Hair bundles of cochlear outer hair cells are shaped to minimize their fluid-dynamic resistance

Abstract

The mammalian sense of hearing relies on two types of sensory cells: inner hair cells transmit the auditory stimulus to the brain, while outer hair cells mechanically modulate the stimulus through active feedback. Stimulation of a hair cell is mediated by displacements of its mechanosensitive hair bundle which protrudes from the apical surface of the cell into a narrow fluid-filled space between reticular lamina and tectorial membrane. While hair bundles of inner hair cells are of linear shape, those of outer hair cells exhibit a distinctive V-shape. The biophysical rationale behind this morphology, however, remains unknown. Here we use analytical and computational methods to study the fluid flow across rows of differently shaped hair bundles. We find that rows of V-shaped hair bundles have a considerably reduced resistance to crossflow, and that the biologically observed shapes of hair bundles of outer hair cells are near-optimal in this regard. This observation accords with the function of outer hair cells and lends support to the recent hypothesis that inner hair cells are stimulated by a net flow, in addition to the well-established shear flow that arises from shearing between the reticular lamina and the tectorial membrane.

Figure 1

Anatomical environment of cochlear hair bundles. (a) Scanning electron microscopy shows hair bundles protruding from the reticular lamina (scale bar 15 μm, tectorial membrane removed). The hair bundles of inner hair cells are planar (asterisk) whereas those of outer hair cells have a characteristic V-shape (double asterisk). (b) Hair bundles of the outer hair cells connect the reticular lamina (RL) to the tectorial membrane (TM). Oscillatory fluid flow occurs in the radial direction, perpendicular to the rows of hair cells (double-sided arrow). (c) Fluid flow around hair bundles can include shear flow as well as net flow. The latter may arise from squeezing of the gap between the reticular lamina and the tectorial membrane.

Figure 2

Analytical approximation of the normalized drag of a hair bundle in a row. (a) Schematic of the analyzed geometry. See text and Methods for parameters. (b) The drag is minimal at an optimal angle θ* (red) of the hair-bundle shape. The optimal angle shifts to larger values for increasing gap size.

Figure 3

Computational analysis of the flow across hair bundles. (a) Geometry of the computational domain and boundary conditions. (b) Triangular discretization of the computational domain. (c) Exemplary computationally obtained solution for the velocity and pressure fields. Here the tip angle is θ = 100°, the gap parameter is γ = 0.33, and the height of the bundle is 4 μm. The slice shown is at mid-height of the hair bundle.

Figure 4

Dependence of the drag on hair-bundle shape for flow across a row of hair bundles. Results are shown from 3D simulations with short (h = 2 μm, triangles) and tall (h = 4 μm, circles) hair bundles, as well as from 2D simulations which are equivalent to infinitely tall hair bundles (squares). (a) The normalized drag for a given gap parameter (here γ = 0.33) is minimal for a hair-bundle shape with tip angle ${\theta }_{\mathrm{num}}^{⁎}\approx {100}^{\circ }$. (b) The minimal drag at the optimal angle emerges from a trade-off between friction drag (red) and pressure drag (blue). (c) The drag increases with decreasing gap parameter γ approximately as γ ⁻² for a hair bundle with an angle $\theta ={100}^{\circ }\approx {\theta }_{\mathrm{num}}^{⁎}$. (d) The reduction in drag at the optimal angle compared to the case θ = 180° is larger for smaller gap sizes. (e) The total drag on a single hair bundle in unbounded flow shows little dependence on the shape of the bundle and varies by only about 3.7%. Significant drag reduction for V-shaped hair bundles as shown in panels a and d is therefore particular to dense rows of hair bundles as found in the cochlea. (f) The numerically-computed optimal angle of the hair bundle’s shape varies slightly with gap sizes (blue, black, and red lines with markers), and so does the range of angles for which the drag is at most 5% higher than at the optimal angle (shading in corresponding colors). All biologically-observed angles of outer hair cells’ hair bundles fall into this range.