A finite element method for mechanical response of hair cell ciliary bundles

This paper describes the development of a methodology for performing a mechanical analysis of hair cell ciliary bundles. The cilia were modeled as shear deformable beams, and interconnections were modeled as two-force members. These models were incorporated into software, which performs a finite element analysis of a user-defined bundle. The algorithm incorporates aspects of the bundle such as geometric realignment and buckling of compressed side links. A sample bundle is introduced and results of modeling it are presented.     

Nonlinear mechanical responses of mouse cochlear hair bundles.

The stiffness of sensory hair bundles of both inner (IHC) and outer (OHC) hair cells was measured with calibrated silica fibres in mouse cochlear cultures to test the hypothesis that the mechanical properties of the hair bundle reflect processes underlying mechanotransduction. For OHCs, the displacement of the hair bundle relaxed with time constants of 6 ms for displacements which open transducer channels and 4 ms for displacements which close the channels. The corresponding values of the time constants for IHCs were 10 ms and 8 ms, respectively. A displacement-dependent change in the stiffness of the hair bundle was not observed when the bundle was displaced orthogonally to the direction of excitation. The stiffness of the hair bundle as a function of nanometre displacements from the resting position was remarkably nonlinear. The stiffness declined to a minimum from the resting stiffness by about 12% for OHCs and 20% for IHCs when the hair bundle was displaced by about 20 nm in the excitatory direction, and it increased by a similar amount when the bundle was displaced by 20 nm in the inhibitory direction. The displacement at which the stiffness reached a minimum was within the most sensitive region of the hair-cell transducer function (receptor potential as a function of hair-bundle displacement), and the displacement at which the stiffness reached a maximum was at the point of saturation of the transducer function in the inhibitory direction. The nonlinear displacement-dependent compliance change is reversibly abolished, and the time constant of relaxation of the bundle for excitatory displacements is reversibly reduced, when mechanotransduction is blocked by the addition of either neomycin sulphate or cobalt chloride to the solution bathing the hair cells. The displacement-dependent compliance change was not apparently reduced when the receptor potential was attenuated through the substitution of sodium in the bathing solution with a less permeant cation, tetraethylammonium. These findings suggest that the nonlinear mechanical properties of the hair bundle are associated with aspects of the hair-cell mechanotransducer process. The mechanical properties of the hair bundle are discussed in relation to the 'gating-spring' hypothesis of hair-cell transduction.     

Lateral mechanical coupling of stereocilia in cochlear hair bundles

For understanding the gating process of transduction channels in the inner ear it is essential to characterize and examine the functional properties of the ultrastructure of stereociliary bundles. There is strong evidence that transduction channels in hair cells are gated by directly pulling at the so-called tip links. In addition to these tip links a second class of filamentous structures was identified in the scanning and transmission electron microscope: the side-to-side links. These links laterally connect stereocilia of the same row of a hair bundle. This study concentrates on mechanical coupling of stereocilia of the tallest row connected by side-to-side links. Atomic Force microscopy (AFM) was used to investigate hair bundles of outer hair cells (OHCs) from postnatal rats (day 4). Although hair bundles of postnatal rats are still immature at day 4 and interconnecting cross-links do not show preferential direction yet, hair bundles of investigated OHCs already showed the characteristic V-shape of mature hair cells. In a first experiment, the stiffness of stereocilia was investigated scanning individual stereocilia with an AFM tip. The spring constant for the excitatory direction was 2.5 +/- 0.6 x 10(-3) N/m whereas a higher spring constant (3.1 +/- 1.5 x 10(-3) N/m) was observed in the inhibitory direction. In a second set of experiments, the force transmission between stereocilia of the tallest row was measured using AFM in combination with a thin glass fiber. This fiber locally displaced a stereocilium while the force laterally transmitted to the neighboring untouched taller stereocilia was measured by AFM. The results show a weak force interaction between tallest stereocilia of postnatal rats. The force exerted to an individual stereocilium declines to 36% at the nearest adjacent stereocilium of the same row not touched with the fiber. It is suggested that the amount of force transmitted from a taller stereocilium to an adjacent one of the same row depends on the orientation of links. Maximum force transmission is expected to appear along the axis of interconnecting side links. In our studies it is suggested that transmitted forces are small because connecting side links are oriented very close to an angle of 90 degrees with respect of the scan direction (excitatory-inhibitory direction).     

Assembly of hair bundles,an amazing problem for cell biology

The hair bundle—the sensory organelle of inner-ear hair cells of vertebrates—exemplifies the ability of a cell to assemble complex, elegant structures. Proper construction of the bundle is required for proper mechanotransduction in response to external forces and to transmit information about sound and movement. Bundles contain tightly controlled numbers of actin-filled stereocilia, which are arranged in defined rows of precise heights. Indeed, many deafness mutations that disable hair-cell cytoskeletal proteins also disrupt bundles. Bundle assembly is a tractable problem in molecular and cellular systems biology; the sequence of structural changes in stereocilia is known, and a modest number of proteins may be involved.     

Sound-induced motions of individual cochlear hair bundles

We present motions of individual freestanding hair bundles in an isolated cochlea in response to tonal sound stimulation. Motions were measured from images taken by strobing a light source at the tone frequency. The tips and bases of hair bundles moved a comparable amount, but with a phase difference that increased by 180 degrees with frequency, indicating that distributed fluid properties drove hair bundle motion. Hair bundle rotation increased with frequency to a constant value, and underwent >90 degrees of phase change. The frequency at which the phase of rotation relative to deflection of the bundle base was 60 degrees was comparable to the expected best frequency of each hair cell, and varied inversely with the square of bundle height. The sharpness of tuning of individual hair bundles was comparable to that of hair cell receptor potentials at high sound levels. These results indicate that frequency selectivity at high sound levels in this cochlea is purely mechanical, determined by the interaction of hair bundles with the surrounding fluid. The sharper tuning of receptor potentials at lower sound levels is consistent with the presence of a negative damping, but not a negative stiffness, as an active amplifier in hair bundles.     

Amiloride causes changes in the mechanical properties of hair cell bundles in the fish lateral line similar to those induced by dihydrostreptomycin

Amiloride is a known blocker of the mechano-electrical transduction current in sensory hair cells. Measurements of cupular motion in the lateral line organ of fish now show that amiloride concurrently changes the micromechanical properties of the hair cell bundles. The effects of amiloride on the mechanics and receptor potentials of the hair cells resemble those previously observed for the aminoglycoside drug dihydrostreptomycin (DHSM) and are similarly antagonized by Ca²⁺. We hypothesize that amiloride and DHSM act on hair cells in two correlated ways which manifest themselves in both the electrical and mechanical properties of the transduction process. One action is the reduction of the transduction current with a concurrent increase of the hair bundle stiffness. The other action is a shift of the hair cell''s operating point on a current–displacement curve, with a concomitant shift along the associated hair bundle stiffness–displacement curve. The latter action has the opposite effect to that of the first and thus may lead, at relatively low blocker concentrations, to both an increase of transduction current and a decrease in hair bundle stiffness.     

Wnt signaling mediates reorientation of outer hair cell stereociliary bundles in the mammalian cochlea

In the mammalian cochlea, stereociliary bundles located on mechanosensory hair cells within the sensory epithelium are unidirectionally oriented. Development of this planar polarity is necessary for normal hearing as stereociliary bundles are only sensitive to vibrations in a single plane; however, the mechanisms governing their orientation are unknown. We report that Wnt signaling regulates the development of unidirectional stereociliary bundle orientation. In vitro application of Wnt7a protein or inhibitors of Wnt signaling, secreted Frizzled-related protein 1 or Wnt inhibitory factor 1, disrupts bundle orientation. Moreover, Wnt7a is expressed in a pattern consistent with a role in the polarization of the developing stereociliary bundles. We propose that Wnt signaling across the region of developing outer hair cells gives rise to planar polarity in the mammalian cochlea.     

Coupling and elastic loading affect the active response by the inner ear hair cell bundles

Active hair bundle motility has been proposed to underlie the amplification mechanism in the auditory endorgans of non-mammals and in the vestibular systems of all vertebrates, and to constitute a crucial component of cochlear amplification in mammals. We used semi-intact in vitro preparations of the bullfrog sacculus to study the effects of elastic mechanical loading on both natively coupled and freely oscillating hair bundles. For the latter, we attached glass fibers of different stiffness to the stereocilia and observed the induced changes in the spontaneous bundle movement. When driven with sinusoidal deflections, hair bundles displayed phase-locked response indicative of an Arnold Tongue, with the frequency selectivity highest at low amplitudes and decreasing under stronger stimulation. A striking broadening of the mode-locked response was seen with increasing stiffness of the load, until approximate impedance matching, where the phase-locked response remained flat over the physiological range of frequencies. When the otolithic membrane was left intact atop the preparation, the natural loading of the bundles likewise decreased their frequency selectivity with respect to that observed in freely oscillating bundles. To probe for signatures of the active process under natural loading and coupling conditions, we applied transient mechanical stimuli to the otolithic membrane. Following the pulses, the underlying bundles displayed active movement in the opposite direction, analogous to the twitches observed in individual cells. Tracking features in the otolithic membrane indicated that it moved in phase with the bundles. Hence, synchronous active motility evoked in the system of coupled hair bundles by external input is sufficient to displace large overlying structures.     

Dynamics of oscillating erythrocyte doublets after electrofusion

Erythrocytes were electrofused with multiple rectangular voltage pulses to show an oscillatory movement, divided into swell phases and pump events. During each swell phase, which lasted from 0.5 s to more than 180 s, the fused cells' (doublets') volume increased by colloid osmotic swelling, and the membrane area was expanded until rupture. Membrane rupture initiated the pump event, where the doublets' volume and membrane area decreased with an almost exponential time course and time constants between 2 ms and 8 ms. Simultaneously, a portion of cytosolic hemoglobin solution was ejected into extracellular space ("jet"). Pump event time constants and swell phase durations decreased with rising chamber temperature, indicating that both parts of the oscillatory movements were determined by physical properties of membrane and liquids. Relative volume change developments express a gradual loss of membrane elasticity during the oscillation, decreasing the elastic forces stored in the membrane. Evidence is given that the first rupture causes a weakening of the membrane at the rupture site. Heat treatment up to 45 degrees C had a negligible effect on swell times, pump time constants, and relative volume changes. A heat treatment of 50 degrees C prevented oscillatory movements. The rupture location accorded with theories of potential induced membrane electropermeabilization.     

Sensory transduction: the 'swarm intelligence' of auditory hair bundles

In vertebrate hair cells, the hair bundle is responsible for the conversion of mechanical vibrations into electrical signals. In a combined experimental and computational tour de force, a group of researchers now presents a quantitative model that explains how the bundle's specific microarchitecture gives rise to its exquisite mechanosensory properties.     

Mechanical properties and consequences of stereocilia and extracellular links in vestibular hair bundles

Although knowledge of the fine structure of vestibular hair bundles is increasing, the mechanical properties and functional significance of those structures remain unclear. In 2004, Bashtanov and colleagues reported the contribution of different extracellular links to bundle stiffness. We simulated Bashtanov's experimental protocol using a three-dimensional finite element bundle model with geometry measured from a typical striolar hair cell. Unlike any previous models, we separately consider two types of horizontal links: shaft links and upper lateral links. Our most important results are as follows. First, we identified the material properties required to match Bashtanov's experiment: stereocilia Young's modulus of 0.74 GPa, tip link assembly (gating spring) stiffness of 5,300 pN/microm, and the combined stiffness of shaft links binding two adjacent stereocilia of 750 approximately 2,250 pN/microm. Second, we conclude that upper lateral links are likely to have nonlinear mechanical properties: they have minimal stiffness during small bundle deformations but stiffen as the bundle deflects further. Third, we estimated the stiffness of the gating spring based on our realistic three-dimensional bundle model rather than a conventional model relying on the parallel arrangement assumption. Our predicted stiffness of the gating spring was greater than the previous estimation.     

Effects of chlorpromazine on mechanical properties of the outer hair cell plasma membrane

An optical tweezers system was used to characterize the effects of chlorpromazine (CPZ) on the mechanical properties of the mammalian outer hair cell (OHC) through the formation of plasma membrane tethers. Such tethers exhibited force relaxation when held at a constant length for several minutes. We used a second-order generalized Kelvin body to model tether-force behavior from which several mechanical parameters were then calculated including stiffness, viscosity-associated measures, and force relaxation time constants. The results of the analysis portray a two-part relaxation process characterized by significantly different rates of force decay, which we propose is due to the local reorganization of lipids within the tether and the flow of external lipid into the tether. We found that CPZ's effect was limited to the latter phenomenon since only the second phase of relaxation was significantly affected by the drug. This finding coupled with an observed large reduction in overall tether forces implies a common basis for the drug's effects, the plasma membrane-cytoskeleton interaction. The CPZ-induced changes in tether viscoelastic behavior suggest that alterations in the mechanical properties of the OHC lateral wall could play a role in the modulation of OHC electromotility by CPZ.     

Dynamics of coupled stomatal oscillators

Stomata are microscopic openings in the leaves of green plants which permit gas exchange. Stomata exhibit oscillatory opening and closing behavior under certain environmental conditions in addition to a daily (diurnal) cycle. In order to explore the effects of coupling between neighboring stomata we present a mathematical model of the dynamics of a system of N coupled stomatal oscillators. An individual stomate is modeled to either remain closed, oscillate periodically, or remain open, depending on the local water potential. Coupling between neighboring stomata is accomplished in the model by taking into account the flow of water in the leaf as well as by oscillator phase coupling.Analysis of the model shows that under certain conditions it exhibits a stable spatially uniform synchronized behavior, referred to here as the in-phase mode. It is also shown that under non-uniform illumination the system may behave in a more complicated fashion.     

Dynamics of coupled thalamocortical modules

We develop a model of thalamocortical dynamics using a shared population of thalamic neurons to couple distant cortical regions. Behavior of the model is determined as a function of the connection strengths with shared and unshared populations in the thalamus, either within a relay nucleus or the reticular nucleus. When the coupling is via the reticular nucleus, we locate solutions of the model where distant cortical regions maintain the same activity level, and regions where one region maintains an elevated activity level, suppressing activity in the other. We locate and investigate a region where both types of solutions exist and are stable, yielding a mechanism for spontaneous changes in global activity patterns. Power spectra and coherence are computed, and marked differences in the coherence are found between the two kinds of modes. When, on the other hand, the coupling is via a shared relay nuclei, the features seen with the reticular coupling are absent. These considerations suggest a role for the reticular nucleus in modulating long distance cortical communication.     

The structure of tip links and kinocilial links in avian sensory hair bundles

Recent studies have indicated that the tip links and kinocilial links of sensory hair bundles in the inner ear have similar properties and share a common epitope, and that cadherin 23 may also be a component of each link type. Transmission electron microscopy was therefore used to study and compare the fine structure of the tip links and kinocilial links in avian sensory hair bundles. Tannic acid treatment revealed a thin strand, 150-200 nm long and 8-11 nm thick, present in both link types. Fourier analysis of link images showed that the strand of both link types is formed from two filaments coiled in a helix-like arrangement with an axial period of 20-25 nm, with each filament composed of globular structures that are approximately 4 nm in diameter. Differences in the radius and period of the helix-like structure may underlie the observed variation in the length of tip and kinocilial links. The similar helix-like structure of the tip links and kinocilial links is in accord with the presence of a common cell-surface antigen (TLA antigen) and similarities in the physical and chemical properties of the two link types. The spacing of the globular structures comprising each filament of the two link types is similar to the 4.3 nm center-to-center spacing reported for the globular cadherin repeat, and is consistent with the suggestion that cadherin 23 is the tip link.     

ATPase activity of myosin in hair bundles of the bullfrog''s sacculus.

Mechanoelectrical transduction by a hair cell displays adaptation, which is thought to occur as myosin-based molecular motors within the mechanically sensitive hair bundle adjust the tension transmitted to transduction channels. To assess the enzymatic capabilities of the myosin isozymes in hair bundles, we examined the actin-dependent ATPase activity of bundles isolated from the bullfrog's sacculus. Separation of 32P-labeled inorganic phosphate from unreacted [gamma-32P]ATP by thin-layer chromatography enabled us to measure the liberation of as little as 0.1 fmol phosphate. To distinguish the Mg(2+)-ATPase activity of myosin isozymes from that of other hair-bundle enzymes, we inhibited the interaction of hair-bundle myosin with actin and determined the reduction in ATPase activity. N-ethylmaleimide (NEM) decreased neither physiologically measured adaptation nor the nucleotide-hydrolytic activity of a 120-kDa protein thought to be myosin 1 beta. The NEM-insensitive, actin-activated ATPase activity of myosin increased from 1.0 fmol x s-1 in 1 mM EGTA to 2.3 fmol x s-1 in 10 microM Ca2+. This activity was largely inhibited by calmidazolium, but was unaffected by the addition of exogenous calmodulin. These results, which indicate that hair bundles contain enzymatically active, Ca(2+)-sensitive myosin molecules, are consistent with the role of Ca2+ in adaptation and with the hypothesis that myosin forms the hair cell's adaptation motor.     

Dynamics of actin filaments during tension-dependent formation of actin bundles

The actin cytoskeleton stress fiber is an actomyosin-based contractile structure seen as a bundle of actin filaments. Although tension development in a cell is believed to regulate stress fiber formation, little is known for the underlying biophysical mechanisms. To address this question, we examined the effects of tension on the behaviors of individual actin filaments during stress fiber (actin bundle) formation using cytosol-free semi-intact fibroblast cells that were pre-treated with the Rho kinase inhibitor Y-27632 to disassemble stress fibers into a meshwork of actin filaments. These filaments were sparsely labeled with quantum dots for live tracking of their motions. When ATP and Ca(2+) were applied to the semi-intact cells to generate actomyosin-based forces, actin meshwork in the protruded lamellae was dragged toward the cell body, while the periphery of the meshwork remained in the original region, indicating that centripetally directed tension developed in the meshwork. Then the individual actin filaments in the meshwork moved towards the cell body accompanied with sudden changes in the direction of their movements, finally forming actin bundles along the direction of tension. Dragging the meshwork by externally applied mechanical forces also exerted essentially the same effects. These results suggest the existence of tension-dependent remodeling of cross-links within the meshwork during the rearrangement of actin filaments, thus demonstrating that tension is a key player to regulate the dynamics of individual actin filaments that leads to actin bundle formation.     

High-purity isolation of bullfrog hair bundles and subcellular and topological localization of constituent proteins.

The small number of hair cells in auditory and vestibular organs severely impedes the biochemical characterization of the proteins involved in mechano-electrical transduction. By developing an efficient and clean "twist-off" method of hair bundle isolation, and by devising a sensitive, nonradioactive method to detect minute quantities of protein, we have partially overcome this limitation and have extensively classified the proteins of the bundles. To isolate hair bundles, we glue the saccular macula of the bullfrog to a glass coverslip, expose the tissue to a molten agarose solution, and allow the agarose to solidify to a firm gel. By rotating the gel disk with respect to the fixed macula, we isolate the hair bundles by shearing them at their mechanically weak bases. The plasma membranes of at least 80% of the stereocilia reseal. To visualize the proteins of the hair bundle, we covalently label them with biotin, separate them by SDS-PAGE, and transfer them to a charged nylon membrane. We can detect less than 500 fg of protein by probing the membrane with streptavidin-alkaline phosphatase and detecting the chemiluminescent product from the hydrolysis of the substrate 3-(4-methoxyspiro-(1,2-dioxetane-3,2'-tricyclo-[3.3.1. 1(3.7)]decan)-4-yl) phenyl phosphate (AMPPD). These techniques reveal a distinct constellation of proteins in and associated with hair bundles. Several proteins, such as calmodulin, calbindin, actin, tubulin, and fimbrin, have previously been described. A second class of proteins in the preparation appears to be derived from extracellular sources. Finally, several heretofore undescribed bundle proteins are identified and characterized by their membrane topology, subcellular localization, and glycosidase and protease sensitivities.     

Surface charges of the membrane and cell adhesion substances determine the structural integrity of hair bundles from the inner ear of fish

Summary The hairs (stereocilia = stereovilli) of sensory cells from the inner ear of vertebrates are interconnected by several types of connectors, whose role is unknown. They appear to stabilize the hair bundle mechanically, and may be directly involved in mechano-electric transduction. Our transmission electron-microscopical investigation of sensory epithelia from two species of fish (Rutilus rutilus, Scardinius erythrophthalmus, both Leuciscidae) has shown that not only the connectors but also the surface charges of the membrane are important factors for determining the shape of the hair bundle and the spatial interrelation of the stereovilli. A reduction of the ionic strength in the medium leads to an increase in distance between the stereovilli. This may be the result of an extension of the spread of the surface potential of the membrane at low ionic strength. The connectors are not broken by the increase in distance between the stereovilli. They are EDTA (ethylene-diamine-tetra-acetic-acid) resistant as are some cell adhesion molecules such as N-CAM (nerve-cell adhesion molecule) and protein A from Dictyostelium discoideum. The connectors do not prevent polycation-induced fusion of adjacent stereovillar membranes.