A model for amplification of hair-bundle motion by cyclical binding of Ca2+ to mechanoelectrical-transduction channels

Abstract

Amplification of auditory stimuli by hair cells augments the sensitivity of the vertebrate inner ear. Cell-body contractions of outer hair cells are thought to mediate amplification in the mammalian cochlea. In vertebrates that lack these cells, and perhaps in mammals as well, active movements of hair bundles may underlie amplification. We have evaluated a mathematical model in which amplification stems from the activity of mechanoelectrical-transduction channels. The intracellular binding of Ca2+ to channels is posited to promote their closure, which increases the tension in gating springs and exerts a negative force on the hair bundle. By enhancing bundle motion, this force partially compensates for viscous damping by cochlear fluids. Linear stability analysis of a six-state kinetic model reveals Hopf bifurcations for parameter values in the physiological range. These bifurcations signal conditions under which the system's behavior changes from a damped oscillatory response to spontaneous limit-cycle oscillation. By varying the number of stereocilia in a bundle and the rate constant for Ca2+ binding, we calculate bifurcation frequencies spanning the observed range of auditory sensitivity for a representative receptor organ, the chicken's cochlea. Simulations using prebifurcation parameter values demonstrate frequency-selective amplification with a striking compressive nonlinearity. Because transduction channels occur universally in hair cells, this active-channel model describes a mechanism of auditory amplification potentially applicable across species and hair-cell types.

Figure 1

Schematic diagram of the active-channel model. Each transduction channel is posited to exist in either a closed form, C, or an open one, O. Application of a positive stimulus force to the hair bundle promotes channel opening, whereas binding of Ca²⁺ favors closure. The rate constants for reactions in the counterclockwise direction are designated k_F, those pertaining to the clockwise direction k_R, and those for the two central transitions k₃₆. During sinusoidal stimulation or limit-cycle oscillation, the occupancy of the channel’s six states cycles in the counterclockwise direction.

Figure 2

Parameter dependence of the system’s eigenvalues (λ). The eigenvalues determined for four values (108, 140, 220, and 300) of the stereociliary number, N_S, are represented on the complex plane. The points designated B, C, and C̄ demonstrate the functional dependence of the system on N_S; the arrows indicate the progression of eigenvalues with increasing values of this parameter. The angular frequency at the Hopf bifurcation for C and C̄, ±32,000 s⁻¹, corresponds to a characteristic frequency of ∼5 kHz. Point D, which represents the conservation of state probability in the channel-gating cycle, remains at the origin. The conjugate pair A and Ā are independent of N_S and also appear stationary. An additional conjugate pair of eigenvalues, those farthest to the left, have been omitted to permit display of the remaining points on an informative scale. The system is characterized by an eighth-degree characteristic polynomial that precludes a closed form for the eigenvalues; numerical evaluation was therefore necessary. The stability analysis and simulation programs were written in mathematica and executed on a Macintosh Quadra 800 computer (Apple Computer, Cupertino, CA) or an Indigo Impact 10000 computer (Silicon Graphics, Mountain View, CA).

Figure 3

Dependence of the model’s frequency selectivity on the number of stereocilia, N_S, and the resting activation energy for forward transitions, ΔG_F^†, expressed in units of the thermal energy kT. Characteristic frequencies are calculated for points at and beyond the Hopf bifurcations, which occur along the ridge. Prebifurcation solutions, which lie to the right of the bifurcation locus, correspond to decaying responses to perturbation. The solutions near the bifurcation imply transient oscillatory components. To the left of the bifurcation locus, postbifurcation solutions represent limit-cycle oscillations.

Figure 4

Modeled responses of a hair bundle to stimulation with a 5-kHz sinusoidal force 0.2 pN in amplitude (Bottom trace). (A) In the active-channel model, the bundle displays oscillations of roughly threshold amplitude upon stimulation. The system’s high Q is revealed by the gradual rise and decline of the response. (B) The response of a similar model without Ca²⁺-induced channel reclosure shows far less amplification and no resonance after stimulation. (C) For a passive model without channel gating, the displacement is roughly the driving force divided by the bundle’s stiffness. In each instance, M_HB = 60 pg, ξ_HB = 100 nN⋅s⋅m⁻¹, K_SP = 9,000 μN⋅m⁻¹, κ_GS = 1,200 μN⋅m⁻¹, N_S = 210, N_T = 184, γ = 0.50, b = 4.5 nm, δ₁₂ = 0.9, δ₃₆ = 0.1, δ₄₅ = 0.1, k_F = 93,500 s⁻¹, k_R = 10,300 s⁻¹, and k₃₆ = 0.88 s⁻¹. The response in B displays an offset of −2.9 nm; the system’s bistability around a resting position at which p_O,REST = 0.5 emerges from fixed-point analysis.

Figure 5

Gain spectrum of the active-channel amplifier. The system’s gain, defined as the ratio of the response’s amplitude to that of the linear passive system, is greatest for stimuli near threshold. With growth of the stimulus amplitude, which is indicated for each trace, the frequency selectivity declines and the system approaches linearity. Peak response amplitudes range from 0.37 nm for 0.2-pN stimulation to 3.4 nm for a 40-pN input.