Fast adaptation of cooperative channels engenders Hopf bifurcations in auditory hair cells

ABSTRACT Since the pioneering work of Thomas Gold published in 1948, it has been known that we owe our sensitive sense of hearing to a process in the inner ear that can amplify incident sounds on a cycle-by-cycle basis. Termed the active process, it uses energy to counteract the viscous dissipation associated with sound-evoked vibrations of the ear’s mechanotransduction apparatus. Despite its importance, the mechanism of the active process and the proximate source of energy that powers it have remained elusive—especially at the high frequencies characteristic of mammalian hearing. This is partly due to our insufficient understanding of the mechanotransduction process in hair cells, the sensory receptors and amplifiers of the inner ear. It has previously been proposed that a cyclical binding of Ca 2+ ions to individual mechanotransduction channels could power the active process. That model, however, relied on tailored reaction rates that structurally forced the direction of the cycle. Here, we ground our study on our previous model of hair-cell mechanotransduction, which relied on the cooperative gating of pairs of channels, and incorporate into it the cyclical binding of Ca 2+ ions. With a single binding site per channel and reaction rates drawn from thermodynamic principles, our model shows that hair cells behave as nonlinear oscillators that exhibit Hopf bifurcations, dynamical instabilities long understood to be signatures of the active process. Using realistic parameter values, we find bifurcations at frequencies in the kilohertz range with physiological Ca 2+ concentrations. In contrast to the myosin-based mechanism, responsible for low-frequency relaxation oscillations in the vestibular hair cells of amphibians, the current model relies on the electrochemical gradient of Ca 2+ as the only energy source for the active process and on the relative motion of cooperative channels within the stereociliary membrane as the single mechanical driver. Equipped with these two mechanisms, a hair bundle proves capable of operating at frequencies in the kilohertz range, characteristic of mammalian hearing. SIGNIFICANCE How the inner ear amplifies incident sounds at audible frequencies of several kilohertz is a key question that has remained unanswered despite decades of research into several candidate mechanisms. Here, we model the behavior of hair cells, the sensory receptors of the inner ear, and show that they can undergo oscillatory instabilities called Hopf bifurcations due to the effect of Ca 2+ on the cooperative opening and closing of mechanotransduction ion channels. Close to the bifurcation point, a hair cell behaves as a nonlinear oscillator that can amplify its input on a cycle-by-cycle basis. We find that our proposed mechanism can operate in the kilohertz range.