Differentiating the two-state free energy of one gated hair-cell channel gives that…

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Biophysics & Medical Physics · Bioelectricity, the senses and the physics of the body

Forces, Machines and Signals, Ch Hearing: from air pressure to a gated channel

Problem

Differentiating the two-state free energy of one gated hair-cell channel gives that channel's contribution to the bundle's stiffness reduction (the gating compliance), Δκ(x)=z2kBTPo(x)(1−Po(x)),\Delta\kappa(x) = \frac{z^2}{k_BT}P_o(x)\big(1-P_o(x)\big), which is greatest at x=x0x=x_0, where Po=1/2P_o=1/2, giving Δκmax⁡=z2/4kBT\Delta\kappa_{\max} = z^2/4k_BT per channel. A bundle has N=50N = 50 such channels acting in parallel, and far from x0x_0 its stiffness is K∞≈1 mN m−1K_\infty \approx 1\,\mathrm{mN\,m^{-1}}. At body temperature kBT=4.28 pN nmk_BT = 4.28\,\mathrm{pN\,nm} (1 pN nm−1=1 mN m−11\,\mathrm{pN\,nm^{-1}} = 1\,\mathrm{mN\,m^{-1}}). ((a)) For a gating force z=0.30 pNz = 0.30\,\mathrm{pN}, find Δκmax⁡\Delta\kappa_{\max} for one channel and the bundle's stiffness dip NΔκmax⁡N\Delta\kappa_{\max}, both in mN m−1\mathrm{mN\,m^{-1}}. ((b)) A stiffer tip-link geometry triples the gating force to z′=0.90 pNz'=0.90\,\mathrm{pN}. Without recomputing from the exponentials, predict the new bundle dip by scaling, compare it with K∞K_\infty, and say what the model then predicts for the bundle's stiffness near x0x_0.

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