§11
Physics Foundations
§11 · Electromechanics and electrochemistry

Key examples — electromechanics

Where the chapter’s machinery shows up across the bookshelf.

Example 1: cell membrane as a capacitor

A neuronal cell membrane is ~5 nm thick lipid bilayer with relative permittivity εr≈5\varepsilon_r \approx 5. The specific capacitance is

C/A  =  ε0εrd  ≈  8.85×10−12×55×10−9  ≈  9 mF/m2.C/A \;=\; \frac{\varepsilon_0 \varepsilon_r}{d} \;\approx\; \frac{8.85\times 10^{-12} \times 5}{5\times 10^{-9}} \;\approx\; 9\,\text{mF/m}^2.

For a 100 mV potential, the surface charge density is σ=CV/A≈1 mC/m2≈6×1015\sigma = CV/A \approx 1\,\text{mC/m}^2 \approx 6\times 10^{15} ions per m² — about 1 ion per 16 nm² of membrane area. This small surface charge is what powers every action potential and every MET-channel current in the bookshelf.

Example 2: the +80 mV endocochlear potential as power source

The Nernst potential of K⁺ across the apical hair-cell membrane is near zero (both compartments have ~150 mM K⁺). The driving force comes entirely from the endocochlear potential — the +80 mV maintained by active K⁺ pumping in the stria vascularis. Combined with the −60 mV cell resting potential, the total electrochemical drive across an open MET channel is +140 mV, supporting ~14 pA of current per channel. This is why hearing collapses when the stria vascularis is damaged (as in age-related “metabolic” presbycusis): the power supply fails, and the receptors lose their driving force. See Hearing Ch 4.1.

Example 3: MET-channel mechanotransduction

Tip-link tension stretches a gating spring with stiffness KTL≈0.5 mN/mK_\text{TL} \approx 0.5\,\text{mN/m} per stereocilium. A 100 nm deflection applies F=KTL x≈50 pNF = K_\text{TL}\,x \approx 50\,\text{pN} of force to the channel. With a gate swing d≈4 nmd \approx 4\,\text{nm} and a zero-bias offset ΔG0≈2 kBT\Delta G_0 \approx 2\,k_BT, the chapter’s Fermi function predicts a sigmoidal Popen(x)P_\text{open}(x) matching cell measurements. See Hearing Ch 4.6.

Example 4: prestin as the cochlear amplifier

Each OHC carries ~10⁷ prestin molecules. Voltage-driven conformational change of each prestin produces a small length change; summed, the cell shortens by up to 4% (about 4 μm out of 100 μm) over the full ±100 mV range. Operating at up to 80 kHz with a phase delay of microseconds, prestin pumps mechanical energy into the basilar-membrane motion exactly where it is needed to overcome viscous damping. The mammalian cochlea’s exquisite frequency selectivity and 100-fold sensitivity gain over passive systems both trace to this single biological piezo. See Hearing Ch 4.5.

Example 5: voltage-gated Ca²⁺ at the ribbon synapse

The hair-cell ribbon synapse has voltage-gated Ca²⁺ channels that open in proportion to membrane depolarisation. Depolarisation from −60 to −40 mV (driven by MET-channel K⁺ influx) opens Ca²⁺ channels with a Fermi-function probability identical in form to the MET gating — only here the bias is Vm−V1/2V_m - V_{1/2} rather than a mechanical displacement. Ca²⁺ entry triggers neurotransmitter release at sub-ms latencies, the highest-speed synaptic transmission in vertebrate physiology. See Hearing Ch 5.1.

Example 6: action potential propagation

The Hodgkin–Huxley model treats each membrane patch as a parallel combination of voltage-gated Na⁺, voltage-gated K⁺, and a leak conductance. The combined GHK equation with time-varying PNa(V,t)P_\text{Na}(V, t) and PK(V,t)P_K(V, t) produces the action-potential waveform — a 1-ms depolarisation from −65-65 to +30 mV+30\,\text{mV} followed by a 2-ms hyperpolarisation. Propagation down an axon is by cable-equation spreading of VV between adjacent membrane patches. The auditory nerve fires action potentials at rates up to 300 Hz, phase-locked to the acoustic stimulus up to ~5 kHz.