A hair-cell transduction channel has two conformational states

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex 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

A hair-cell transduction channel has two conformational states, closed and open, at the hair cell's operating temperature of

310\,\mathrm{K}

, where

k_BT = 4.28\,\mathrm{pN\,nm}

. At zero bundle displacement the open state's free energy exceeds the closed state's by an unknown amount

\Delta G_0 = G_{\mathrm{open}} - G_{\mathrm{closed}}

. A displacement

x

of the bundle's tip stretches the tip link and does mechanical work

zx

on the channel, lowering the open state's free energy by this amount, where the gating force

z

is also unknown. Measurements on the bundle give an open probability of

0.10

x = 24.0\,\mathrm{nm}

and

0.90

x = 61.6\,\mathrm{nm}

. ((a)) Using the Boltzmann factor for a two-state system, derive the open probability

P_o(x)

, and show it has the logistic form

P_o(x) = 1/(1+e^{-z(x-x_0)/k_BT})

, giving

x_0

in terms of

\Delta G_0

and

z

. ((b)) From the two measurements, find

z

and

x_0

. ((c)) Hence find

\Delta G_0

in units of

k_BT

, and the channel's open probability when the bundle is undisplaced,

x=0

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