A small charged bead of diffusion coefficient and drag coefficient is confined to a…

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Biophysics & Medical Physics · Physical foundations of the living cell

Forces, Machines and Signals, Ch Diffusion: friction, capture and the crowded cytoplasm

Problem

A small charged bead of diffusion coefficient

D

and drag coefficient

\zeta

is confined to a channel of length

L = 1.0\,\mathrm{\mu m}

with reflecting (impermeable) walls at

x=0

and

x=L

, at

T = 298\,\mathrm{K}

. A uniform steady force of magnitude

F

pushes the bead in the

-x

direction (towards the wall at

x=0

). In steady state the net flux

J = -D\,\mathrm{d}c/\mathrm{d}x - (F/\zeta)c

is zero everywhere, so a concentration profile

c(x)

builds up along the channel. ((a)) Show that the steady-state profile has the form

c(x) = c(0)\,e^{-Fx/k_BT}

, independent of

D

and

\zeta

separately. ((b)) Evaluate the ratio

c(L)/c(0)

for three forces such that the dimensionless combination

FL/k_BT

equals

0.1

1

and

10

in turn. ((c)) State in words how the shape of the profile changes as

FL/k_BT

is raised from

0.1

10

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