A dilute suspension of a small ribonucleoprotein particle is spun in a centrifuge at…

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 dilute suspension of a small ribonucleoprotein particle is spun in a centrifuge at constant angular velocity

\omega

. In the rotating frame each particle of buoyant mass

m_b = m(1-\bar v\rho)

(the mass corrected for the buoyancy of the displaced fluid) feels an outward force

F(r) = m_b\omega^2 r

at radius

r

from the axis, playing the part that

m_b g

plays for a particle settling under gravity. ((a)) Write the outward drift flux

J_{\rm drift}(r) = c(r)F(r)/\zeta

and the inward diffusive flux

J_{\rm diff}(r) = -D\,dc/dr

for the spinning suspension. Setting

J_{\rm drift}+J_{\rm diff}=0

once the profile has become steady, show that the resulting

c(r)

matches the Boltzmann distribution

c(r)\propto \exp[m_b\omega^2 r^2/(2k_BT)]

only if

D\zeta = k_BT

, independent of

\omega

r

and the particle's shape. ((b)) A run at

\omega = 5.0\times10^{2}\,\mathrm{rad\,s^{-1}}

and

T=293\,\mathrm{K}

reaches a steady profile with

c(r_2) = 22\,c(r_1)

for

r_1 = 6.80\,\mathrm{cm}

and

r_2 = 7.00\,\mathrm{cm}

. Use the Boltzmann profile of part (a) to find the buoyant mass

m_b

of one particle. ((c)) Electron microscopy gives the particle a radius

a=6.0\,\mathrm{nm}

. Taking the drag coefficient in water (

\eta = 1.0\times10^{-3}\,\mathrm{Pa\,s}

293\,\mathrm{K}

) as

\zeta = 6\pi\eta a

(quoted, for a sphere), use

D\zeta = k_BT

from part (a) to find the particle's diffusion coefficient

D

. ((d)) The same particle now diffuses towards a fixed absorbing site of radius

A = 50\,\mathrm{nm}

(large enough that the particle can be treated as a point particle). Find the diffusion-limited rate constant for capture,

k = 4\pi DA

, in

\mathrm{m^3\,s^{-1}}

per particle and convert it to

\mathrm{M^{-1}\,s^{-1}}

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