A repressor protein must find its single operator site on the chromosome of a…

PHYSICS · Multi-part · Fermi University Physics · Biophysics & Medical Physics · Physical foundations of the living cell · Physical foundations of the living cell

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

Problem

A repressor protein must find its single operator site on the chromosome of a bacterium of cytoplasmic volume

V = 1\,\mathrm{\mu m^3}

, which contains one copy of the operator. In the cytoplasm the repressor has

D = 4\times10^{-11}\,\mathrm{m^2\,s^{-1}}

(slowed by crowding relative to its value in dilute buffer), and the operator itself, ignoring any sliding, behaves as a Smoluchowski target of radius

a = 1\,\mathrm{nm}

. ((a)) Estimate the mean time

\tau_{3D}

for the repressor to find the operator by three-dimensional diffusion alone. ((b)) Suppose the repressor can slide along nonspecific DNA for about

30

base pairs (

0.34\,\mathrm{nm}

per base pair) before dissociating. Taking the effective capture radius to be enlarged from

1\,\mathrm{nm}

to this sliding length, estimate the search time

\tau_{\rm slide}

with sliding included.

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