A morphogen protein X is produced at the anterior pole of an insect embryo of length…

PHYSICS · Multi-part · Fermi University Physics · Biophysics & Medical Physics · Genetic circuits, regulation and biological information · Genetic circuits, regulation and biological information

Forces, Machines and Signals, Ch Genetic circuits and biological pattern

Problem

A morphogen protein X is produced at the anterior pole of an insect embryo of length

600\,\mathrm{\mu m}

and forms a gradient that nuclei use to read out position along the embryo. Its steady-state profile is measured to fall to

1/e

of its anterior value at a distance

\lambda_{\rm obs}=70\,\mathrm{\mu m}

from the source. Three further, independent measurements are available: a mean protein lifetime

\tau = 25\,\mathrm{min}

from a pulse-chase degradation assay; a diffusion coefficient

D_1 = 4.0\,\mathrm{\mu m^2\,s^{-1}}

from fluorescence correlation spectroscopy (FCS) on single molecules; a diffusion coefficient

D_2 = 0.15\,\mathrm{\mu m^2\,s^{-1}}

from fluorescence recovery after photobleaching (FRAP) on the bulk pool. ((a)) Using the source-diffusion-degradation model, find the diffusion coefficient

D

that would be needed to give the observed

\lambda_{\rm obs}

. ((b)) Compute the gradient length predicted by each measured diffusion coefficient (using the same

\tau

) and compare each with

\lambda_{\rm obs}

. ((c)) State which measurement, if either, is compatible with the observed gradient to within about a factor of two, and suggest which assumption of the simple model a genuine mismatch might be violating.

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