A motile bacterium of radius senses a chemoattractant of diffusion coefficient…

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 motile bacterium of radius

a = 0.50\,\mathrm{\mu m}

senses a chemoattractant of diffusion coefficient

D = 5.0\times10^{-10}\,\mathrm{m^2\,s^{-1}}

present at bulk concentration

c_\infty = 1.0\,\mathrm{\mu M}

, using receptor patches of radius

s=2.0\,\mathrm{nm}

scattered uniformly over its surface. The rate at which ligand molecules are captured by

N

patches is

I = 4\pi Dac_\infty\,\frac{Ns}{Ns+\pi a}

(quoted). ((a)) Design the number of receptor patches

N

the bacterium needs to capture at least

90\%

of the diffusion-limited maximum rate

I_{\max}=4\pi Dac_\infty

(the rate a fully absorbing sphere would achieve). ((b)) What fraction of its surface area do these patches cover, using

Ns^2/4a^2

(quoted) as the coverage? ((c)) Evaluate

I_{\max}

and the actual capture rate

I

achieved with the

N

from part (a), in molecules per second.

Browse the problem library