Use the same uptake and use as the spherical cell of the supply-limit problem: each…

PHYSICS · Subjective · Fermi University Physics · Scientific Practice & Experimental Methods · Biophysics & Medical Physics · Physical foundations of the living cell

Life at Thermal Energy, Ch The scales of life and the thermal energy

Problem

Use the same uptake and use as the spherical cell of the supply-limit problem: each square micrometre of membrane lets in at most

1.0\times10^{6}

oxygen molecules per second, and each cubic micrometre of the interior uses

1.5\times10^{6}

per second. Now let the cell be a rod of radius

r

and straight length

h

with hemispherical ends, the shape the chapter uses for E. coli, which grows by getting longer at nearly fixed width. (a) List the assumptions of this estimate. (b) Using

A = 2\pi rh + 4\pi r^2

and

V = \pi r^2h + \tfrac43\pi r^3

, find

A/V

for

r = 0.5\,\mathrm{\mu m}

h = 0

1\,\mathrm{\mu m}

and

100\,\mathrm{\mu m}

, and the value it approaches for a very long rod. (c) Does the supply condition limit how long a rod of radius

0.5\,\mathrm{\mu m}

can grow? What is the largest radius a very long rod can have and still be supplied? (d) Compare with the largest supplied sphere, radius

2.0\,\mathrm{\mu m}

, and with the measured width of E. coli, and say what this shows about shape.

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