A growth cone at the tip of an extending axon is modelled as a sphere of radius…

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Forces, Machines and Signals, Ch Numbers and noise in a small system

Problem

A growth cone at the tip of an extending axon is modelled as a sphere of radius

a=5\,\mu\mathrm{m}

sampling a diffusible guidance cue of concentration

c=1\,\mathrm{nM}

and diffusion coefficient

D=1.0\times10^{-10}\,\mathrm{m^2\,s^{-1}}

(smaller than a small metabolite's, because the cue is a secreted protein). The growth cone integrates its reading of the local concentration over

\tau=10\,\mathrm{s}

before deciding which way to turn. ((a)) List the assumptions behind the counting-limit estimate

\delta c/c \sim (Dac\tau)^{-1/2}

as applied to this growth cone. ((b)) Estimate the relative precision

\delta c/c

with which the growth cone can read the cue concentration. ((c)) In a shallow guidance gradient the concentration over the front half of the growth cone exceeds that over the back half by

5\%

c

. Suppose the growth cone compares the counts on its two halves, each half collecting independently half of the count in (b). Estimate the rms error in the difference of the two readings, as a fraction of

c

. Taking a difference of two rms errors as the least the growth cone can trust, say whether one

10\,\mathrm{s}

window is enough, and if not, how long it would have to average.

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