A cell that is surrounded by chemoattractant molecules will sense or 'measure' the number of molecules, $N$, in roughly its own volume $a^{3}$, where $a$ is the radius of the cell. Suppose the chemoattractant aspartate is in concentration $c=1 \mu \mathrm{M}$ and has diffusion constant $D=$ $10^{-5} \mathrm{~cm}^{2} \mathrm{~s}^{-1}$. For E. coli, $a \approx 10^{-6} \mathrm{~m}$.
(a) Compute $N$.
(b) The error or noise in the measurement of a quantity like $N$ will be approximately $\sqrt{N}$. Compute the precision $P$ of E. coli's measurement, which is the noise/signal ratio $P=\sqrt{N} / N$.
(c) Compute the 'clearance time' $T_{c}$ that it takes for a chemoattractant molecule to diffuse roughly the distance $a$. This is the time over which $E$. coli makes its 'measurement.'
(d) If the cell stays in place for $m$ units of time $T=m \tau_{c}$, to make essentially $m$ independent measurements, the cell can improve its precision. Write an expression for the precision $P_{m}$ of the cell's measurement after $m$ time steps, as a function of $a, D, c$, and $T$.
(e) Given $\tau=1 \mathrm{~s}$, compute $P_{m}$.