= Solution
The bacterium samples its surroundings in time. A sustained increase in <chemoattractant> makes a favorable run persist by reducing the chance of tumbling; declining <chemoattractant> makes reorientation more likely. Individual tumbles need not point up the gradient. Biasing \b[run durations] can generate net migration even if reorientation itself is random. This is temporal comparison, not a direct measurement of a <concentration> gradient across the short cell body.
Compare the swimming time over length $\ell$, $t_a=\ell/v$, with <diffusion> over the same scale, $t_D\sim\ell^2/D$. Their ratio is the <Péclet number> $\mathrm{Pe}=v\ell/D$. The crossover estimate is
$$
\boxed{\ell\sim D/v=\frac{10^3}{30}\ \mathrm{\mu m}\simeq33\ \mathrm{\mu m},
\qquad t_{\rm run}\sim\ell/v\simeq1.1\ \mathrm s.}
$$
Order-one factors depend on the chosen diffusion-distance convention. A hypothetical stirrer of extent $d$ must instead have $Ud/D$ of order one, giving
$$
\boxed{U\sim D/d=10^3\ \mathrm{\mu m\,s^{-1}}=1\ \mathrm{mm\,s^{-1}}.}
$$
These are transport-scale estimates, not a proof of a particular increase in uptake. Around a one-micrometre body, $v d/D\simeq0.03$: local diffusive supply is already rapid. Actual uptake enhancement depends on the nutrient field and the cell's uptake boundary condition, neither supplied. The long run can help the organism reach a better nutrient region without implying that it greatly increases the flux from a uniform bath. <E. coli> normally uses <bacterial flagella>; the <cilium> in this comparison is a hypothetical generic stirring appendage, not its actual locomotor organelle.
Back to article page
