= Solution
A small swimming member of the <bacteria> samples too little spatial separation across its own body to compare weak chemical gradients accurately. <Temporal sensing in bacterial chemotaxis> instead compares recent receptor activity with an adapted memory. Straight runs carry it between locations; tumbles reorient it. When conditions improve, tumbling becomes less frequent and favorable runs last longer. An unfavorable direction is abandoned sooner. Repetition creates a biased <random walk> without requiring the cell to identify the best direction at each tumble. <Rotational diffusion> and environmental obstacles eventually destroy the benefit of maintaining a direction indefinitely.
An order-of-magnitude transport estimate compares the time $\ell/v$ to swim a distance $\ell$ with the time $\ell^2/D$ for an attractant to diffuse that distance. Their crossover is a <Péclet number> of order one:
$$
\frac{\ell}{v}\sim\frac{\ell^2}{D},\qquad
\boxed{\ell\sim\frac Dv=\frac{1000}{30}\,\mathrm{\mu m}\simeq33\,\mathrm{\mu m},\qquad
\tau\sim\frac{D}{v^2}\simeq1.1\,\mathrm s.}
$$
This estimates a useful run scale: over much shorter distances, nutrient <diffusion> rapidly replenishes the local environment compared with the organism's displacement. Runs of order a second also permit comparison along a substantial path before reorientation. The given $D$ and $v$ determine this scale, not an exact biological mean run length or a complete sensory optimization. If a one-coordinate root-mean-square diffusion length $\sqrt{2D\tau}$ is equated with $v\tau$, the resulting prefactor is two; a three-dimensional root-mean-square convention gives six. Such constants cannot be fixed by a scaling argument alone.
For a cilium of length $d$, compare its advective time $d/u$ with the local <diffusion> time $d^2/D$. Significant stirring relative to <diffusion> requires $\mathrm{Pe}_d=ud/D\gtrsim1$, so
$$
\boxed{u\gtrsim\frac Dd=10^3\,\mathrm{\mu m\,s^{-1}}=1\,\mathrm{mm\,s^{-1}}\qquad(d=1\,\mathrm{\mu m}).}
$$
This is about $33$ times the stated swimming speed. If the tip travelled on a circular path of radius $d$, that speed would correspond to an illustrative frequency $u/(2\pi d)\simeq160\,\mathrm{Hz}$. That geometric conversion is not a specified beat pattern. At microscopic scales the flow has small <Reynolds number>; useful transport depends on the actual nonreciprocal motion and resulting flow, and rapid motion alone does not guarantee increased uptake. The scaling explains why simply stirring with a very short appendage is demanding when molecular <diffusion> is already fast.
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