Life at low Reynolds number

Purcell takes his audience into the world of very low Reynolds numbers — the world inhabited by microorganisms. At micron scales, Reynolds numbers are ~10⁻⁴–10⁻⁵; inertia is irrelevant, and motion is determined entirely by forces exerted at each moment. A swimming bacterium that stops pushing coasts only ~0.1 Å and halts in under a microsecond.

The central insight is the scallop theorem: at low Reynolds number, any reciprocal motion — deforming a body into a shape and then returning through the same sequence in reverse — produces zero net displacement. A scallop, with only one hinge, cannot swim in this regime. To move, an organism needs a non-reciprocal cycle, which requires at least two degrees of freedom in configuration space.

Real microorganisms solve this with flexible oars (which bend differently on each half-stroke) or corkscrew drives. Purcell shows that E. coli swims by continuously rotating its helical flagellum — a genuine rotary motor — confirmed by Berg's tracking experiments and Silverman & Simon's tethered-cell assays. The propulsion matrix for a helix is symmetric (only three independent constants), and efficiency is low — roughly 1%. But that barely matters: the energy budget is only ~0.5 W/kg, a small fraction of metabolism. "They're driving a Datsun in Saudi Arabia."

The real constraint at low Reynolds number is diffusion. Stirring is useless locally — the "stirring number" (Sherwood number) lv/D is ~10⁻² at these scales, so transport of nutrients and wastes is entirely diffusion-controlled. A bacterium doesn't swim to scoop up more food (that would require speeds 20× faster than it can achieve); it swims to find greener pastures. To outrun diffusion and meaningfully sample its environment, it must travel a distance of at least D/v ≈ 30 µm — which is exactly the distance Berg observed in E. coli tracks.

Berg also discovered the chemotaxis algorithm: when things are getting better, don't stop so soon. When things are getting worse, path lengths don't shorten — Purcell speculates that's because there's a bedrock length set by the diffusion-outrunning distance, below which sampling makes no sense.

Originally a talk honoring Victor Weisskopf, the paper is a model of physical storytelling — playful, personal, and built from elementary physics applied to an unfamiliar domain. It shows that at low Reynolds number, what matters isn't propulsion efficiency or energy, but information: how an organism navigates a world it cannot stir.