The bacterial flagellar motor is a rotary nanomachine embedded in the cell envelope. Protons (H⁺) flow down their electrochemical gradient — the proton-motive force, PMF — through membrane-spanning stator units (MotA₅MotB₂ complexes). Each conducting proton drives a conformational cycle that pushes against charged residues on the rotor ring (FliG), generating torque, much like pistons turning a crankshaft:
Steady-state speed: τ_motor(ω) = γ · ω
Torque–speed curve: τ(ω) = τ₀ , ω ≤ ω_knee (many stators bound, high torque)
τ(ω) = τ₀ · (ω_max − ω) / (ω_max − ω_knee) , ω_knee < ω ≤ ω_max
τ₀ ∝ N_stators (stall torque scales with engaged stator count)
ω_max, ω_knee ∝ PMF (speed scales with the proton flux driving each stator)
The motor's operating point is wherever its torque curve meets the viscous load line τ = γω — a light bead load runs fast near ω_max in the torque-falling regime, while the real flagellar bundle's high drag pins the motor in the flat, torque-generating regime at lower speed. This matches tethered-cell and bead-assay measurements from the Berg lab (Berry & Berg 1997; Chen & Berg 2000; Yuan & Berg 2008) — the exact constants here are illustrative/order-of-magnitude, not one specific dataset.
- Stator units — up to 11 MotA/MotB complexes can dock around the rotor at once; each one adds roughly equal torque, so stall torque scales with how many are engaged.
- PMF — the transmembrane voltage/pH gradient that powers proton flux through each stator; collapsing it (e.g. with an uncoupler) slows and eventually stops the motor.
- Load — a lone motor spinning a small bead runs near its unloaded top speed; the real helical filament bundle drags far more, so live cells mostly run in the flat, high-torque part of the curve.
- Switch — binding of phosphorylated CheY to the FliM/FliN switch complex flips the rotor from default counter-clockwise (CCW, filaments bundle into a coherent left-handed helix → smooth swimming) to clockwise (CW, the bundle flies apart into a "tumble" that re-randomizes heading) — the mechanical basis of the run-and-tumble behavior chemotaxis relies on.