Module 4 · Behaviour

Motility & Chemotaxis

A swimming E. coli moves at ~30 μm/s — ~30 body lengths per second — through a fluid environment whose Reynolds number is ~10⁻⁵. At these scales viscosity dominates, inertia is irrelevant, and stopping happens within a body length when propulsion ceases. The bacterial flagellum is the evolutionary answer.

1. Life at Low Reynolds Number

The Reynolds number compares inertial to viscous forces:\( Re = \rho v L / \eta \). For E. coli: \(\rho = 10^3\) kg/m³, \(v = 30\,\mu\)m/s, \(L = 2\,\mu\)m, \(\eta = 10^{-3}\) Pa·s, giving \(Re \approx 6 \times 10^{-5}\).

This is a famously alien regime. Edward Purcell (Nobel 1952 for NMR) gave a memorable 1976 lecture titled “Life at low Reynolds number” showing that any reciprocal motion (e.g., scallop opening and closing) cannot propel; you need a non-time-reversible stroke. The flagellar helix, rotating continuously, satisfies this requirement.

2. The Flagellar Rotary Motor

The bacterial flagellum is a ~30 nm protein turbine driven by the proton motive force (in some species the sodium gradient). Components:

Filament — a helical polymer of ~20,000 flagellin (FliC) subunits, ~5–10 μm long, 20 nm wide.
Hook — a flexible universal joint (FlgE) connecting filament to motor.
Rotor — the C-ring (FliM/FliN/FliG) at the base; FliG carries the charged residues that interact with stator current.
Stator — ~11 MotA₄MotB₂ complexes; each conducts protons across the membrane and torques the rotor.

The motor rotates at ~100–300 Hz, can switch direction in < 1 ms, and develops a torque of ~4500 pN·nm. Each revolution couples to ~1000 protons translocated — a remarkably tight stoichiometry that can be measured by tethering the cell at the filament and observing rotation by microscopy.

3. Run-and-Tumble Navigation

When the motor rotates counter-clockwise (CCW), the multiple flagella bundle together and propel the cell forward in a smooth run (~1 s, ~30 μm). When one or more motors switch to clockwise (CW), the bundle fails and the cell tumbles (~0.1 s) to a new heading. Run-tumble-run-tumble describes a biased random walk.

When the cell senses an increasing concentration of attractant during a run, it extends the run. The result is a directed migration up the gradient. The cell measures temporal differences in concentration over ~1–3 s; its biochemical memory is implemented by the methylation state of the chemoreceptors.

4. The CheA–CheY Cascade

Chemotaxis signal transduction is a paragon of bacterial sensory wiring:

MCPs (methyl-accepting chemotaxis proteins; Tar, Tsr, Trg, Tap, Aer): transmembrane receptors clustered at the cell pole. Bind ligands like aspartate, serine, sugars.
CheA — histidine kinase. Activity decreases on attractant binding.
CheY — response regulator. CheY-P diffuses to the flagellar motor, binds FliM, biases the motor toward CW (tumble).
CheZ — phosphatase that dephosphorylates CheY-P (~10 s lifetime).
CheR / CheB — methylation enzymes that adapt the receptor sensitivity, providing “memory” that lets the cell compare current vs past concentration.

The system has remarkable exact adaptation: after a step in attractant, tumble frequency returns precisely to baseline regardless of the absolute concentration. Barkai & Leibler (1997) showed this is a robust property of the network architecture, not parameter tuning — an early triumph of systems biology.

5. The Berg-Purcell Limit

Howard Berg & Edward Purcell’s 1977 paper derived a fundamental limit on how accurately a small cell can measure a concentration \(c\) by counting molecules diffusing into its surface receptors over time\(\tau\):

\( \frac{\delta c}{c} \sim \frac{1}{\sqrt{D a c \tau}} \)

with \(D\) the diffusion constant and \(a\) the cell radius. For E. coli measuring 1 μM aspartate over 1 s, the limit is ~1% — and the measured behavioural threshold is ~3%. The chemotaxis system is operating near the physical limit.

6. Other Modes of Motility

Twitching motility — mediated by type-IV pili that extend, attach, and retract (driven by PilT ATPase) to drag the cell along surfaces (Pseudomonas, Neisseria).
Gliding motility — Myxococcus xanthus: focal-adhesion-like traction sites driven by the proton motive force.
Spirochete swimming — flagella inside the periplasm (e.g., Borrelia, the Lyme spirochete). Rotation deforms the cell body, which corkscrews through viscous gels — remarkable for an organism that has to traverse connective tissue.

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