E. M. Purcell's “Life at Low Reynolds Number”

E. M. Purcell’s “Life at Low ReynoldsNumber”Random Walks in Biology SeriesEric PetersonMarch 8, 2005

Outline• The Consequences of Low Reynolds Number• How do microscopic organisms Swim at Low Reynolds Number?-Inertia and reciprocal motion-Mechanics of flagella• Why do these organisms Swim at Low Reynolds Number?-Stirring versus diffusion-Runs and twiddles

Reynolds NumberR ≡ Ratio of InertialForces to Viscous Forcesav#R ="a ! dimensional elementv ! velocity# ! density = 1g/mLfor waterMichael Phelps: R ≈ 10 4www.speedo.ruE. coli.: R ≈ 10 -4www.nature.com" !viscosity= 10- 2 Pfor waterGuppy: R ≈ 10 2www.aquariumfish.net

Inertia at Low Reynolds Number• At R ≈ 10 -4 viscous forces >> inertial forces• F = ma is irrelevant!• Organisms on this scale do not coast in solutiona = 1 µmDrag F = 6πηavv = 30 µm/sη = 1 cp R = 3 x 10 -5How Far does it Coast?

Animals that SwimTorroidsFlagellated MicroorganismsHow do E. coli flagella work?

Answer… They RotateFlagellar StructureFlagellar Rotational FrequencyMutant E. Coli with no flagella areadhered to a glass slide by their hooksvia anti-hook antigens.V xand V yare voltage readings from atransducer following the movementof a reference point on a bacterium.Tortora, G. J.; Berdell, R. F.; Case, C. L.; Microbiology: anIntroduction, Benjamin Cummings p. 83 (2002)Berg, H. C.; Nature 249, 77-79 (1974)

Physics of Flagellar PropulsionHead rotates at angular velocity ΩTail rotates at angular velocity ω-ΩBacteriummoves withvelocity vForces and velocities at two differentpositions on the helix:Normal to axis: F Ω and v ΩParallel to axis: F v and v vNormal to tangent of helix: F n and v nParallel to tangent of helix: F p and v pBerg, H. C.; Random Walks in Biology, Princeton UniversityPress p. 79 (1983)

The Purpose of Swimming: Stirring VersusDiffusionCan microorganisms gather resources faster by swimming?Time of transport for stirring:Time of transport by diffusion:t s=lvlD2t D=For stirring to matter :ttDs! Sherwood Number (S)ttDslv=D> 1

Stirring Versus Diffusion ContinuedFor l = 1 µmD = 10 -5 cm 2 /sv = 10 µm/sS = 10 -2 : Stirring does nothing!To increase the capture rate by10% the bacterium would have togo 700 µm/s.

The Purpose of Swimming: To Find GreenerPasturesBacteria cannot gather foodfaster by swimming, but they canswim to areas with more food.E. coli Random Walks:“Runs” and “Twiddles”Berg, H. C.; Brown, D. A. Nature 239, 500-504 (1972)

The Length of the “Runs”Swimming Distance•Each Run must be long enoughto beat diffusion.•If conditions are getting better,the runs get longer.•Even if conditions get worse,runs are never shorter than thethreshold length (i.e. 30 µm).

Summary• At low Reynolds Number viscous forces dominate andinertia is meaningless.• In order for microbes to swim, they must produce acontinuous force: reciprocal motion will get them nowhere.• For moving objects on a micron scale, transport bydiffusion dominates over transport by stirring.• Bacteria swim to find areas with more food, not to gatherfood more quickly out of solution.