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The Scallop theorem states that to achieve propulsion at low Reynolds number in Newtonian fluids a swimmer must deform in a way that is not invariant under time-reversal. Edward Mills Purcell stated this theorem in his 1977 paper Life at Low Reynolds Number explaining physical principles of aquatic locomotion. The theorem is named for the scallop, the only bivalve able to actively swim distances. However, its technique is not applicable to viscous fluids, thus the name of the theorem.
Although the movement of animal cells is usually studied as they migrate, it seems likely that many motile cells can also swim. Thus, human granulocytes are able to migrate towards a source of a chemoattractant, the tripeptide FMLP, whilst suspended in a uniformly-dense (isodense) medium. They swim at the same speed as they would crawl on a solid surface. Likewise, Dictyostelium discoideum amoebae swim towards a chemical attractant, in this case cyclic AMP. The actual mechanism that these neutrophils or amoebae use to produce a thrust against the medium to propel themselves is uncertain; however, how they do so must be consistent with physical principles. To swim they must transmit a force against the viscous fluid in order to propel themselves forward. Different mechanisms by which they might do so were presented by Ed Purcell  in a famous talk he gave celebrating the 80th birthday of his friend Viki Weisskopf.
In this he developed his “scallop theorem”: a normal scallop moves by opening its shells slowly and shutting them quickly. In the latter step it quickly squeezes the fluid between the shells backwards and, using the momentum of the water, pushes itself forward. Purcell realised that a microorganism trying to do the same would simply move forwards on shutting its shells and move backwards to its original position on opening them. The set of movements is “reciprocal”: it appears the same if viewed forwards or backwards in time. He concluded that microorganisms cannot move by a reciprocal mechanism: to move, they must exert some thrust against the medium and do so in a non-reciprocal manner. He suggested various ways in which an organism could swim:
- They could do so with a flagellum, which rotates, pushing the medium backwards — and the cell forwards — in much the same way that a ship’s screw moves a ship. This is how some bacteria move; the flagellum is attached at one end to a complex rotating motor held rigidly in the bacterial cell surface
- They could use a flexible arm: this could be done in many different ways. For example, mammalian sperm have a flagellum which, whip-like, wriggles at the end of the cell, pushing the cell forward. Cilia are quite similar structures to mammalian flagella; they can advance a cell like paramecium by a complex motion not dissimilar to breast stroke.
- A hypothetical toroidal cell could move by rotating its surface through the central hole, thereby creating a surface flow. The surface drag on the outer edges of the cell could provide the thrust against the medium needed to move the cell forward. This is related to the membrane flow model B, above, except in that scheme the surface flow is achieved by removing surface from the rearward end of the cell and transporting it as vesicles through the cell interior to the cell's front.
The manner in which cells swim, and therefore move, suggests that it is membrane flow which is the motor for movement.
- Barry NP & Bretscher MS (2010). "Dictyostelium amoebae and neutrophils can swim". Proc. Natl. Acad. Sci. USA 107: 11376–11380.
- Purcell EM (1977). "Life at low Reynolds number". Am J Physics 45: 3–11.
- Berg HC & Anderson RA (1973). "Bacteria swim by rotating their flagellar filaments". Nature 245 (5425): 380–382. doi:10.1038/245380a0. PMID 4593496.
- Silverman M & Simon M (1974). "Flagellar rotation and the mechanism of bacterial motility". Nature 249 (100): 73–74. doi:10.1038/249073a0. PMID 4598030.
- Brokaw CJ (1991). "Microtubule Sliding in Swimming Sperm Flagella: Direct and Indirect Measurements on Sea Urchin and Tunicate Spermatozoa". J Cell Biol 114 (6): 1201–1215.
- E.M. Purcell. Life at Low Reynolds Number, American Journal of Physics vol 45, p. 3-11 (1977)
- A presentation on Purcell's theory
- Kinematic Reversibility and the Scallop Theorem
- Video of a swimming robot unable to propel in viscous fluid due to the Scallop theorem
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