Geometry of ciliary dynamics.

Cilia are motile biological appendages that are driven to bend by internal shear stresses between tubulin filaments. A continuum model of ciliary material is constructed that incorporates the essential ciliary constraints: (i) one-dimensional inextensibility of filaments, (ii) three-dimensional incompressibility, and (iii) shear strain only longitudinally along filaments. It is shown that twist of filaments about each other is not an independent degree of freedom under ciliary constraints. The constraint on twist appears in the equations of motion for cilia as a term not previously recognized. As another application of the same geometrical idea, a general approach to the polymorphism of bacterial flagella is proposed.