Abstract
We study the dynamics of an elastic rod-like filament in two dimensions, driven by internally generated forces. This situation is motivated by cilia and flagella which contain an axoneme. These hair-like appendages of many cells are used for swimming and to stir surrounding fluids. Our approach characterizes the general physical mechanisms that govern the behaviour of axonemes and the properties of the bending waves generated by these structures. Starting from the dynamic equations of a filament pair in the presence of internal forces we use a perturbative approach to systematically calculate filament shapes and the tension profile. We show that periodic filament motion can be generated by a self-organization of elastic filaments and internal active elements, such as molecular motors, via a dynamic instability termed Hopf bifurcation. Close to this instability, the behaviour of the system is shown to be independent of many microscopic details of the active system and only depends on phenomenological parameters such as the bending rigidity, the external viscosity and the filament length. Using a two-state model for molecular motors as an active system, we calculate the selected oscillation frequency at the bifurcation point and show that a large frequency range is accessible by varying the axonemal length between 1 and 50 µm. We discuss the effects of the boundary conditions and externally applied forces on the axonemal wave forms and calculate the swimming velocity for the case of free boundary conditions.
Highlights
Many small organisms and cells swim in a viscous environment using the active motion of cilia and flagella
We are interested in those flagella and cilia which contain force generating elements integrated along the whole length of the elastic filamentous structure. They represent rod-like elastic structures which move and bend as a result of internal stresses. Examples for these systems are paramecium which has a large number of cilia on its surface; sperm, which use a single flagellum to swim; and chlamydomonas which uses two flagella to swim (Bray 1992)
An example is the kinocilium which exists in many hair bundles of mechanosensitive cells and has the ability to beat periodically (Rusch and Thurm 1990)
Summary
Many small organisms and cells swim in a viscous environment using the active motion of cilia and flagella. This implies an instability of the initial straight state with respect to a wave-like mode It has been demonstrated using a simple model for molecular motors that a large number of motors working against an elastic element can generate oscillations via a Hopf bifurcation by a generic mechanism (Julicher and Prost 1995, Julicher and Prost 1997). This suggests that a system consisting only of molecular motors and semiflexible filaments can in general undergo self-organized oscillations. We discuss the relevance of our simple model to real axonemal cilia and flagella and propose experiments which could be performed to test predictions that follow from our work
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