Abstract
Biological cilia pump the surrounding fluid by asymmetric beating that is driven by dynein motors between sliding microtubule doublets. The complexity of biological cilia raises the question about minimal systems that can re-create similar patterns of motion. One such system consists of a pair of microtubules that are clamped at the proximal end. They interact through dynein motors that cover one of the filaments and pull against the other one. Here, we study theoretically the static shapes and the active dynamics of such a system. Using the theory of elastica, we analyse the shapes of two filaments of different lengths with clamped ends. Starting from equal lengths, we observe a transition similar to Euler buckling leading to a planar shape. When further increasing the length ratio, the system assumes a non-planar shape with spontaneously broken chiral symmetry after a secondary bifurcation and then transitions to planar again. The predicted curves agree with experimentally observed shapes of microtubule pairs. The dynamical system can have a stable fixed point, with either bent or straight filaments, or limit cycle oscillations. The latter match many properties of ciliary motility, demonstrating that a two-filament system can serve as a minimal actively beating model.
Highlights
Cilia and flagella are cellular appendages that can spontaneously beat in an asymmetric or undulatory fashion in order to transport the surrounding fluid or propel a swimming microorganism [1]
We recently reported on the bottom-up assembly of a minimal synthetic axoneme consisting of two microtubules, growing from a common seed, and a patch of self-assembled dynein motors on one of them [25]
We have shown that two filaments with one clamped end and motors acting between them show a variety of static and dynamical regimes
Summary
Cilia and flagella are cellular appendages that can spontaneously beat in an asymmetric or undulatory fashion in order to transport the surrounding fluid or propel a swimming microorganism [1]. The beating is powered by axonemal dynein motors that induce a shearing force between pairs of doublet microtubules. The control mechanism that activates the dynein motors and maintains the beating, is not yet well understood. An attractive hypothesis is that the dynein motors react to the sliding motion of the filaments with effective negative damping at a certain frequency [3,4]. There are several processes by which motor proteins can induce spontaneous oscillations [5]. An alternative proposition is the ‘geometric clutch’ model, which was initially based on the qualitative notion that a buckled filament loses contact with the motors, which become inactivated [6]. It has been proposed that the motors are controlled by transverse stress, which is coupled to the curvature in helically twisted axonemes [7]
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