Max Planck Institute for Dynamics and Self-Organization (MPIDS), Göttingen 37077, Germany.
Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3PU, UK.
J R Soc Interface. 2022 Jan;19(186):20210693. doi: 10.1098/rsif.2021.0693. Epub 2022 Jan 5.
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.
生物纤毛通过由滑动微管二联体之间的动力蛋白马达驱动的不对称拍打来泵送周围的流体。生物纤毛的复杂性引发了一个问题，即是否存在可以再现类似运动模式的最小系统。这样的系统由一对在近端被夹住的微管组成。它们通过覆盖一个纤维丝并对另一个纤维丝施加拉力的动力蛋白马达相互作用。在这里，我们从理论上研究了这种系统的静态形状和主动动力学。我们使用弹性理论分析了具有固定端的两个不同长度的纤维丝的形状。从相等的长度开始，我们观察到类似于欧拉屈曲的转变，导致平面形状。当进一步增加长度比时，系统在二次分叉后呈现出非平面形状，并且在再次过渡到平面后自发打破手性对称。预测的曲线与实验观察到的微管对的形状相吻合。动力学系统可以具有稳定的固定点，具有弯曲或直的纤维丝，或者具有极限环振荡。后者与纤毛运动的许多特性相匹配，证明了双纤维丝系统可以作为一个最小的主动拍打模型。
