Bayly P V, Wilson K S
Mechanical Engineering and Materials Science, Washington University in Saint Louis, 1 Brookings Drive, Box 1185, Saint Louis, MO 63130, USA
Mechanical Engineering and Materials Science, Washington University in Saint Louis, 1 Brookings Drive, Box 1185, Saint Louis, MO 63130, USA.
J R Soc Interface. 2015 May 6;12(106). doi: 10.1098/rsif.2015.0124.
The mechanisms underlying the coordinated beating of cilia and flagella remain incompletely understood despite the fundamental importance of these organelles. The axoneme (the cytoskeletal structure of cilia and flagella) consists of microtubule doublets connected by passive and active elements. The motor protein dynein is known to drive active bending, but dynein activity must be regulated to generate oscillatory, propulsive waveforms. Mathematical models of flagellar motion generate quantitative predictions that can be analysed to test hypotheses concerning dynein regulation. One approach has been to seek periodic solutions to the linearized equations of motion. However, models may simultaneously exhibit both periodic and unstable modes. Here, we investigate the emergence and coexistence of unstable and periodic modes in three mathematical models of flagellar motion, each based on a different dynein regulation hypothesis: (i) sliding control; (ii) curvature control and (iii) control by interdoublet separation (the 'geometric clutch' (GC)). The unstable modes predicted by each model are used to critically evaluate the underlying hypothesis. In particular, models of flagella with 'sliding-controlled' dynein activity admit unstable modes with non-propulsive, retrograde (tip-to-base) propagation, sometimes at the same parameter values that lead to periodic, propulsive modes. In the presence of these retrograde unstable modes, stable or periodic modes have little influence. In contrast, unstable modes of the GC model exhibit switching at the base and propulsive base-to-tip propagation.
尽管纤毛和鞭毛的协调摆动具有重要的基础意义，但其潜在机制仍未被完全理解。轴丝（纤毛和鞭毛的细胞骨架结构）由通过被动和主动元件连接的微管双联体组成。已知动力蛋白驱动主动弯曲，但动力蛋白的活性必须受到调节才能产生振荡的推进波形。鞭毛运动的数学模型产生了定量预测，可对其进行分析以检验有关动力蛋白调节的假设。一种方法是寻求线性化运动方程的周期解。然而，模型可能同时呈现周期模式和不稳定模式。在这里，我们研究了鞭毛运动的三个数学模型中不稳定模式和周期模式的出现与共存情况，每个模型基于不同的动力蛋白调节假设：（i）滑动控制；（ii）曲率控制；（iii）通过双联体间分离进行控制（“几何离合器”（GC））。每个模型预测的不稳定模式被用于严格评估其潜在假设。特别是，具有“滑动控制”动力蛋白活性的鞭毛模型允许存在具有非推进性、逆行（从尖端到基部）传播的不稳定模式，有时在导致周期性推进模式的相同参数值下也会出现。在这些逆行不稳定模式存在的情况下，稳定或周期模式的影响很小。相比之下，GC模型的不稳定模式在基部表现出切换以及从基部到尖端的推进传播。
