最适弹性鞭毛
The optimal elastic flagellum
摘要：
能动的真核细胞在粘性液体中通过鞭毛传递弯曲变形波来推动自己。一个无限长的鞭毛实现流体动力学最佳的低雷诺数运动时，其本地切线和游泳方向之间的角度保持恒定沿其长度。因此，最适鞭毛在三维中采用螺旋形（光滑），在二维中采用螺旋形（不光滑）。从物理上讲，生物有机体（或工程微型游泳者）必须消耗内能，以产生负责运动的变形波。在这里，我们提出了一个物理动机的最佳鞭毛形状的推导。我们确定的鞭毛波的形状，导致最快的游泳，同时最大限度地减少了适当定义的能量消耗的分析和数值。我们的新方法是定义一种能量，它不仅包括对周围流体的功，而且包括（1）在鞭毛弯曲时弹性储存的能量，（2）在负责产生弯曲波（微管）的聚合物细丝的内部滑动中弹性储存的能量，以及（3）由于内部流体的存在而产生的粘性耗散。这种方法以流体动力学效率的小损失为代价，正则化了二维变形的最佳形状。有限大小的鞭毛的最佳波形取决于旋转运动和弯曲成本之间的竞争，我们观察到一个令人惊讶的偏向半整数波数。它们的最终流体动力学效率高于6%，显著大于游泳细胞的流体动力学效率，因此表明进一步生物调谐的可用空间。
英文摘要：
Motile eukaryotic cells propel themselves in viscous fluids by passing waves of bending deformation down their flagella. An infinitely long flagellum achieves a hydrodynamically optimal low-Reynolds number locomotion when the angle between its local tangent and the swimming direction remains constant along its length. Optimal flagella therefore adopt the shape of a helix in three dimensions (smooth) and that of a sawtooth in two dimensions (non-smooth). Physically, biological organisms (or engineered micro-swimmers) must expend internal energy in order to produce the waves of deformation responsible for the motion. Here we propose a physically-motivated derivation of the optimal flagellum shape. We determine analytically and numerically the shape of the flagellar wave which leads to the fastest swimming while minimizing an appropriately-defined energetic expenditure. Our novel approach is to define an energy which includes not only the work against the surrounding fluid, but also (1) the energy stored elastically in the bending of the flagellum, (2) the energy stored elastically in the internal sliding of the polymeric filaments which are responsible for the generation of the bending waves (microtubules), and (3) the viscous dissipation due to the presence of an internal fluid. This approach regularizes the optimal sawtooth shape for two-dimensional deformation at the expense of a small loss in hydrodynamic efficiency. The optimal waveforms of finite-size flagella are shown to depend upon a competition between rotational motions and bending costs, and we observe a surprising bias towards half-integer wave-numbers. Their final hydrodynamic efficiencies are above 6%, significantly larger than those of swimming cells, therefore indicating available room for further biological tuning.
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