Spiroplasma is a small helical bacterium that swims and performs chemotaxis in a nonconventional way. Equipped with a flexible flat ribbon cytoskeleton, Spiroplasma deforms its body into a helix that periodically changes its chirality. The helical transformation produces a kink that propagates along the cell body and propels the microorganism in the opposite direction. Based on experimental observations, we develop a hydrodynamic model to describe Spiroplasma motility. We obtain expressions for the total linear and angular displacements of the cell body per swimming stroke. We show mathematically that the cell body does not reorient at the end of one period, which allows us to define an effective swimming speed, $v_s$. We then use $v_s$ to calculate the energy dissipated in one stroke, $\dot{W}$, and to define a hydrodynamic efficiency, $\eta ={\dot{W}}_0/\dot{W}$, where ${\dot{W}}_0$ is the power spent by a straight filament moving at the same speed $v_s$. We show that the helical shape of the cytoskeleton that maximizes both $v_s$ and $\eta$ are helices of pitch angles $\varphi$ close to that of Spiroplasma, $\varphi \simeq {35}^{\circ }$, in agreement with experimental observations and with previous numerical simulations.

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螺旋体运动性的流体力学模型

螺旋体是一种小的螺旋细菌，其以非常规方式游动并执行趋化作用。配备有柔性扁平带状细胞骨架，螺原体变形其身体成周期性地改变其手性螺旋。螺旋形转变产生一个扭结，该扭结沿着细胞体传播并以相反的方向推动微生物。基于实验观察，我们开发了描述螺旋体运动性的流体动力学模型。我们获得了每个游泳冲程中细胞体的总线性和角位移的表达式。我们从数学上证明，一个周期结束时细胞体不会重新定向，这使我们能够定义有效的游泳速度，$v_s$。然后我们使用$v_s$ 计算一次消耗的能量， $\dot{\mathrm{w\; ^}}$，并定义水动力效率， $\eta ={\dot{\mathrm{w\; ^}}}_0/\dot{\mathrm{w\; ^}}$，在哪里 ${\dot{\mathrm{w\; ^}}}_0$ 是直丝以相同速度移动所消耗的功率 $v_s$。我们表明，使两者都最大化的细胞骨架的螺旋形状$v_s$ 和 $\eta$ 是俯仰角的螺旋 $\varphi$接近螺旋体，$\varphi \simeq {35}^{\circ }$，与实验观察结果和先前的数值模拟一致。