A minimal model for Spiroplasma chemotaxis

\textit{Spiroplasma} is a small helical bacterium that swims and performs chemotaxis in a non conventional way. Instead of actuating flagella, it swims by progressively shifting the chirality of its body. The change in geometry gives rise to a wall domain - a kink - which propagates along the cell body. The chirality is then reverted in a similar fashion completing a swimming stroke. The whole deformation is non-reciprocal in time, therefore movement at low Reynolds number is achieved with the bacterium moving in the direction opposite to the kink pair propagation. Based on experimental observations, we develop a minimal model to describe \textit{Spiroplasma} chemotaxis. We start by a simple resistive force theory model of the bacterium swimming gait. Using symmetry arguments we show how to calculate the net rotation and translation of the bacterium during one full stroke. We obtain expressions for the linear displacement as a function of the time between kinks, $\tau_k$, that compare favourably with numerical computations. Using our theoretical results, we then construct a random walk model for \textit{Spiroplasma} and we obtain expressions for its diffusivity $D_e$ and chemotactic drift velocity $v_d$ as a function of $\tau_k$ and the kink angle $\theta$.