Rigid-Body rotation versus transverse bending wave swimming of magnetically-functionalized sperm cells

Cell membrane potential affects the electrostatic self-assembly of magnetizable nanoparticles around the flagellum of sperm cells, leading to the formation of biohybrid microrobots (i.e. IRONSperm) with various bending stiffness. Here we explain the influence of bull sperm cell membrane potential on the formation of two types of IRONSperm samples that are produced by electrostatic self-assembly. The first is a proximal-coated soft body with nanoparticles concentrated on the head to maintain high flexibility of the flagellum and create a passively propagating transverse bending wave under the influence of an external rotating magnetic field. The second is a rigid-body with nanoparticles approximately uniformly distributed along the length to provide arbitrary geometry that maintains a constant chiral shape and propel by rotation about its long axis. We present a magneto-elastohydrodynamic model to predict the swimming speed at low Reynolds number for rigid IRONSperm with arbitrary shapes, and show that decreasing the bending stiffness allows the model to capture the behavior of its soft counterpart. While the response of a rigid chiral IRONSperm is distinguished by a greater swimming speed with a smooth decay with frequency, the benefit of a soft flagellum in certain scenarios would present a much smaller range of frequencies for wireless actuation.