Fluid dynamic model of invertebrate sperm chemotactic motility with varying calcium inputs

In a marine environment, invertebrate sperm are able to adjust their trajectory in response to a gradient of chemical factors released by the egg in a process called chemotaxis. In response to this chemical factor, a signaling cascade is initiated that causes an increase in intracellular calcium (Ca2+). This increase in Ca2+ causes the sperm flagellar curvature to change, and a change in swimming direction ensues. In previous experiments, sperm swimming in a gradient of chemoattractant have exhibited Ca2+ oscillations of varying peaks and frequency. Here, we model a simplified sperm flagellum with mechanical forces, including a passive stiffness component and an active bending component that is coupled to the time varying Ca2+ input. The flagellum is immersed in a viscous, incompressible fluid and we use a fluid dynamic model to investigate emergent trajectories. We investigate the sensitivity of the model to the frequency of Ca2+ oscillations. In this coupled model, we observe that longer periods of Ca2+ oscillation corresponds to circular paths with greater drift. In contrast, shorter periods of Ca2+ oscillations corresponded to tighter search patterns. These outcomes shed light on the relation between Ca2+ oscillations and different searching trajectories and strategies that invertebrate sperm may utilize to reach and fertilize the egg in a marine environment.