Title: Optimal Feeding of Swimming and Attached Ciliates

Abstract: Ciliated microorganisms near the base of the aquatic food chain either swim to encounter prey or attach at a substrate and generate feeding currents to capture passing particles. Here, we represent attached and swimming ciliates using a popular spherical model in viscous fluid with slip surface velocity that afford analytical expressions of ciliary flows. We solve an advection-diffusion equation for the concentration of dissolved nutrients, where the P'eclet number (Pe) reflects the ratio of diffusive to advective time scales. For a fixed hydrodynamic power expenditure, we ask what ciliary surface velocities maximize nutrient flux at the microorganism's surface. We find that surface motions that optimize feeding depend on Pe. For freely swimming microorganisms at finite Pe, it is optimal to swim by employing a "treadmill" surface motion, but in the limit of large Pe, there is no difference between this treadmill solution and a symmetric dipolar surface velocity that keeps the organism stationary. For attached microorganisms, the treadmill solution is optimal for feeding at Pe below a critical value, but at larger Pe values, the dipolar surface motion is optimal. We verified these results in open-loop numerical simulations, asymptotic analysis, and using an adjoint-based optimization method. Our findings challenge existing claims that optimal feeding is optimal swimming across all P'eclet numbers, and provide new insights into the prevalence of both attached and swimming solutions in oceanic microorganisms.