Spontaneous polarization and locomotion of an active particle with surface-mobile enzymes*

We study the locomotion of active colloids whose surface is uniformly coated with mobile enzymes, which can be realized experimentally by attaching enzymes to the surface of vesicles, polymerosomes or other particles with fluid-like surfaces. The enzymes can migrate and diffuse laterally along the surface and they and catalyze a reaction that generates surface phoretic flows. Since the  distribution of enzymes along the surface is homogeneous the locomotion of the active colloid is forbidden by symmetry. However, we find that the ability of the enzymes to migrate over the surface, combined with self-phoresis, can lead to a spontaneous symmetry-breaking instability whereby the homogeneous distribution of enzymes polarizes and the active particle propels. The instability is driven by the advection of enzymes by the phoretic flows and occurs above a critical Péclet number. The transition to polarized motile states occurs via a supercritical or subcritical pitchfork bifurcations, the latter of which enables hysteresis and coexistence of uniform and polarized states.