I. INTRODUCTION Understanding the properties of colloidal systems which are intrinsically out-of- equilibrium as they are active, is an important emerging goal in soft matter physics, driven both by the potential for modelling motile cells as active colloidal “swimmers”, and for designing self-propelled particles with novel nanotechnological applications [1–7]. Several mechanisms for colloidal self-propulsion have been proposed: these can be broadly grouped into those where the colloid exerts force on the surrounding fluid (as for swimming bacteria or other motile cells) [1–5], and those in which the colloid changes the chemical properties of the surrounding medium in an asymmetric way by catalysing a chemical reaction on its surface [8–10] or by secreting some product [11–15]. In this paper, we consider a “chucker”: a spherical colloidal particle which produces smaller “solute” particles at its surface. Our model colloid is simpler than most cases considered thus far [12–15], since it “chucks” solute particles isotropically, and therefore does not display directional motion. However, the chucking drives our system out of equilibrium, giving it interesting dynamical properties. We find that the effective diffusion constant of the chucker increases with the chucking rate k$_{c}$ for low k$_{c}$ , and eventually decreases for high k$_{c}$ due to self-crowding. We further simulate the dragging of a chucker, to obtain a measurement of its effective mobility. Comparing this to the effective diffusion constant allows us to show that the fluctuation-dissipation theorem does not hold in this non-equilibrium system, and that the deviations from the FDT are non-monotonic in the chucking rate. Finally, we consider the steady-state positional probability distribution function for a chucker which sediments against a planar surface, in the case where the surface is “hard”, and in the case where the surface absorbs the solute particles. Although our model is designed to be generic, our work is inspired by the observa- tion that many bacterial cells secrete extracellular polysaccharides [16, 17], in some cases in large quantities (for example the bacterium Xanthomonas campestris can produce xan- than polymer at a rate as high as 10 4 polymer molecules per cell per second [18, 19]). These exopolysaccharides can have important effects on the collective properties (e.g. phase behaviour) of assemblies of bacteria [20]. Our simulations suggest that in some cases ex- opolysaccharide production might also affect the diffusive motion of single bacterial cells. We note that the case of a chucker sedimenting onto an absorbing surface might be relevant 2