Actin growth is a fundamental biophysical process and it is, at the same time, a prototypical example of diffusion-mediated surface growth. We consider a one dimensional growth model for actin: a rod-like element composed of actin monomers is fixed at one end and connected to an elastic device at the other. The mechanical behaviour of the rod, the diffusion of free actin monomers in a surrounding solvent and the kinetic growth laws at the accreting/ablating ends are accounted for. The constitutive behaviour of actin is prescribed by a convex strain energy density. The existence of treadmilling solutions, characterized by a constant length of the continuously evolving body, is investigated. It is shown that the present model admits at most one such solution and that it is always stable.
