Treadmilling stability of a one-dimensional actin growth model

Actin growth is a fundamental biophysical process and it is, at the same time, a prototypical example of diffusion-mediated surface growth. We formulate a coupled chemo-mechanical, one-dimensional growth model encompassing both material accretion and ablation. A solid bar composed of bound actin monomers is fixed at one end and connected to an elastic device at the other. This spring-like device could, for example, be the cantilever tip of an atomic force microscope. The compressive force applied by the spring on the bar increases as the solid grows and affects the rate of growth. The mechanical behaviour of the bar, 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 response of actin is modeled by a convex but otherwise arbitrary elastic strain energy density function. Treadmilling solutions, characterized by a constant length of the continuously evolving body, are investigated. Existence and stability results are condensed in the form of simple formulas and their physical implications are discussed.