A mathematical model for axonal transport of large cargo vesicles.

In this study, we consider axonal transport of large cargo vesicles characterised by transient expansion of the axon shaft. Our goal is to formulate a mathematical model which captures the dynamic mechanical interaction of such cargo vesicles with the membrane associated periodic cytoskeletal structure (MPS). It consists of regularly spaced actin rings that are transversal to the longitudinal direction of the axon and involved in the radial contraction of the axon. A system of force balance equations is formulated by which we describe the transversal rings as visco-elastic Kelvin-Voigt elements. In a homogenisation limit, we reformulate the model as a free boundary problem for the interaction of the submembranous MPS with the large vesicle. We derive a non-linear force-velocity relation as a quasi-steady state solution. Computationally we analyse the vesicle size dependence of the transport speed and use an asymptotic approximation to formulate it as a power law that can be tested experimentally.