A one-dimensional moving-boundary model for tubulin-driven axonal growth

TL;DR

This paper introduces a one-dimensional continuum-mechanical model for axonal growth driven by tubulin assembly, analyzing steady states, stability, and concentration distributions to understand axon elongation dynamics.

Contribution

It develops a coupled differential equation model for tubulin transport and axon elongation, providing explicit steady-state solutions and stability analysis based on biological parameters.

Findings

Multiple steady states can exist for axon length.

The shorter steady state is unstable, the longer is stable.

Free tubulin concentration is often lower in the axon than in soma and growth cone.

Abstract

A one-dimensional continuum-mechanical model of axonal elongation due to assembly of tubulin dimers in the growth cone is presented. The conservation of mass leads to a coupled system of three differential equations. A partial differential equation models the dynamic and spatial behaviour of the concentration of tubulin that is transported along the axon from the soma to the growth cone. Two ordinary differential equations describe the time-variation of the concentration of free tubulin in the growth cone and the speed of elongation, respectively. All steady-state solutions of the model are categorized. Given a set of the biological parameter values, it is shown how one easily can infer whether there exist zero, one or two steady-state solutions and directly determine the possible steady-state lengths of the axon. Explicit expressions are given for each stationary concentration distribution. It is thereby easy to examine the influence of each biological parameter on a steady state. Numerical simulations indicate that when there exist two steady states, the one with shorter axon length is unstable and the longer is stable. Another result is that, for nominal parameter values extracted from literature, in a large portion of a fully grown axon the concentration of free tubulin is lower than both concentrations in the soma and in the growth cone.