  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.
