An exact solution of transient equations describing slow axonal transport

An exact analytical solution of equations describing slow axonal transport of cytoskeletal elements (CEs) injected in an axon is presented. The equations modelling slow axonal transport are based on the stop-and-go hypothesis. The simplest model implementing this hypothesis postulates that CEs switch between pausing and running kinetic states, and that the probabilities of CE transition between these two states are described by first-order rate constants. It is assumed that initially CEs are injected such that they form a uniform pulse of a given width. All injected CEs are initially attributed to the pausing state. It is shown that within 30 s kinetic processes redistribute CEs between pausing and running states; after that the process occurs under quasi-equilibrium conditions. The parameter accessible to experiments is the total concentration of CEs (pausing plus running). As the initial rectangular-shaped pulse moves, it changes its shape to become a bell-shaped wave that spreads out as it propagates. The wave's amplitude is decreasing during the wave's propagation. It is also shown that the system forgets its initial condition, meaning that if one starts with pulses of different widths, after sometime they converge to the same bell-shaped wave.