Mathematical formulation and analysis of a continuum model for tubulin-driven neurite elongation

Abstract

A partial–differential–equation model of neurite growth is developed. This model is the first of its kind and uses a continuum mechanical approach to model the effects of active transport, diffusion and species degradation of the oligomer tubulin, which is used in the elongation of a single neurite. The model problem is mathematically difficult since it must be solved on a dynamically growing domain. The development and implementation of a spatial transformation to a neurite length coordinate simplifies the problem. Existence and uniqueness of solutions to the steady–state problem are found and shown to be equivalent to solving a nonlinear equation for the steady–state length. This expression is not directly solvable except in certain degenerate cases. However, one system parameter is naturally small and permits solutions in terms of asymptotic series. We identify three growth regimes analytically and verify them numerically. It is then evident that a neuron may easily regulate the extent of its own neuritic growth by increasing or decreasing its tubulin production relative to the active transport/degradation fraction.