Neurite outgrowth (dendrites and axons) should be a stable, but easily regulated process to enable a neuron to make its appropriate network connections during development. We explore the dynamics of outgrowth in a mathematical continuum model of neurite elongation. The model describes the construction of the internal microtubule cytoskeleton, which results from the production and transport of tubulin dimers and their assembly into microtubules at the growing neurite tip. Tubulin is assumed to be largely synthesised in the cell body from where it is transported by active mechanisms and by diffusion along the neurite. It is argued that this construction process is a fundamental limiting factor in neurite elongation. In the model, elongation is highly stable when tubulin transport is dominated by either active transport or diffusion, but oscillations in length may occur when both active transport and diffusion contribute. Autoregulation of tubulin production can eliminate these oscillations. In all cases a stable steady-state length is reached, provided there is intrinsic decay of tubulin. Small changes in growth parameters, such as the tubulin production rate, can lead to large changes in length. Thus cytoskeleton construction can be both stable and easily regulated, as seems necessary for neurite outgrowth during nervous system development.

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.

No abstract

No abstract

No abstract