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.

Modeling mitochondrial dynamics during in vivo axonal elongation

“…The result is in qualitative agreement with studies that observed a decline in transported mitochondria in growing axons (Miller and Samuels, 1997; Morris and Hollenbeck, 1993). This flux gradient is present in models of transport which consider protein degradation (McLean and Graham, 2004) and is absent in models which do not (Friedman and Craciun, 2005). Smith and Simmons reported a steady flux in their unidirectional transport model and a linearly declining flux in their bidirectional transport model (Smith and Simmons, 2001).…”

Stochastic models of intracellular transport

“…(It is possible that short, freshly nucleated microtubles are also actively transported into axons (Baas and Buster, 2004)). For the sake of illustration, consider a continuum model of the active transport of tubulin (Graham et al, 2006;McLean and Graham, 2004). Let c(x, t) denote the concentration of tubulin at position x along the axon at time t. Suppose that at time t the axon has length l(t) so that x ∈ [0, l(t)].…”

“…Axon elongation is a consequence of the interplay between force generation at the growth cone that pulls the axon forward, pushing forces due to microtubule and actin polymerization and depolymerization, the rate of protein synthesis at the cell body, and the action of cytoskeletal motors (Baas and Ahmad, 2001;Goldberg, 2003;Lamoureux et al, 1989;Mitchison and Kirschner, 1988;O'Toole et al, 2008;Suter and Miller, 2011). Several models of axonal elongation have focused on the sequence of processes based on the production of tubulin dimers at the cell body, the active transport of these proteins to the the tip of the growing axon, and microtubule extension at the growth cone (Graham et al, 2006;Kiddie et al, 2005;McLean and Graham, 2004;Miller and Samulels, 1997;van Veen and van Pelt, 1994). One motivation for identifying the polymerization of microtubules as a rate limiting step is that axonal growth occurs at a similar rate to the slow axonal transport of tubulin, namely, around 1mm per day.…”

Analytical solution of equations describing slow axonal transport based on the stop-and-go hypothesis

“…For Fig. 6b, the following PDF values at the boundaries [selected to satisfy the integral condition (18)] are utilized:…”

“…In particular, defects in slow axonal transport of NFs may lead to aggregation of NFs in certain regions in an axon [14]; such aggregation has been linked to human motor neuron diseases [15,16]. The study of NF transport is also important for understanding the dynamics of axon elongation for growing axons [17,18].…”