Bruce Graham

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  Mathematical models of intracellular transport-limited neurite elongation and branching
  Bruce Graham

  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 (McLean & Graham, Proc. R. Soc. Lond. A, 460:2437-2456, 2004; Graham et al, J. Comput. Neurosci., 20:43-60, 2006). 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.

  
  

  In a model variant, we demonstrate competitive growth between two neurite branches being supplied by the same source of tubulin (van Ooyen et al, Neurocomputing, 38-40:73-78, 2001). The faster growing neurite can completely inhibit elongation in the other neurite. Such apparent competition has been observed in real neuron outgrowth.

  
  

  In a different model formulation, the propensity of neurite branching is assumed to depend on the amount of tubulin reaching the growth cone (Graham & van Ooyen, J. Theor. Biol., 230:421-432, 2004). This transport-limited branching yields matches to the characteristic dendritic morphologies from different neuronal types.