Efficient simulations of tubulin-driven axonal growth

TL;DR

This paper introduces an efficient numerical scheme for simulating tubulin-driven axonal growth, enabling detailed analysis of growth dynamics and parameter dependencies with improved computational speed and accuracy.

Contribution

The paper presents a novel application of the Peaceman--Rachford splitting scheme combined with scalings for simulating a nonlinear moving-boundary PDE model of axonal growth.

Findings

Initial axon growth is very rapid.

Active transport dominates over diffusion in growth velocity.

Polymerization rate does not influence final axon length.

Abstract

This work concerns efficient and reliable numerical simulations of the dynamic behaviour of a moving-boundary model for tubulin-driven axonal growth. The model is nonlinear and consists of a coupled set of a partial differential equation (PDE) and two ordinary differential equations. The PDE is defined on a computational domain with a moving boundary, which is part of the solution. Numerical simulations based on standard explicit time-stepping methods are too time consuming due to the small time steps required for numerical stability. On the other hand standard implicit schemes are too complex due to the nonlinear equations that needs to be solved in each step. Instead, we propose to use the Peaceman--Rachford splitting scheme combined with temporal and spatial scalings of the model. Simulations based on this scheme have shown to be efficient, accurate, and reliable which makes it possible to evaluate the model, e.g.\ its dependency on biological and physical model parameters. These evaluations show among other things that the initial axon growth is very fast, that the active transport is the dominant reason over diffusion for the growth velocity, and that the polymerization rate in the growth cone does not affect the final axon length.