axonalGrowth

Developing a phase-field model for axonal growth. We have interest on how mRNA traffic affects the growth cone navigation.

Phase-field model for axonal growth

The growth cone is described by the order parameter $\phi$ as a droplet of radius $R_{GC}$ with velocity $\vec{v}$

where $\alpha_V$ is the Lagrange multiplier with volume target is the instantaneous volume of the growth cone. The velocity is written as a function of the concentration of the Nerve Growth Factor (NGF)

The neuron and axon are modeled by the Cahn-Hilliard equation plus a phenomenological proliferative term coupled with the growth cone movement

where the first term in the right-hand side maintains the interface and the second term is responsible by introducing material at the interface of $\phi$ and $\psi$. This proliferative term depends on the concentration of NGF.

We model the mRNA with two phenotypes: free $m_f$ and linked $m_l$ mRNA. The mRNA dynamic is described by the following reaction-diffusion equations