Volume-surface reaction-diffusion systems arise in many applications such as cell-biology, chemistry, fluid dynamics, crystal growth, etc.

A particular example is asymmetric cell division, which is an important mechanism in generating diversity of cell fates during the development of animals, where specific proteins, so-called cell fate determinants, are localised to one side of the cell cortex and, thus, segregated into only one daughter cell, making that cell different from its sibling.

We shall discuss recent results concerning the mathematical modelling, the analysis, the large-time behaviour and the numerical simulation of volume-surface reaction-diffusion models describing the asymmetric localisation of the protein Lgl (Lethal giant larvae) in Drosophila SOP precursor cells.

In particular, we shall present entropy methods for volume-surface reaction diffusion systems and a conservative, positivity preserving first order finite element discretisation.

Using a linear stability analysis, we identify which structure parameters make the membrane wall unstable around a spherical shape. In particular, we show that the coupling between the two equations has to be highly non-linear to make the system unstable around the spherical shape. Finally, we present numerical simulations which confirm this theoretical stability analysis.