Cell polarisation in a bulk-surface model can be driven by both classic and non-classic Turing instability

The GTPase Cdc42 is the master regulator of cell polarisation. During this process the active form of Cdc42 is accumulated at a particular site on the cell membrane called the pole. It is believed that the accumulation of the active Cdc42 resulting in a pole is driven by a combination of activation-inactivation reactions and diffusion. It has been proposed using mathematical modelling that this is the result of diffusion-driven instability, originally proposed by Alan Turing. In this study, we develop a 3D bulk-surface model of the dynamics of Cdc42. We show that the model can undergo both classic and nonclassic Turing instability. We thoroughly investigate the parameter space for which this occurs. Using simulations we show that the model can be used to simulate polarisation and to predict a number of relevant quantitative measures, including pole size and time to polarisation.