Pattern formation in a one-dimensional MARCKS protein cyclic model with spatially inhomogeneous diffusion coefficients

We analytically investigate the conditions for the wave instability in a reaction-diffusion system describing the nonlinear dynamics of the myristoylated alanine-rich C kinase substrate (MARCKS) between cytosol and cytoplasmic membrane. Taking into account the effect of spatial inhomogeneous diffusion coefficients, and by applying the discrete multiple scale expansion method, we show that the nonlinear generic model can be transformed into a one-dimensional discrete nonlinear Schrödinger equation. We perform a linear stability analysis on the plane wave solutions to derive the criterion of the modulational instability (MI) phenomenon. This analysis reveals that the critical amplitude of the plane wave is highly influenced by the phosphorylation rate and weakly influenced by the inhomogeneous diffusion coefficients. The exact analytical solutions show that the system exhibits traveling waves and periodic array of patterns. The results seem to indicate the features of synchronization in the collective dynamics. In homogenous state, we obtained a spatial pattern of horizontal stripes. By considering the spatial inhomogeneity effect, we obtain a spatial pattern of oblique stripes. We also notice that an increase in wavenumber induces the increase in the number of stripes in the model.