THE aim of the work is the analysis of the stability properties of stationary front solutions to a Mc Kean caricature of  a spatially extended model of mutual interaction between cytosolic kinase molecules and membrane receptors.  The starting point of our considerations is the model analyzed in [1], which in turn is a generalization of the models introduced in [2],[3] and [4]. It is assumed in these models that non-active kinase molecules are activated by active receptors on the boundary and diffuse freely in the cytosol. The stream of activated kinase molecules resulting from their phosphorylation by the surface receptors can be modelled by nonlinear Robin-type boundary conditions in which the diffusional flux of activated kinase, according to the law of mass action, is proportional to the product of the concentration of active receptors and the concentration of non-active kinases. The activated receptors as well as the activated kinases may be dephosphorylated by the action of phosphatases, in general of different kinds. Here we are interested in the situation when all the kinases live on the membrane of the cell. In fact, this corresponds to the assumption that  the cytosolic kinases are only read-out transducers of the membrane activation phenomena.

We find an explicit expression for  standing front solutions  to the equations of the simplified model on the sphere. In fact, we consider the equation describing the diffusion of kinases on the membrane (by the Laplace-Beltrami operator) and the kinase-receptor  interaction together with their  dephosphorylation by the phosphatases (by means of a nonlinear bistable piecewise continuous function).  We prove that the standing front solutions can be expressed in terms of hypergeometric functions. Examples of such solutions for various parameters of the model, as a function of the polar angle, are shown  below.

We proved, among others, that the standing front solution u₀(x) is unstable i.e., there exist initial data arbitrarily close to u₀ such that the solution to the time dependent problem tends to one of the stable states approximately as a traveling wave.

[1] Crooks E, Kazmierczak B, Lipniacki T (2013) A spatially extended model of kinase-receptor interaction, SIAM J Appl Math, 73, 374–400

[2] Kazmierczak B, Lipniacki T (2009), Regulation of kinase activity by diffusion and feedback.  J Theor  Biol 259,  291-296.

[3] Kazmierczak B, Lipniacki T (2010) Spatial gradients in kinase cascade regulation,  IET Sys  Biol 4,  348-355

[4] Hat B., Kazmierczak B, Lipniacki T (2011)  B cell activation triggered by the formation of the small  receptor cluster: a computational study.  PLoS Comp Biol 5, e1000448