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Abstract:
We perform a mathematical analysis of a spatially extended model describing mutual phosphorylation of cytosolic kinases and membrane receptors. The analyzed regulatory system is a part of signal transduction mechanisms, which enables communication of the cell with its extracellular environment or other cells. The mutual receptor-kinase interaction is characteristic for immune receptors and Src family kinases. From the mathematical viewpoint, the considered system is interesting because it couples differential equations defined in a domain $\Omega$ and on its boundary $\partial\Omega$ via nonlinear Robin boundary conditions. Assuming a spherically symmetric framework, our approach is to consider an auxiliary problem in which the Robin boundary condition on the external boundary of the spherical shell $\Omega$ is replaced by a uniform Dirichlet boundary condition. This method allows us to find the stationary spherically symmetric solutions, both stable and unstable. Interestingly, numerical computations suggest also the existence of nonspherically symmetric unstable stationary solutions to the spherically symmetric problem. These conjectured solutions appear to lie between super- and subsolutions that converge in time to two different stable spherically symmetric steady states. The study is completed by the proof of an existence theorem for a general version of the original system encompassing the earlier models. The theorem holds in domains of arbitrary shape, with a number of holes that may represent various organelles impenetrable to the considered kinases. The last result ensures that the problem in which the flux of active kinases (which replaces the source term) is determined by the Robin-type boundary condition at the cell membrane is well posed. The considered generalized model may thus serve as a template for intracellular signal transduction analysis.

Keywords:
reaction-diffusion systems, Robin-type boundary conditions, bistability, molecular signal transduction