An Asymptotic Analysis of Intracellular Signaling Gradients Arising from Multiple Small Compartments

Abstract. Intracellular signaling gradients naturally arise through a local activation of a diffusible signaling molecule, e.g. by a localized kinase, and a subsequent deactivation at a distant cellular site, e.g. by a cytosolic phosphatase. Here, we consider a spherical cell containing a finite number of small spherical compartments where a signaling molecule becomes activated by a localized enzyme. For the activation rate two cases are considered: a saturated enzyme with a constant rate and an unsaturated enzyme with a linear rate. Using the method of matched asymptotic expansions we derive approximate solutions of the steady-state diffusion equation with a linear deactivation rate to obtain the three dimensional concentration profile of activated signaling molecules inside the cell. Depending on the diffusion length of the signaling molecule the profiles either decay exponentially or algebraically, where the mode of decay is described by an associated Green’s function. Our analysis provides simple expressions for the local concentration profile in the neighborhood of a signaling compartment, which can be used to estimate the amplitude and the extent of the gradients in the respective regimes. The global concentration profiles also depend on the cell size and the particular spatial arrangement of the compartments relative to each other and to the cell boundary.