Turing instabilities in a mathematical model for signaling networks

Abstract

GTPase molecules are important regulators in cells that continuously run through an activation/deactivation and membrane-attachment/membrane-detachment cycle. Activated GTPase is able to localize in parts of the membranes and to induce cell polarity. As feedback loops contribute to the GTPase cycle and as the coupling between membrane-bound and cytoplasmic processes introduces different diffusion coefficients a Turing mechanism is a natural candidate for this symmetry breaking. We formulate a mathematical model that couples a reaction-diffusion system in the inner volume to a reaction-diffusion system on the membrane via a flux condition and an attachment/detachment law at the membrane. We present a reduction to a simpler non-local reaction-diffusion model and perform a stability analysis and numerical simulations for this reduction. Our model in principle does support Turing instabilities but only if the lateral diffusion of inactivated GTPase is much faster than the diffusion of activated GTPase.

Exhibiting cross-diffusion-induced patterns for reaction-diffusion systems on evolving domains and surfaces

“…Our final example is again inspired by biological observations which support that in many processes where surface growth is involved, growth is driven partly by chemical species resident on the cell-membrane surface An example is that of cell polarization in cell biology due to responses to external signals through the outer cell membrane [51,52]. To model such concentration-driven surface evolution, we assume that a spherical surface is evolving according to the following evolution law:…”

A Highly Anisotropic Nonlinear Elasticity Model for Vesicles I. Eulerian Formulation, Rigidity Estimates and Vanishing Energy Limit

“…One can find solid phases, gel phases and liquid phases. Let us mention the existence of models for 2D membranes with phase coexistence [14,5] and of a model for 2D membranes in gel phase which includes the local orientation of the lipids [1]. In contrast, in the present articles we consider thick membranes in liquid phase.…”

Turing-type instabilities in bulk–surface reaction–diffusion systems

“…The mathematical modelling of these processes leads us to a system of reaction-diffusion equations for the concentrations of membrane bound active and inactive GTPase molecules and of cytosolic inactive GTPase molecules [4]. This reaction-diffusion system is a bulk-surface system of PDE's due to the different dimensionalities of the involved quantities and it is related to a model in [5] with more chemical species under consideration and scaling factors accounting for the different dimensions of membrane and cytosolic cell volume.…”