Symmetry breaking in a bulk–surface reaction–diffusion model for signalling networks

Abstract

Signaling molecules play an important role for many cellular functions. We investigate here a general system of two membrane reaction-diffusion equations coupled to a diffusion equation inside the cell by a Robin-type boundary condition and a flux term in the membrane equations. A specific model of this form was recently proposed by the authors for the GTPase cycle in cells. We investigate here a putative role of diffusive instabilities in cell polarization. By a linearized stability analysis we identify two different mechanisms. The first resembles a classical Turing instability for the membrane subsystem and requires (unrealistically) large differences in the lateral diffusion of activator and substrate. The second possibility on the other hand is induced by the difference in cytosolic and lateral diffusion and appears much more realistic. We complement our theoretical analysis by numerical simulations that confirm the new stability mechanism and allow to investigate the evolution beyond the regime where the linearization applies.

Diffuse-Interface Approximations of Osmosis Free Boundary Problems

“…The "diffuse domain method" has originally been used for the numerical treatment of PDE's and corresponding boundary conditions in domains with possibly complicated boundaries and for the simulation of coupled bulk-surface PDE's [9,10,11,24], see also [27,16] for the evolving surface case. In this contribution it is used for a new diffuseinterface approximation of a Stefan-like free boundary problem for osmotic swelling, which has not been numerically investigated yet.…”

“…In such a diffuse-interface approximation the interface is implicitly given by a phase-field function, which is obtained, roughly speaking, by smearing out the indicator function of Ω + on a short length O(ε) for some small parameter ε > 0. The diffuse domain method [9,10,11] offers a powerful tool for the approximation of free boundary problems [23,25] and coupled bulk-surface PDE-systems [27,24]. It will be used here, in order to numerically simulate the one-and two-phase models previously introduced.…”

Analysis and Simulations of Coupled Bulk-surface Reaction-Diffusion Systems on Exponentially Evolving Volumes

“…For the case of circular and spherical domains, if r = 1, then k 2 l,m = l(l + 1). Taking x ∈ B, y ∈ Γ , then writing in polar coordinates x = ry, r ∈ (0, 1) we can define, for all l ∈ N 0 , m ∈ Z, |m| ≤ l, the following power series solutions [35,36] …”

“…The coupling of bulk and surface dynamics through the use of partial differential equations (PDEs) in multi-dimensions is prevalent in many cellular biological systems and fluid dynamics [3,5,8,15,28,29,32,33,35,36] as well as in other exotic areas of solid mechanics such as topological insulator thin films [9] and in large-and small-scale atmospheric and coupled ocean-atmosphere models for air-sea interactions [4]. In the areas of cellular and developmental biology, in many of these applications and processes, the formation of heterogeneous distributions of chemical substances emerge through symmetry breaking of morphological instabilities [17].…”

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

“…In Section 3, we review the results of the analysis of the linear stability of spatially homogeneous stationary states for the above mentioned reduced non-local model [4,7] (see also [8]), and we find two scenarios for an instability [7]. While the first requires different diffusion constants for the two membrane bound species -which is common to classical Turing instabilities, but not a reasonable assumption in this application -the second is also possible for equal lateral diffusion coefficients due to the non-local nature of the model.…”