Quasi-steady-state approximation and numerical simulation for a volume-surface reaction-diffusion system

Abstract

Abstract. The asymmetric stem-cell division of Drosophila SOP precursor cells is driven by the asymmetric localisation of the key protein Lgl (Lethal giant larvae) during mitosis, when Lgl is phosphorylated by the kinase aPKC on a subpart of the cortex and subsequently released into the cytoplasm.In this paper, we present a volume-surface reaction-diffusion system, which models the localisation of Lgl within the cell cytoplasm and on the cell cortex. We prove well-posedness of global solutions as well as regularity of the solutions. Moreover, we rigorously perform the fast reaction limit to a reduced quasi-steady-state approximation system, when phosphorylated Lgl is instantaneously expelled from the cortex. Finally, we apply a suitable first order finite element scheme to simulate and discuss interesting numerical examples, which illustrate i) the influence of the presence/absence of surface-diffusion to the behaviour of the system and the complex balance steady state and ii) the dependency on the release rate of phosphorylated cortical Lgl.

Entropy Methods and Convergence to Equilibrium for Volume-Surface Reaction-Diffusion Systems

“…Recently, a rapidly increasing amount of attention has been devoted to the mathematical theory of VSRD systems arising from such different applications as fluid mechanics [16,21], ecology [1,2,3], crystal growth [20] or, especially, cell biology, see e.g. [9,11,13,17].…”

“…This paper aims to investigate the large time behaviour of two particular linear VSRD systems arising from two different application backgrounds in cell biology: the first being a model for the localisation of the key-protein Lgl during the asymmetric stem cell division of SOP precursor cells in Drosophila (see e.g. [11]), the second one being a model on the so-called JAK2/STAT5 signalling pathway, see [13]. In both examples, the cell cytoplasm constitutes the volume domain and the surrounding cell-membrane/cortex constitutes the surrounding boundary.…”

“…Concerning the existence of global weak solutions of (1.1), we refer to [11], where also the quasi-steady-state approximation in the limit ξ → +∞ and numerical simulations were carried out.…”

Error estimates for the finite element approximation of bilinear boundary control problems

“…with certain constants γ, τ > 0. The author is aware that there are weaker assumptions which are sufficient for local minima, for instance one could formulate (15) for all directions v from a critical cone. However, with this assumption the convergence proof for the postprocessing approach presented in Section 5.3 requires some more careful investigations, in particular the construction of a modified interpolant onto U ad .…”

“…As an application of bilinear boundary control problems we mentioned the identification of an unknown Robin coefficient from a given measurement y d of the state quantity. This is for instance of interest in the modeling of stem cell division processes [15], where u is the unknown parameter describing the chemical reactions between proteins from the cell interior and the cell cortex. For further applications, u can be interpreted as a heat-exchange coefficient in thermodynamics or as a quantity for corrosion damage in electrostatics.…”