Communications in Mathematical Sciences

Volume 14 (2016)

Number 6

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

Pages: 1553 – 1580

DOI: https://dx.doi.org/10.4310/CMS.2016.v14.n6.a5

Authors

Klemens Fellner (Institute of Mathematics and Scientific Computing, NAWI Graz, University of Graz, Austria)

Stefan Rosenberger (Institute of Mathematics and Scientific Computing, NAWI Graz, University of Graz, Austria)

Bao Quoc Tang (Institute of Mathematics and Scientific Computing, NAWI Graz, University of Graz, Austria; and Faculty of Applied Mathematics and Informatics, Hanoi University of Science and Technology, Hanoi, Vietnam)

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.

Keywords

reaction-diffusion equations, global existence, surface diffusion, quasi-steady-state approximation, asymmetric stem cell division, finite element method

2010 Mathematics Subject Classification

35B40, 35K57, 92C45

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Published 12 August 2016