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

TL;DR

This paper models the cellular localization of Lgl protein during Drosophila stem cell division using a volume-surface reaction-diffusion system, proving well-posedness, deriving a quasi-steady-state approximation, and performing numerical simulations.

Contribution

It introduces a novel volume-surface reaction-diffusion model for Lgl localization, proves mathematical properties, and applies numerical methods to analyze system behavior.

Findings

Surface diffusion influences system behavior and steady states.

The quasi-steady-state approximation accurately captures rapid reactions.

Numerical simulations reveal how phosphorylation rates affect protein localization.

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.