Entropy methods and convergence to equilibrium for volume-surface reaction-diffusion systems

TL;DR

This paper applies entropy methods to two biologically motivated volume-surface reaction-diffusion systems, demonstrating exponential convergence to inhomogeneous equilibria with explicit rates, advancing understanding of spatially compartmentalized biochemical processes.

Contribution

It extends entropy method techniques to complex balanced volume-surface systems with inhomogeneous equilibria, providing explicit convergence rates.

Findings

Exponential convergence to equilibrium established for both systems.

Explicit estimates for convergence constants and rates derived.

Applicable to biologically relevant cell biology models.

Abstract

We consider two volume-surface reaction-diffusion systems arising from cell biology. The first system describes the localisation of the protein Lgl in the asymmetric division of Drosophila SOP stem cells, while the second system models the JAK2/STAT5 signalling pathway. Both model systems have in common that i) different species are located in different spatial compartments, ii) the involved chemical reaction kinetics between the species satisfies a complex balance condition and iii) that the associated complex balance equilibrium is spatially inhomogeneous. By using recent advances on the entropy method for complex balanced reaction-diffusion systems, we show for both systems exponential convergence to the equilibrium with constants and rates, which can be explicitly estimated.