Delayed Pattern Formation in Two-Dimensional Domains

TL;DR

This paper explores how gene expression delay and domain size influence pattern formation in reaction-diffusion systems, revealing a linear delay-pattern relationship and a complex size-pattern emergence dynamic in two dimensions.

Contribution

It extends the ligand internalisation model to two dimensions and uncovers the non-monotonic relationship between domain size and pattern formation time.

Findings

Linear relationship between gene delay and pattern formation time in 1D and 2D.

Non-monotonic correlation between domain size and pattern emergence.

Computed a two-parameter chart of pattern dynamics.

Abstract

This study investigates how the interaction between gene expression time delay and domain size governs spatio-temporal pattern formation in a reaction-diffusion system. To investigate these phenomena, we utilize a modified version of the Schnakenberg model called the ligand internalisation (LI) model. In a one-dimensional domain, a linear relationship has been observed between the gene expression time delay and the time it takes for patterns to form. We extend the model to the two-dimensional domain and confirm that a similar relationship holds there as well. However, our exploration reveals a non-monotonic correlation between domain size and the time required for pattern emergence. To unravel these dynamics, we consider a range of initial conditions, including random perturbations of the spatially homogeneous steady state and initial conditions from its unstable manifold. We compute a two-parameter chart of patterns with respect to time delay and domain size.