Delayed Pattern Formation in Two-Dimensional Domains

TL;DR

This work extends the LI variant of the Schnakenberg reaction-diffusion system to a two-dimensional domain with a gene-expression time delay $\tau$, examining how delay and domain size $L_x,L_y$ govern Turing pattern formation. Through linear stability analysis, the authors derive a delay-augmented characteristic equation $D_{k_xk_y}(\lambda,\tau)=0$ and define $\alpha$ as the maximal real part of the eigenvalues, mapping out the 2D Turing space and its dependence on domain geometry. They complement the analysis with extensive simulations on a finite grid, showing a robust linear relationship between the time-to-pattern $\mathcal{T}$ and $\tau$, with the slope modulated by $L_x$, and a non-monotonic dependence of $\mathcal{T}$ on domain size. The results demonstrate that delays slow pattern formation and that domain size can qualitatively alter the emergent patterns (stripes vs circles vs honeycomb) and the timing of their onset, with the initial condition playing a significant role. Overall, the study highlights the critical impact of gene-expression delays and spatial confinement on the kinetics and morphology of pattern formation in reaction-diffusion systems, providing quantitative links between delay, domain geometry, and pattern emergence.

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.

Figures (10)

• Figure 1: Heatmap of the maximal real part of the eigenvalues, $\alpha$, as defined in \ref{['eq:alpha1']}, computed over $a$ and $b$ parameter space by solving the characteristic equation (\ref{['eq:ch1']}) without delay ($\tau = 0$). The stability curves for the spatially homogeneous (solid black curve) and spatially inhomogeneous (black dotted curve) conditions are shown. The area between these two curves signifies the Turing region. Here $d_{u} = 0.01$, $d_{v} = 0.2$ and $L_{y} = 0.2$. For (a)--(d), $L_{x} = 0.2, 0.38, 0.5, 1$ respectively. As $L_{x}$ increases, the Turing space expands.
• Figure 2: (a)--(d) $\alpha_{k_x,0}$ as defined in \ref{['eq:alpha1']} is plotted as a function of $L_{x}$ for different modes for equation \ref{['eq:ch1']} for a fixed delay. In (a), $L_{y} = 0.2$ and (b), $L_{y} = 3$; $\tau = 0$ in both cases. In (c), $L_{y} = 0.2$ and (d) $L_{y} = 3$; $\tau = 1$ in both cases. $a = 0.1, b = 0.9$, $d_{u} = 0.01$, and $d_{v} = 0.2$ through (a--d). A significant variability in the dominant mode is noticed for smaller domain sizes. As we increase $\tau$ to 1, the eigenvalues tend toward zero, leading to a higher density of lines.
• Figure 3: Plot of the maximal real part of the eigenvalues against time delay. In (a)--(d), $\alpha$ as defined in \ref{['eq:alpha1']} is evaluated for fixed delay for \ref{['eq:ch1']} as $\tau$ varies from $0$ to $2$. In the case (a), $(a, b) = (0.1, 0.9)$, pattern formation can be predicted (since $\alpha > 0$). Conversely, in case (b), $(a, b) = (0.4, 0.4)$, no pattern formation can be anticipated (since $\alpha < 0$). In both (a) and (b), $L_{y} = 0.2$, $L_{x} = 3$. In (c), $(a, b) = (0.17, 0.5)$, $L_{y} = 0.2$, $L_{x} = 0.38$. In (d), $(a, b) = (0.18, 0.4)$, $L_{y} = 0.2$, $L_{x} = 0.5$. In both (c) and (d), equilibrium becomes stable after a certain $\tau$. In (d), a mode switch is also apparent at $\tau \approx 1.33$. $d_{u} = 0.01$, and $d_{v} = 0.2$ is fixed through (a)--(d).
• Figure 4: (a)--(d) $\alpha_{k_x,0}$ as defined in \ref{['eq:alpha1']} is plotted as a function of $L_{x}$ for different modes for equation \ref{['eq:ch1']} for a fixed delay. In (a), $L_{y} = 0.2$ and (b), $L_{y} = 3$; $\tau = 0$ in both cases. In (c), $L_{y} = 0.2$ and (d) $L_{y} = 3$; $\tau = 1$ in both cases. $a = 0.1, b = 0.9$, $d_{u} = 0.01$, and $d_{v} = 0.2$ through (a--d). A significant variability in the dominant mode is noticed for smaller domain sizes. As we increase $\tau$ to 1, the eigenvalues tend toward zero, leading to a higher density of lines.
• Figure 5: (a) Predicted time to pattern with respect to delay computed from the characteristic equation \ref{['eq:ch1']} showing a linear trend for different $L_{x}$ = $0.5$ (black, dashed), $1$ (red, dashed), $1.2$ (green, dashed), $1.5$ (blue, dashed). (b) The change in the slope with $L_{x}$ for the curves in (a). (c) Simulated Time to pattern for the initial condition in \ref{['initial_2']} for $L_{x} = 0.5$ (black, dashed), $1$ (red, dashed), $1.2$ (green, dashed), and $1.5$ (blue, dashed). (d) Time to pattern simulated considering random initial conditions \ref{['initial_1']} for $L_{x} = 0.5$ (black dots), $1$ (red dots), $1.2$ (green dots) and $1.5$ (blue dots). In all the cases, $a=0.1$, $b =0.9$, $d_{u} = 0.01$, $d_{v} = 0.2$, $L_{y}= 0.2$.