Imperfect Turing Patterns: Diffusiophoretic Assembly of Hard Spheres via Reaction-Diffusion Instabilities

Abstract

Turing patterns are stationary, wave-like structures that emerge from the nonequilibrium assembly of reactive and diffusive components. While they are foundational in biophysics, their classical formulation relies on a single characteristic length scale that balances reaction and diffusion, making them overly simplistic for describing biological patterns, which often exhibit multi-scale structures, grain-like textures, and inherent imperfections. Here, we integrate diffusiophoretically-assisted assembly of finite-sized cells, driven by a background chemical gradient in a Turing pattern, while also incorporating intercellular interactions. This framework introduces key control parameters, such as the Péclet number, cell size distribution, and intercellular interactions, enabling us to reproduce strikingly similar structural features observed in natural patterns. We report imperfections, including spatial variations in pattern thickness, packing limits, and pattern breakups. Our model not only deepens our understanding but also opens a new line of inquiry into imperfect Turing patterns that deviate from the classical formulation in significant ways.

Figures (5)

• Figure 1: Natural patterns in fish and proposed framework for replicating these patterns. (a) Skin patterns of a male ornate boxfish, showcasing distinct hexagonal and striped structures. Magnified views of the hexagonal and striped patterns on the fish skin are shown on the right. These patterns are characterized by two length scales: the pattern size ($\lambda_C$) and the pattern thickness ($\lambda_N$). The patterns exhibit deviations from perfect geometries, featuring imperfections and defects alongside spatial variations in thickness. Additionally, these patterns possess "grain-like" features. (b,c) Schematic of the proposed simulation framework: Diffusiophoretic assembly of cells in response to the reaction-diffusion of biochemical molecules in the background. (b) The simulation starts by a random distribution of two or more types of cells, with different mobilities and sizes. (c) The solute concentrations are determined through continuum reaction-diffusion descriptions (equation (1)), on top of which the cells assemble via diffusiophoresis and form Turing patterns. The interaction between cells is included via a hard-sphere potential. (d) Simulated hexagon and stripe patterns obtained by diffusiophoretic assembly of two types of cells on top of chemical patterns generated by using the Brusselator model prigogine1967pena2001. The simulations effectively reproduce the natural skin patterns, including the grain-like feature and imperfections. Details of the simulation method and parameters are provided in the Methods section. Photo courtesy of the Birch Aquarium at the Scripps Institution of Oceanography.
• Figure 2: Comparison between the particle-level and continuum models in simulating cell assembly. Steady-state cell concentration fields computed from (a) the continuum model and (b) the particle-level simulation for different Péclet numbers $Pe$.(c) Comparison between the two models in predicting azimuthally averaged cell concentration, $\overline{\mathcal{N}}$, as a function of radial distance $r$ from the center of a hexagon. Significant differences emerge between the continuum and particle-level models in the high Péclet number regime ($Pe \gg 1$) where the former predicts a collapse of all particles to the center of the hexagons while the latter yields finite-size clusters. Shaded regions represent the standard deviation of the numerical data points. See the Methods section for details on the derivation of $\overline{\mathcal{N}}$, corresponding error estimates, and the parameters used in the simulations.
• Figure 3: Master curve showing dependence of cell pattern thicknesses $\lambda_N$ on the Péclet number $Pe$ in the limit $d/\lambda_C \ll 1$. Comparison between continuum and particle-level simulation results over a wide range of $Pe$, for both stripe and hexagonal patterns. The prediction from the continuum model is independent of $\phi$, while the particle-level simulations show a notable dependence on $\phi$. To confirm that the results are within the limit of small cell sizes compared to pattern thickness, we perform simulations with smaller cells and observe quantitatively similar results. Error bars represent the standard deviation of the local pattern thickness. Details of the simulation method and parameters are provided in the Methods section.
• Figure 4: Impact of cell size on emerging stripe and hexagonal patterns: Cell assemblies in $\phi = 15\%$ polydisperse systems are examined for different Damköhler numbers $Da_c$. (a-e) Hexagon and (f-j) stripe pattern type using the same cell size distribution for all simulations with cell sizes (diameters) randomly assigned within the range $[0.5d, 1.4d]$ where $d$ is the characteristic particle size of the simulation. Note that the pattern size $\lambda_c$ decreases with increasing Damköhler numbers $Da_c$, following the scaling relation$\lambda_C \sim Da_c^{-1/2}$, thereby reducing the ratio between pattern features and cell size. Imperfections emerge at higher $Da_c$ as the relative particle size approaches the characteristic thickness of the pattern, eventually preventing the formation of a complete pattern when particles become too large. The same parameters detailed in the Methods are used for solute species, except that the Damköhler number ($Da_c$) is varied.
• Figure 5: Effect of cell size on pattern edge thickness for hexagonal patterns. Comparison between the pattern thickness $\lambda_N$ computed by particle-level simulations and the asymptotic prediction $\lambda_{N,\text{pack}}$ (equation (4)) in the high Péclet number limit ($Pe = 100$).As the asymptotic prediction $\lambda_{N, pack}$ approaches the particle size $d$, the $\lambda_N$ computed from the particle-level simulations exhibits large variance, signaling notable imperfections and incomplete patterns. We set the same model parameters detailed in Methods to simulate the solute concentrations for the hexagonal pattern shown in Fig. \ref{['fig:compareC']}.