Intermediate physical interactions induce spatiotemporal dynamics in Turing patterns

TL;DR

This work shows that physical A–I interactions coupled to nonlinear reactions qualitatively alter pattern formation beyond classical Turing behavior. By analyzing a two-component RD system with a Flory–Huggins free energy, the authors identify three regimes—stationary Turing patterns, chemically active droplets, and a dynamical DP regime with growth–fission cycles—controlled by the interaction strength χ and reaction nonlinearity h. They derive distinct length-scale scalings, ℓ_{TP} ∼ k^{-1/2} and ℓ_{AD} ∼ k^{-1/3}, with DP exhibiting intermediate exponents, illustrating a continuum between diffusion-dominated and phase-separation-dominated dynamics. The findings highlight how cross-diffusion and phase separation mediated by reactions can generate persistent spatiotemporal dynamics with potential implications for soft matter and biological patterning.

Abstract

Turing patterns are a central paradigm for describing spatial patterns in nature. The corresponding theory of reaction-diffusion dynamics combines ideal diffusion with nonlinear reactions, resulting in patterns when species diffuse at different rates and reactions are sufficiently nonlinear. However, real systems are more complex and particularly involve physical interactions between constituents. While such interactions can promote patterns, we here show that they can also induce dynamic, chaotic patterns. These patterns exhibit well-defined length and time scales, which result from cycles of droplet coarsening and fission. The dynamical patterns combine properties of traditional Turing patterns and chemically active droplets, which emerge for strong physical interactions. Our analysis thus reveals three qualitatively different regimes that emerge when two components interact physically and undergo nonlinear reactions.

Figures (8)

• Figure 1: A dynamical regime (DP) separates stationary Turing patterns (TP) from active droplets (AD). (A--C) Activator volume fraction $\phi_A$ as a function of the one spatial coordinate $x$ and time $t$ for the Turing pattern regime (TP, $\chi=0$), the dynamical pattern regime (DP, $\chi=6$), and the active droplet regime (AD, $\chi=8$). (E--G) Activator fraction $\phi_A$ (blue) and inhibitor fraction $\phi_I$ (orange) as a function of $x$ at $t=10^7 \, k_0^{-1}$ corresponding to panels A--C. (D, H) State diagrams indicating the three regimes as a function of interaction strength $\chi$ and reaction nonlinearity $h$ (panel D) and diffusivity ratio $D_I/D_A$ (panel H). Homogenous states are stable in the white region. (A--H) Model parameters are $h = 5$, $D_I/D_A = 10$, $k/k_0 = 10^{-3}$, $k_0 = D_A/w^{2}$, $\phi_0 = 0.2$, and system size $L=2000\,w$; Numerical details are described in Appendix \ref{['app:numerics']}.
• Figure 2: Dynamical regime exhibits cycles of growth and fission. Snapshots of the activator fraction $\phi_A$ of a two-dimensional system for the indicated times $t$. Model parameters are $\chi = 6$, $h = 5$, $D_I/D_A = 3$, $k = 0.01\,k_0$, $k_0 = D_A/w^2$, and $\phi_0= 0.2$.
• Figure 3: Growth of a single droplet reveals fission mechanism. (A--D) Volume fractions of activator ($\phi_A$, blue lines) and inhibitor ($\phi_I$, orange lines) at four indicated times $t$. Red dashed lines mark the instability boundary that is crossed in panel C. (E) Phase diagram of system without reactions ($k=0$) as a function of $\phi_A$ and $\phi_I$. Homogeneous states are unstable in the red region (Appendix \ref{['app:stability_analysis']}). Blue line shows trajectory of $\phi_A(t)$ and $\phi_I(t)$ at the droplet center at $x = 100\,w$ with points marking states shown in panels A--D. (A--E) Model parameters are $\chi = 4.5$, $k = 0.01\,k_0$, $D_I/D_A = 10$, $h = 5$, $\phi_0 = 0.2$, and $k_0 = D_A/w^2$.
• Figure 4: Interactions and reaction rates control pattern length scale. (A) Pattern length scale $\ell$ (measured from time-averaged structure factor; see Appendix \ref{['app:length_scale']}) as a function of interaction strength $\chi$ and reaction rate $k$. White dotted line corresponds to panel B. (B) $\ell$ as a function of $\chi$ for $k = 2 \cdot 10^{-3}\, k_0$. (C) Scaling exponent $\alpha$ extracted by fitting $\ell \sim k^{-\alpha}$ on a log-log scale to data in panel A as a function of $\chi$ (inset shows data for $\chi=5.5$). (A--C) Model parameters are $D_I/D_A = 10$, $h = 6$, $\phi_0 = 0.2$, $k_0 = D_A/w^2$, and $tk_0 = 10^7$.
• Figure S1: Length scale determination in dynamic pattern regime requires time averaging. (A) Activator volume fraction $\phi_A$ as a function of the single spatial coordinate $x$ and time $t$. (B) Pattern length scale $\ell$ (defined as mean structure factor) of data in panel A as a function of $t$. Model parameters are $k = 10^{-3}\,k_0$, $\chi = 5.5$, $D_I/D_A = 10$, $h = 5$, $\phi_0 = 0.2$, and $k_0 = D_A/w^2$.