Physical interactions enable energy-efficient Turing patterns

TL;DR

The paper develops a thermodynamically consistent framework for Turing-type pattern formation by coupling diffusive and reactive fluxes to chemical potentials in a ternary fluid with activator $A$, inhibitor $I$, and solvent $S$. It demonstrates that repulsive interactions, quantified by a positive $\chi$ in a Flory-Huggins free-energy, generate cross-diffusion that anti-correlates $A$ and $I$ and significantly reduces the energy required to sustain patterns, with an optimal driving $\Delta\mu$ near the pattern-threshold. Key contributions include showing that energy efficiency increases with $\chi$ along fixed pattern length, that anti-correlated profiles emerge from cross-diffusion, and that cross-diffusion is the dominant mechanism for reducing $\dot E$ in this thermodynamically consistent setting. The findings imply that physical interactions can be central to natural pattern formation and may inform design of energy-efficient, chemically active materials and biological patterning systems.

Abstract

Patterns are ubiquitous in nature, but how they form is often unclear. Turing developed a seminal theory to explain patterns based on reactions that counteract the equalizing tendency of diffusion. These reactions require continuous energy input since the system otherwise would proceed to equilibrium, but what systems are energy-efficient is currently unclear. To address this question, we introduce a thermodynamically-consistent model of a Turing system. We reveal that repulsive interactions between the stereotypical activator and inhibitor reduce energy requirements significantly. Interestingly, efficient patterns occur for weak activity, albeit at reduced amplitude. Our results suggest that physical interactions might be central in forming natural patterns.

Figures (4)

• Figure 1: Physical interactions reduce power requirements of Turing patterns. (A) Pattern length scale $\ell$ determined from the number of peaks in the stationary profile $\phi_A(x)$ in a finite system as a function of interaction parameter $\chi$ and reaction rate $k$. (B) Power $\dot{E}$ given by Eq. \ref{['Edot_eqn']} as a function of $\chi$ and $k$. (C) $\dot{E}$ as a function of $\chi$ along the iso-contour for $\ell=20 w$ (white lines in A and B). (A--C) Model parameters are $\Delta \mu = 5$, $h = 5$, $D_I/D_A = 10$, $\phi_0 = 0.2$, and $k_0 = D_A/w^2$. Simulations were performed on period grids of size $L = 2000\,w$.
• Figure 2: Interactions induce anti-correlation between activator and inhibitor. (A, B) Amplitude $\max(\phi_i) - \min(\phi_i)$ of activator profile $\phi_A$ (panel A) and inhibitor profile $\phi_I$ (panel B) for the data in Fig. \ref{['fig:Contoursdmu5']}. (C) Activator profile $\phi_A$ (blue) and inhibitor profile $\phi_I$ (orange) as a function of space $x$ for two interaction parameters $\chi$. (D) Covariance $\langle \phi_A \phi_I \rangle - \langle \phi_A \rangle \langle \phi_I \rangle$ for the data in Fig. \ref{['fig:Contoursdmu5']}.
• Figure 3: Cross-diffusion aids pattern formation. Spatial profiles of the three terms on the right hand side of Eq. \ref{['Eqn_CrossDiffExplicit']} for the activator (upper row) and inhibitor (lower row) as a function of space $x$ for two interaction parameters $\chi$ for the data in Fig. \ref{['fig:Contoursdmu5']}, $k= 0.01$. The equalizing self-diffusion (blue dashed lines) is often opposed by cross-diffusion (blue dotted lines) and the reactions (red lines).
• Figure 4: Weak drive leads to efficient patterns with small amplitudes. Activator amplitude ($\max(\phi_A) - \min(\phi_A)$, panel A), covariance between activator and inhibitor ($\langle \phi_A \phi_I \rangle - \langle \phi_A \rangle \langle \phi_I \rangle$, panel B), pattern length scale $\ell$ (panel C), and power $\dot E$ (Eq. \ref{['Edot_eqn']}, panel D) as functions of interaction $\chi$ and chemical driving force $\Delta\mu$. Simulation parameters are $k = 0.01\, k_0$, $h = 8$, $D_I/D_A = 15$, $\phi_0 = 0.2$, and $k_0 = D_A/w^2$. Simulations were performed on periodic grids of size $L = 2000\,w$.