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Annual Review of Cell and Developmental Biology

Volume 39, 2023

The Unreasonable Effectiveness of Reaction Diffusion in Vertebrate Skin Color Patterning

Abstract

In 1952, Alan Turing published the reaction-diffusion (RD) mathematical framework, laying the foundations of morphogenesis as a self-organized process emerging from physicochemical first principles. Regrettably, this approach has been widely doubted in the field of developmental biology. First, we summarize Turing's line of thoughts to alleviate the misconception that RD is an artificial mathematical construct. Second, we discuss why phenomenological RD models are particularly effective for understanding skin color patterning at the meso/macroscopic scales, without the need to parameterize the profusion of variables at lower scales. More specifically, we discuss how RD models () recapitulate the diversity of actual skin patterns, () capture the underlying dynamics of cellular interactions, () interact with tissue size and shape, () can lead to ordered sequential patterning, () generate cellular automaton dynamics in lizards and snakes, () predict actual patterns beyond their statistical features, and () are robust to model variations. Third, we discuss the utility of linear stability analysis and perform numerical simulations to demonstrate how deterministic RD emerges from the underlying chaotic microscopic agents.

Keyword(s): cellular automata, phenomenological models, reaction diffusion, sequential patterning, skin color patterns, Turing patterns
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2023-10-16
2025-06-15
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The Unreasonable Effectiveness of Reaction Diffusion in Vertebrate Skin Color Patterning
Annual Review of Cell and Developmental Biology 39, 145 (2023); https://doi.org/10.1146/annurev-cellbio-120319-024414
/content/journals/10.1146/annurev-cellbio-120319-024414
/content/journals/10.1146/annurev-cellbio-120319-024414
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Supplementary Data

  • Download the Supplemental Appendices (PDF). Includes the Supplemental Figures.

    Supplemental Video 1: Initial perturbation dependence in sequential patterning (see Figure 2b). Steady-state striped patterns on a zebra 3D model are generated with different initial conditions: (a) random perturbations across the whole domain, (b) a horizontal line of perturbation on each flank, and (c) a vertical line of perturbation around the flanks and back. The simulation in panel a is slowed down 4x relative to the two simulations on the right.

    Supplemental Video 2: Topological defects emerge from collisions between wave fronts (see Figure 2c).

    Supplemental Video 3: Sequential patterning with sustained oscillations (see Figures 3b and 3d).

    Supplemental Video 4: Cellular-automaton dynamics emerge from RD in 2D when diffusion coefficients are reduced at the 1D borders of skin scales (see Figure 5a).

    Supplemental Video 5: Scale-by-scale patterns emerge from RD in a 3D geometry with super-Gaussian bumps (right panels in Figure 5b).

    Supplemental Video 6: Different wave-like dynamics in RD systems. (a) Space-filling traveling waves (see Supplemental Figure 2a). (b) Solitary traveling waves (see Supplemental Figure 2b).

    Supplemental Video 7: Colliding hard spheres in a box (see Supplemental Figures 14a, 14b, and 14c).

    Supplemental Video 8: Colliding hard spheres in a box with extreme initial condition: all particles have the same initial momentum (same norm and same direction) (see Supplemental Figures 14a, 14b, and 14c).

    Supplemental Video 9: Diffusion behaviour of a subset of colliding hard spheres (see Supplemental Figure 13d).

    Supplemental Video 10: Stochastic model for Gray-Scott reaction-diffusion (see Supplemental Figures 14c and 14d).

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