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We have previously demonstrated [1-6] that the skin color patterning of lizards with monochromatic skin scales can be modeled computationally with at least two different mathematical frameworks: Turing’s reaction-diffusion model (RD) —relying on interaction data among colored cells— and the Lenz-Ising model —based on effective mesoscopic interactions among individual skin scales. The RD model provides a biological interpretation in terms of the interactions among colored cells via diffusive signals (diffusive morphogens and/or random cellular movements), whereas the Lenz-Ising model is interpreted through effective mesoscopic/macroscopic interactions among individual skin scales. However, a mechanistic connection between these very different descriptions remained elusive. Here, we develop a computational approach to bridge this gap in a series of evolutionary divergent lizard species. After performing a mapping between the RD and a four-parameter extension of the classic Lenz-Ising model, we show that the latter acquires a cell-interaction interpretation by observing how its parameters change when tuning the RD parameters. Taken together, our findings establish for the first time quantitative links between, on one hand, the strength and length scales of microscopic interactions among colored cells and, on the other hand, the scale-by-scale lizard skin color patterns visible to the naked eye.
[1] Nature 544, 7649 : 173–179 (2017)
[2] Nature Communications 12 : 2433 (2021)
[3] Physical Review Letters 128, 48102 (2022)
[4] Current biology 32 : 5069-5082.e13 (2022)
[5] Annual Review of Cell and Developmental Biology 39 : 145-174 (2023)
[6] Physical Review X 13, 41011 (2023)
Fig. 1: The dominant interaction in our extended Lenz-Ising model differs from species to species: the first nearest-neighbor interaction in the Ocellated Lizard enables a labyrinthine pattern of single-scale thickness, whereas the second nearest-neighbor interaction in the Gila Monster enables a thicker labyrinthine pattern. In the Black and white Tegu, both first and second neighbor interactions combine to generate localized blotches.
Image credits: M. Ibrahimi, S. Zakany & M.C. Milinkovitch.
Much additional information is available in the original article:
Cell-biology effective interpretation of the Ising model describing skin color patterning
Ibrahimi M, Zakany S & M.C. Milinkovitch
Physical Review Research 7: 023093 (2025) — DOI: 10.1103/PhysRevResearch.7.023093
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