Cell-biology effective interpretation of the Ising model describing skin color patterning - Muhamet Ibrahimi; Szabolcs Zakany; Michel C. Milinkovitch (Physical Review Research)
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 (relying on interaction data among colored cells), and the Lenz-Ising model (based on effective mesoscopic interactions among individual skin scales). However, a mechanistic connection between these very different descriptions remains elusive. Capitalizing both on the biological interpretability of the reaction-diffusion model and on the geometric interpretability of the Lenz-Ising model, we develop a computational approach to bridge this gap in a series of evolutionary divergent lizards. First, by combining mathematical and computational methods to perform a mapping between the two models, we show that a four-parameter extension of the classic Lenz-Ising model efficiently captures statistical aspects of color patterning in all species. Second, we show that the Lenz-Ising model parameters acquire an effective interpretation, in terms of cell interactions, by observing how these parameters change when tuning the reaction-diffusion parameters optimized for each species studied here. Taken together, our findings establish quantitative links between the strength and length scales of microscopic interactions among colored cells, and the scale-by-scale lizard skin color patterns visible to the naked eye.