Cell-biology effective interpretation of the Ising model describing skin color patterning

Abstract

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.