Emergence of hyperuniformity from reaction-diffusion interactions in Turing patterns

Self-organized Turing patterns, arising from the dynamic interplay of reaction and diffusion processes, are instrumental in modeling natural morphogenesis and deciphering the biological functions of diverse structures. These patterns exhibit complex spatial arrangements that can be analyzed through the lens of statistical mechanics, particularly in relation to hyperuniformity-a state characterized by suppressed long-range density fluctuations. Despite this, the connection between Turing patterns and hyperuniformity remains relatively unexplored. In this study, we examine the spatial distribution of generic Gray-Scott patterns and Arabidopsis thaliana trichome patterns, linking their morphogenesis to disordered hyperuniform patterns. Our findings demonstrate that hyperuniformity emerges in Turing patterns as a solution to a reaction-diffusion equation system, reaching its apex in a specific region of the Gray-Scott phase space marked by dotlike spatial patterns. Mapping the hyperuniformity classes in this region reveals patterns with class III hyperuniformity surrounding special class I solutions. By extending this framework to biological systems currently under investigation for mechanosignalling triggrered immunity, we show that A. thaliana trichomes form class III disordered hyperuniform systems. Such exotic organization, as described by Turing patterning, has a strong potential to influence mechanically triggered immune response, offering novel insights into plant mechanoperception.