Quantifying the intricacies of biological pattern formation: A new perspective through complexity measures

In this study we examine the emergence of complex biological patterns through the lens of reaction-diffusion systems. We introduce two novel complexity metrics — Diversity of Number of States (DNOS) and Diversity of Pattern Complexity (DPC)— which aim to quantify structural intricacies in pattern formation, enhancing traditional linear stability analysis methods. We demonstrate this approach to different systems including the linear Turing, Gray-Scott and FitzHugh-Nagumo models. These measures reveal insights into nonlinear dynamics, multistability, and the conditions under which complex biological patterns stabilize. We then apply the approach to gene regulatory networks, including models of the toggle switch in developmental biology, demonstrating how diffusion and self-activation contribute to robust spatial patterning. Additionally, simulations of the Notch-Delta-EGF signaling pathway in Drosophila neurogenesis highlight the role of gene regulation and parameter variations in modulating pattern complexity and state diversity. Overall, this work establishes complexity-based approaches as valuable tools for exploring the conditions that drive diverse and stable biological pattern formation, offering a pathway for future applications in synthetic biology and tissue engineering.