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

Abstract

Nonlinear pattern-forming systems can generate spatial structure through many mechanisms, but comparing the patterns that are actually realized after the dynamics unfold remains difficult. Linear stability analysis identifies when spatial modes can grow and which length scales are favored near onset, but it does not directly measure the state diversity, structural organization, or dynamical route of the resulting patterns. Here we present a comparative operational framework, mechanism-agnostic under a fixed measurement protocol, based on two coordinates: the Diversity of Number of States (DNOS), an occupied-state measure of effective alphabet breadth, and the Diversity of Pattern Complexity (DPC), a lossless-compression measure of structural complexity. We apply the framework to a linear reaction--diffusion benchmark, the Gray--Scott and FitzHugh--Nagumo models, toggle-switch gene-regulatory systems with and without self-activation, and a Notch--Delta--EGF model of Drosophila neurogenesis. Across these systems, DNOS--DPC landscapes identify realized pattern regimes not determined by linear onset analysis alone. Robustness checks against bin count, entropy-based DNOS diagnostics, compression settings, image noise, clustering method, and graph, spectral, and wavelet descriptors support the stability of the comparative summaries under the stated protocol. In the Drosophila model, endpoint, field-resolved, multichannel, and time-resolved analyses distinguish pathway-level Notch/Delta signaling collapse from partial downstream proneural-output persistence, and suggest an EGF-dependent entry/progression bias into organized proneural-wave states. The framework provides reproducible coordinates for comparing realized pattern diversity, structural complexity, and developmental trajectories across nonlinear reaction--diffusion and regulatory systems.