Information bounds the robustness of self-organized patterns

Self-organized systems, from developing organisms to synthetic nanostructures, are made up of individual components which collectively arrange and alter themselves to establish functional structures. Unlike macroscopic deterministic physical systems, these small collectives are governed by stochastic dynamics. Yet, remarkably, they are often able to reproducibly organize themselves in physical space in response to externally imposed or internally generated cues. Are there limits to how robust a pattern can be in these finite, noisy, mesoscopic systems? How can systems maximize their pattern-forming accuracy? Studying the emergence of collective behavior in minimal microscopic models of self-organization, we make these questions precise using the positional information, an information-theoretic measure of pattern emergence and robustness. We find that short-range correlated systems with discrete states are fundamentally limited in their positional information capacity and require fine-tuned transport parameters to reach this maximum. By providing long-range correlations, the existence of global constraints can bypass this maximum capacity and reduce the need for fine-tuning by providing effective integral feedback. Our work thus identifies design principles for synthetic self-organized systems. It also suggests that in biological systems the ubiquity of scale separation and hierarchical structures, which introduce long-range correlations, could be due to the superior capacity of these motifs for robust spatial patterning.