Universal Hyperuniform Organization in Looped Leaf Vein Networks

Leaf vein network is a hierarchical vascular system that transports water and nutrients to the leaf cells. The thick primary veins form a branched network, while the secondary veins develop closed circuits forming a well-defined cellular structure. Through extensive analysis of a variety of distinct leaf species, we discover that the apparently disordered cellular structures of the secondary vein networks exhibit a universal hyperuniform organization and possess a hidden order on large scales. Disorder hyperuniform (DHU) systems lack conventional long-range order, yet they completely suppress normalized large-scale density fluctuations like crystals. Specifically, we find that the distributions of the geometric centers associated with the vein network loops possess a vanishing static structure factor in the zero-wavenumber limit, i.e., $S(k) \sim k^{\alpha}$, where $\alpha \approx 0.64$, providing an example of class III hyperuniformity in biology. This hyperuniform organization leads to superior efficiency of diffusive transport, as evidenced by the much faster convergence of the time-dependent spreadability $\mathcal{S}(t)$ to its long-time asymptotic limit, compared to that of other uncorrelated or correlated disordered but non-hyperuniform organizations. Our results also have implications for the discovery and design of novel disordered network materials with optimal transport properties.