Surface Minimization Model Explains Angle Diversity in Physical Networks

A study proposes a surface minimization model to account for the observed diversity in branching angles within various physical networks, specifically addressing deviations from established rules such as Steiner's solution. The model introduces a unique circumference constraint, denoted as w_i, for each network link.

Model Framework and Predictions

By setting w_1 = w_2 = w and w_3 = w', and varying the ratio ρ = w'/w, the model predicts the minimal manifold connecting three nodes (Fig. 4a,b). This model identifies two distinct regimes separated by a threshold value, ρ_th:

• Branching Regime (ρ > ρ_th): The steering angle Ω_1→2 is predicted to exhibit an approximately linear dependence on ρ, specifically Ω_1→2 ≈ k(ρ − ρ_th) (Fig. 4e,f,g). This behavior is suggested to explain the range of angles observed in certain network structures (Fig. 1f).
• Sprouting Regime (ρ < ρ_th): When links 1 and 2 have comparable diameters, the model predicts they form a straight path with Ω_1→2 = 0. The thinner link 3 is predicted to emerge perpendicularly, with Ω_1→3 ≈ Ω_2→3, a behavior consistent with orthogonal sprouting (Fig. 4c,d,g). This prediction suggests that 90° solutions are optimal for any ρ < ρ_th, contrasting with earlier geometric approaches that posited convergence to 90° only in the ρ → 0 limit.

Empirical Validation of Predictions

To test these predictions, researchers analyzed bifurcation motifs across a database of networks, focusing on branches where w_1 ≈ w_2. Measurements of Ω(ρ) = Ω_1→2 as a function of observed ρ indicated that most bifurcations for ρ < ρ_th were sprout-like, characterized by small Ω(ρ) values. The cumulative value of observed angles supported the model's predictions, showing an approximate (ρ_th − ρ)^1 behavior for ρ < ρ_th and a quadratic (ρ − ρ_th)^2 behavior for ρ > ρ_th, consistent with Fig. 4g (Fig. 4i–n).

Prevalence and Functional Role of Orthogonal Sprouts

The surface minimization model predicts a common occurrence of orthogonal sprouts when ρ < ρ_th. Investigations across various physical networks confirmed their presence:

• Blood Vessels: 25.6% of w_1 ≈ w_2 cases exhibited a third branch perpendicular to the main branches.
• Other Networks: Sprouts were observed in tropical trees (12.9%), corals (52.8%), arabidopsis (11.2%), fruit fly neurons (13.8%), and human neurons (18.4%).

Certain biological systems appear to utilize sprouting for functional advantages. In the human connectome, 3,911 out of 4,003 identified sprouts (98%) terminated in a synapse (Fig. 4h). This suggests neuronal systems may employ orthogonal sprouts as dendritic spines to form synaptic connections efficiently. Similarly, perpendicular sprouting in plant roots and fungal hyphae is noted for facilitating broader exploration of soil for resources with minimal material expenditure.

Angle Distribution (P(Ω)) Predictions

Further predictions concern the P(Ω) angle distributions, conditioned on observed ρ values:

• Sprouting Regime (ρ < ρ_th): The model predicts Ω = 0 independent of ρ, leading to a sharp peak of P(Ω) at Ω = 0. Empirical data align with this prediction (left side of Fig. 5a–f).
• Branching Regime (ρ > ρ_th): A broad distribution with high variance for P(Ω) is predicted due to the linear relationship shown in Fig. 4g. Empirical data also support this prediction (right side of Fig. 5a–f).

These findings contrast with Steiner's prediction, which posits a sharp peak of P(Ω) irrespective of ρ (thin grey lines in Fig. 5a–f).