Minimal branching and fusion morphogenesis approaches biological multi-objective optimality

TL;DR

The paper addresses how multi-task transport networks can emerge from simple local growth rules rather than global optimization. It introduces a minimal two-dimensional morphogenesis model with stochastic growth, branching, stopping, and fusion, and analyzes the resulting morphospace and multi-objective performance. By evaluating robustness $ ilde{R}$, space exploration $\|A\|$, and conductance $\|G\|$, the study shows that synthetic networks lie on a curved Pareto front closely aligned with empirical fungal networks, indicating near-biological multi-objective optimality without explicit optimization. This suggests that simple local growth rules can generate diverse, efficiently performing architectures and may apply to other elongation-based transport systems.

Abstract

Many biological networks grow by elongation of filaments that can branch and fuse -- typical examples include fungal mycelium or slime mold. These networks must simultaneously perform multiple tasks such as transport, exploration, and robustness under finite resources. Yet, how such multi-task architectures emerge from local growth processes remains poorly understood. Here, we introduce a minimal model of spatial network morphogenesis based solely on stochastic branching, fusion, and stopping, during elongation. Despite the absence of global optimization or feedback, the model generates a broad morphospace from tree-like, to loopy, as well as hybrid architectures. By quantifying multiple functional objectives, we show that (i) these synthetic structures occupy similar regions of performance space than evolved empirical fungal networks, and (ii) that their Pareto front of optimal trade-offs lies close to that of these same fungal networks. Our results show that biological architectures approaching multi-objective optimality can arise from simple local growth rules, and identify branching and fusion as fundamental ingredients shaping the architecture of living transport networks.

Figures (4)

• Figure 1: Minimal model of spatial network growthA. The four local processes of the model: each active tip node (red) can grow, branch, stop, or fuse with an existing edge. B. Typical network growth generated by the model. Active tips (red nodes) form a travelling front, and core edge (grey) density saturates. C. Zoomed-in versions of the network from B (right), compared to an empirical network from a filamentous fungus aguilar-trigueros2022network (left).
• Figure 2: Morphospace of branching networks. To explore the morphospace associated with the model, we simulate networks for a range of parameter values ($p_{\rm branch}$, $p_{\rm stop}$), with 3 realizations each. A. Morphospace defined by network density $\rho$ and loopiness $1-f_b$. Each point represents one simulated network with total length $10^4$. B, C. The position in this morphospace is controlled by the two parameters and $k$, the sensing distance factor. D. Examples of three network architectures, from sparse and tree-like (left) to dense and loopy (right), and an intermediate hybrid architecture (middle). E. Zoomed-in versions of the above networks.
• Figure 3: Multi-objective optimality with Pareto analysis.A. Performance space defined by three objectives formalized by network metrics: robustness to damage $\tilde{R}$, space exploration $\| A \|$, and conductance $\| G \|$ for transport efficiency. Each point (grey) represents one simulated network with total length $10^4$ (same data as in \ref{['fig:morphospace']}A). The points fall on a curved surface. Networks that are non-dominated define the set of optimal solutions, called Pareto front (red). B. Two-dimensional projections of performance space. C. Visualization of the Pareto front solutions (red dots) in morphospace. The triangle (black) is the best fitting polygon following the method in shoval2012evolutionary.
• Figure 4: Model-grown networks approach empirical multi-objective optimality. In the same 2D performance spaces as in \ref{['fig:pareto']}, we show the region occupied by simulated networks (grey shade), the associated Pareto front (PF), the 253 empirical networks (blue crosses) and their PF (blue line), and the grids' PF. The PF front of the simulated networks lies close to the other two.