Phenotypes of Vascular Flow Networks

Complex distribution networks are pervasive in biology. Examples include nutrient transport in the slime mold Physarum polycephalum as well as mammalian and plant venation. Adaptive rules are believed to guide development of these networks and lead to a reticulate, hierarchically nested topology that is both efficient and resilient against perturbations. However, as of yet, no mechanism is known that can generate such networks on all scales. We show how hierarchically organized reticulation can be constructed and maintained through spatially correlated load fluctuations on a particular length scale. We demonstrate that the network topologies generated represent a trade-off between optimizing transport efficiency, construction cost, and damage robustness and identify the Pareto-efficient front that evolution is expected to favor and select for. We show that the typical fluctuation length scale controls the position of the networks on the Pareto front and thus on the spectrum of venation phenotypes.

Figure 1

(a) Network model of liquid transport. Edges $e$ of length $L_e$ carry currents $F_e$. At each node $i$, a net current $S_i$ is drawn from the network. The net current $S_i$ models local sources and sinks. (b) The Pareto front (orange) is the set of points out of all possible phenotypes (gray) for which performance cannot be improved at both objectives simultaneously. For any point not on the Pareto front [e.g., (i)] a different point can be found [e.g., (ii)] that has better performance at both objectives. For a point on the Pareto front, like (iii), this is not possible. The endpoints of the Pareto front (stars) are functional archetypes. (c) Leaf veins of Acer platanoides near the “reticulate archetype” identified in this Letter. (d) Leaf veins of Protium dawsonii show many freely ending veinlets, similar to what is found near the “tree archetype” identified in this Letter.

Figure 2

The variety of network phenotypes that can be produced with a locally adaptive fluctuating load model. All examples lie on the Pareto front of efficient networks (Fig. 3), thus representing different trade-offs between baseline power dissipation, cost, and damage robustness. The number of loops and thus damage robustness increases to the right, the value of $\sigma$ increases from 0.5 to 4.0 to the right. The Pareto front corresponds to the whole spectrum of “natural” reticulate networks, from highly hierarchical trees, fragile but cheap, to highly robust reticulate, expensive networks. (a)–(e) The inlet is at the center. (f)–(j) The inlet is at the left side.

Figure 3

Geometry of the Pareto front of adaptive distribution networks. We plot the phenotypic space of networks obtained from parameter values $\rho \in \{1,10,100\}$, $\kappa \in \{1,0.1,0.01\}$, $\sigma \in [0.1,5]$, $\alpha =1,\beta =2/3$ as an example of the phenotypic space that can be reproduced using the model. We calculate the Pareto front for simultaneous minimization of power dissipation, network cost, and percolation penalty. The data were scaled to zero mean and unit variance in each objective. (a) The dataset; colors indicate the value of $\sigma$. The Pareto front is in red, and the non-Pareto networks from Fig. 4 are in blue. (b) Principal component analysis (PCA) embedding of the Pareto front from (a). 91% of the variance is encoded in the first PCA coordinate, suggesting that the front is approximately one-dimensional. The first PCA coordinate (PCA 1) approximately parametrizes the Pareto front. Red points correspond to the networks from Figs. 2. (c) For all combinations $(\rho ,\kappa ,\sigma )$ on the Pareto front $P$ we hold $\rho$ and $\kappa$ fixed and vary $\sigma$. For a wide range of $\sigma$, the average distance $d_P(x)={\min }_{p\in P}\parallel x-p\parallel$ from the Pareto front is well below the mean $⟨d_P⟩$, suggesting that phenotypes remain close to the Pareto front (blue curve, shaded region is one standard deviation over combinations of $\rho$, $\kappa$). Varying $\sigma$ moves linearly along the Pareto front parametrized by PCA 1 (orange curve). Thus, $\sigma$ approximately parametrizes the Pareto front. Similarly varying $\rho$ (d) or $\kappa$ (e) while holding the other parameters fixed may lead to phenotypes close to the Pareto front for large $\rho$ and all $\kappa$, but the position on the front PCA 1 is random. Thus, $\rho$ and $\kappa$ cannot be used to parametrize the Pareto front.

Figure 4

Network phenotypes not lying on the Pareto front show less hierarchical organization for the same amount of reticulation than their Pareto optimal counterparts. In both networks, $\kappa =1.0$, $\rho =1.0$. (a) $E=6.0$, $A=0.48$, $C=47.0$. (b) $E=10.8$, $A=7.73$, $C=32.8$.