Les Houches Lectures on Flow Networks in Biology

TL;DR

This work analyzes how biological flow networks transport resources efficiently by combining low-Reynolds-number fluid mechanics with local radius-adaptation rules. It develops a framework where tube radii evolve under shear-stress feedback and force-balance, and introduces a generalized metabolic-cost scaling with parameter $\gamma$, yielding explicit optimal conductances $C_{mn}$ and adaptation dynamics $dC/dt$. It shows distinct architectural regimes: for $\gamma<1$ networks tend toward hierarchical trees to minimize dissipation, while for $\gamma>1 they become loopy with reduced hierarchy; memory emerges for $\gamma=\tfrac{1}{2}$ as past inflows imprint in the network via vanishing low-conductance tubes. The analysis also covers transport and absorption in single tubes through Taylor dispersion and explores homogeneous supply in leaves, deriving an optimal inflow rate $Q_{in}^*=2\pi a l M \nu$ that yields uniform resource distribution. Overall, the work highlights how physical transport laws and local adaptive rules together shape the diverse architectures of biological flow networks with implications for vascular and venation systems.

Abstract

Flows are essential to transport resources over large distances. As soon as diffusion becomes time-limiting, flows are needed. Flows are key for the function of multiple human organs, from the blood vasculature to the lungs, the digestive tract, the lymphatic system, and many more. While physics governs the flow dynamics, biology's response to flows governs the flow network architecture. We start with the fluid physics of Stokes flow, the prerequisite to describe the flows in biological flow networks. Then we explore how the network adaptation dynamics of biological flow networks reorganize network architecture to minimize flow dissipation or homogenize transport, storing memories of past flows along the way.

Figures (13)

• Figure 1: Flow profile in a cylindrical tube. In a cylindrical tube of radius $a$, length $l$ the pressure drop $\Delta P = P_1-P_2$ drives a parabolic flow profile known as Poiseuille flow.
• Figure 2: Murray's law exemplified at a three tube network node. The tube radius $a_0$ of the tube with incoming flow $Q_0$ is related to the radii of the tubes $a_1$ and $a_2$ of outgoing flow $Q_1$ and $Q_2$, respectively, by Murray's law according to $a_0^3=a_1^3+a_2^3$, thereby minimizing both dissipation and metabolic cost.
• Figure 3: Shear stress induces tube radius adaptation with a time delay in the slime mold Physarum polycephalum. (a) Time-averaged shear rate $\langle\tau\rangle/\mu$ dynamics preceed time-averaged tube radius dynamics $d\langle a\rangle /dt$ with a time delay $t_{\text{delay}}$. (b) Deriving the local shear rate within a flow network, the non-linear dynamics of sensed shear rate $\tau_s$ and tube radius $\langle a\rangle$ are mapped out in a phase portrait. Two stable fixed points (light yellow) constrain tube trajectories to circle in spirals (blue) or shrink away (pink). Reproduced from (Marbach et. al. (2023) marbach_vein_2023).
• Figure 4: Minimal dissipation networks with fluctuating outflows transition from tree to loopy networks as metabolic cost becomes more and more expensive for large tube radii. In each network, the inflow is at the lower left corner and all other nodes are outlets with uncorrelated fluctuating flows. The radius of each tube is proportional to the square root of its conductance. (a) Tree-like network ($\gamma = 0.25$). (b) Hierarchical network with loops ($\gamma=0.75$). (c) Network with many loops and no hierarchical organization ($\gamma=1.25$). Reproduced from Corson (2010) corson_fluctuations_2010.
• Figure 5: Fluctuations introduce loops also with local adaptation dynamics. In each network, the inflow is at the lowest node and distributed throughout all nodes serving as outlets in the remaining network, $\gamma=1/2$. (a) Fixed and equal outflow at all nodes results in a tree architecture (b) Outlet fluctuate as the tube radii are adapting, resulting in loops interconnecting the tree architecture. Reproduced from Hu and Cai (2013) hu_adaptation_2013.