Experimental models for Murray’s law

Abstract

Transport networks are ubiquitous in multicellular organisms and include leaf veins, fungal mycelia and blood vessels. While transport of materials and signals through the network plays a crucial role in maintaining the living system, the transport capacity of the network can best be understood in terms of hydrodynamics. We report here that plasmodium from the large, single-celled amoeboid Physarum is able to construct a hydrodynamically optimized vein-network when evacuating biomass from confined arenas of various shapes through a narrow exit. Increasingly thick veins developed towards the exit, and the network spanned the arena via repetitive bifurcations to give a branching tree. The Hausdorff distance from all parts of the plasmodium to the vein network was kept low, whilst the hydrodynamic conductivity from distal parts of the network to the exit was equivalent, irrespective of the arena shape. This combination of spatial patterning and differential vein thickening served to evacuate biomass at an equivalent rate across the entire arena. The scaling relationship at the vein branches was determined experimentally to be 2.53-3.29, consistent with predictions from Murray's law. Furthermore, we show that mathematical models for self-organised, adaptive transport in Physarum simulate the experimental network organisation well if the scaling coefficient of the current-reinforcement rule is set to 3. In simulations, this resulted in rapid development of an optimal network that minimised the combined volume and frictional energy in comparison with other scaling coefficients. This would predict that the boundary shear forces within each vein are constant throughout the network, and would be consistent with a feedback mechanism based on a sensing a threshold shear at the vein wall.

Automated analysis of Physarum network structure and dynamics

“…Evacuation networks were formed by allowing an excised sheet of protoplasm to exit an enclosed rectangular arena through a single channel in response to addition of a food source according to [1]. Time lapse images were obtained with transmitted light with an (x, y) pixel spacing of 6.2µm, at 1 h intervals and imported into a MATLAB GUI for all subsequent processing steps.…”

“…We have previously examined the scaling relationship between parent and daughter vessel radii at three-way junctions [1] to test whether Physarum networks obey Murray's law [38]. This predicts that the radius r 0 of a parent vessel is related to the radii of the daughter vessels r 1 and r 2 by the relationship:…”

Automated analysis ofPhysarumnetwork structure and dynamics

“…If the original binary image correctly covers the complete vein width, the Euclidean Distance Map (EDM, [25]) or the segmented areato-length ratio [4], can be used to estimate the width. Alternatively, the original intensity image can be interrogated directly using calibrated measurements of light transmission following the Lambert-Beer law [1], or using greyscale granulometry techniques [39].…”

A mathematical model for adaptive vein formation during exploratory migration ofPhysarum polycephalum: routing while scouting

“…It has been found that taking µ in an appropriate regime is crucial to reproducing these behaviours [7]. Based on the analysis in [32], we take µ = 1. 3 The time scales of actomyosin contraction and migration are determined by τ 1 and τ 2 .…”