The physics of transportation networks in nature

An international team of researchers has described how loops, crucial for the stability of such networks, occur in transport networks found in nature. The researchers observed that when a branch of the network reaches the system boundary, the interactions between the branches change dramatically. Previously repulsive branches begin to attract each other, leading to the sudden formation of curls.

The results were published in the journal Proceedings of the National Academy of Sciences. The process described appears in a surprisingly large number of systems, from electrical discharge networks to instabilities in fluid mechanics to biological transport networks such as the canal system of the jellyfish Aurelia aurita.

Nature provides us with a wide spectrum of spatial transportation networks, from networks of blood vessels in our bodies to electrical discharges during a storm.

“These networks take various forms,” explains Stanislaw Żukowski, a Ph.D. student of the University of Warsaw and Paris Cité University and main author of the publication.

“They can have a tree geometry, where the branches of the network only divide and repel each other during growth. In other cases, when the branches attract each other and reconnect during growth, we are dealing with structures in a loop.”

Networks with many loops are widespread in living organisms, where they actively transport oxygen or nutrients and remove metabolic wastes. An important advantage of loop networks is their reduced vulnerability to damage; In networks without loops, destroying a branch can cut off all connected branches, whereas in networks with loops there is always another connection to the rest of the system.

Recently, researchers from the Faculty of Physics of the University of Warsaw described the mechanism responsible for the stability of already existing loops. However, the dynamic process leading to their formation remains unclear.

How are curls formed?

Many transport networks develop in response to a diffusive field, such as the concentration of a substance, the pressure in the system, or the electrical potential. The fluxes of such a field are transported much more easily through the branches of the network than through the surrounding environment.

This affects the distribution of the field in space: lightning rods attract electric discharges precisely because they have a lower resistance than the ambient air. The large difference in resistance between the network and the environment surrounding it leads to competition and repulsion between the branches.

However, the attraction of branches in growing networks, leading to the formation of loops, has long remained unexplained. The first attempt to understand the formation of loops in such systems was made a few years ago by the group of Professor Piotr Szymczak from the Faculty of Physics at the University of Warsaw.

“We showed that a small difference in resistance between the network and the support can lead to attraction between growing branches and the formation of loops,” says Szymczak.

The work gave rise to a joint project, in the form of Żukowski’s joint doctorate, carried out in Szymczak’s group and that of Annemiek Cornelissen, researcher at the Matter and Complex Systems Laboratory.

“In our laboratory, we study the morphogenesis of the gastrovascular network in jellyfish. This is a great example of a transport network with many loops,” explains Cornelissen.

“When I saw Annemiek’s presentation at a conference in Cambridge a few years ago, I immediately thought that our models could be applied to duct growth in jellyfish,” adds Piotr.

Breakthrough in curl formation

“The formation of loops when one of the branches reaches the system boundary – a phenomenon we describe in our latest publication – was first noticed in the channel network of the gastrovascular system of jellyfish,” explains Żukowski.

“By analyzing the development of these channels over time, I noticed that when one of them connects to the jellyfish’s stomach (the boundary of the system), then the shorter channels are immediately attracted towards him and form loops.”

The same phenomenon was observed by scientists during gypsum fracture dissolution experiments carried out at the University of Warsaw by Florian Osselin; in the so-called Saffman-Taylor experiment, in which the boundary between two fluids is unstable and transforms into finger-shaped patterns; and also encountered in the literature on electric discharges.

“The wealth of systems in which we discovered very similar dynamics convinced us that there must be a simple physical explanation for this phenomenon,” explains Cornelissen.

In their publication, the researchers presented a model describing the interactions between branches. They focused on how these interactions change when one of the branches approaches the system boundary and a breakthrough occurs.

“The competition and repulsion between branches then disappear and attraction appears,” explains Stéphane Douady. “This inevitably leads to the formation of loops.”

“Our model predicts that attraction between neighboring branches after a breakthrough occurs regardless of the geometry of the network or the difference in resistance between the network and the surrounding medium,” explains Szymczak.

“In particular, we have shown that near-breakthrough loops can form in systems with a very large difference in resistance, which was previously thought to be impossible. This explains why this phenomenon is so widespread in physical and biological systems. “

“In cases where the growth mechanisms are not yet clear, this will be a strong indication that the dynamics of the system is controlled by diffusive flows,” adds Żukowski. “We are extremely curious to see in which other systems we will observe the formation of near-breakthrough loops.

The team includes researchers from the Faculty of Physics of the University of Warsaw, the Matter and Complex Systems Laboratory and the Orléans Institute of Earth Sciences.