How to pare a pair: Topology control and pruning in intertwined complex networks

Abstract

Recent work on self-organized remodeling of vasculature in slime molds, leaf venation systems, and vessel systems in vertebrates has put forward a plethora of potential adaptation mechanisms. All these share the underlying hypothesis of a flow-driven machinery, meant to alter rudimentary vessel networks in order to optimize the system's dissipation, flow uniformity, or more, with different versions of constraints. Nevertheless, the influence of environmental factors on the long-term adaptation dynamics as well as the network structure and function have not been fully understood. Therefore, interwoven capillary systems such as those found in the liver, kidney, and pancreas present a novel challenge and key opportunity regarding the field of coupled distribution networks. We here present an advanced version of the discrete Hu-Cai model, coupling two spatial networks in three dimensions. We show that spatial coupling of two flow-adapting networks can control the onset of topological complexity in concert with short-term flow fluctuations. We find that both fluctuation-induced and spatial coupling induced topology transitions undergo curve collapse, obeying simple functional rescaling. Further, our approach results in an alternative form of Murray's law, which incorporates local vessel interactions and flow interactions. This geometric law allows for the estimation of the model parameters in ideal Kirchhoff networks and respective experimentally acquired network skeletons.

Quantification of nematic cell polarity in three-dimensional tissues

“…This biaxial liquid-crystal order may represent a self-organization principle of liver tissue that solves two conflicting design requirements [ 15 ]: every hepatocyte must be connected to both the sinusoidal network and the bile canaliculi network to fulfill its metabolic functions. One the other hand, the mutual distance between the two intertwined networks of sinusoids and bile canaliculi should be maximized [ 35 ], since oxygen- and nutrient-rich blood transported by sinusoids and toxic bile containing digestive enzymes transported by bile canaliculi should never mix. The layered organization of the sinusoidal network together with the nematic alignment of the ring axis ( a 2 ) of apical cell polarity parallel to the sinusoidal plane axis ( s 2 ) implies that ring-like surface patterns of apical cell membrane are approximately co-planar to the sinusoid layers: this maximizes the mutual distance between the sinusoidal and the bile canaliculi network ( Fig 5B ).…”

On biological flow networks: Antagonism between hydrodynamic and metabolic stimuli as driver of topological transitions

“…Hence we think that the current models of antagonistic adaptation mechanisms, flux or demand-supply based, are in need of further improvement: In terms of cost-optimization models we propose a leap-frog-style adaptation where a solute uptake optimization follows a conventional dissipation-volume optimizing system. Considering the conventional stress driven adaptation schemes with cost rescaling [44] and stochastic flow patterns [48,49] one may still generate topological complex, space-filling structures, or reach those by different dynamic environments [21,50]. Yet, instead of readjusting radii directly given the tissue's metabolic demands in terms of oxygen, glucose etc., we suggest that once a space-filling perfusion is reached, a secondary optimization takes place adjusting β * .…”

Resilience of three-dimensional sinusoidal networks in liver tissue

“…We expect that also the low-current edges contribute to the supply of hepatocytes, provided these are connected to high-current edges by short distances. Future work will address the self-organization of pairs of space-filling, mutually repulsive, intertwined networks 50 , and study their transport and resilience properties, inspired by the bile and sinusoidal network in liver tissue.…”