  The plant hormone auxin controls many aspects of the development of plants.
One striking dynamical feature is the self-organisation of leaf venation
patterns which is driven by high levels of auxin within vein cells. The auxin
transport is mediated by specialised membrane-localised proteins. Many venation
models have been based on polarly localised efflux-mediator proteins of the PIN
family. Here, we investigate a modeling framework for auxin transport with a
positive feedback between auxin fluxes and transport capacities that are not
necessarily polar, i.e.\ directional across a cell wall. Our approach is
derived from a discrete graph-based model for biological transportation
networks, where cells are represented by graph nodes and intercellular
membranes by edges. The edges are not a-priori oriented and the direction of
auxin flow is determined by its concentration gradient along the edge. We prove
global existence of solutions to the model and the validity of Murray's law for
its steady states. Moreover, we demonstrate with numerical simulations that the
model is able connect an auxin source-sink pair with a mid-vein and that it can
also produce branching vein patterns. A significant innovative aspect of our
approach is that it allows the passage to a formal macroscopic limit which can
be extended to include network growth. We perform mathematical analysis of the
macroscopic formulation, showing the global existence of weak solutions for an
appropriate parameter range.
