Abstract
The plant hormone auxin controls many aspects of the development of plants. One striking dynamical feature is the self-organization of leaf venation patterns which is driven by high levels of auxin within vein cells. The auxin transport is mediated by specialized membrane-localized proteins. Many venation models have been based on polarly localized efflux-mediator proteins of the PIN family. Here, we investigate a modelling 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.
Highlights
The hormone auxin is important for a multitude of developmental processes in plants [23, 26, 28, 14, 35, 34, 33]
Auxin is present at high levels in the forming vein cells compared to neighboring tissues, and the membrane localized PIN-FORMED (PIN) family of auxin transport mediators have been shown to be essential for the correct patterning of the vein network [31, 35]
It has been suggested that the pattern could be the result of a canalisation mechanism where auxin flux itself feed back to a polarised transport connecting sources and sinks of auxin[32, 21, 22], and this idea has been revisited in models including polarised PIN transporters [29, 11, 10]
Summary
The hormone auxin is important for a multitude of developmental processes in plants [23, 26, 28, 14, 35, 34, 33]. PIN proteins are involved in several patterning processes in plants, and alternative models, not based on auxin flux, have been proposed for example for producing turing-like dynamics in the context of phyllotaxis [17, 38, 4], and for single cell polarity resulting in planar polarity [1]. We analyse an alternative model for leaf venation, which is not dependent on polar localisation of auxin transport mediators. It is based on a recent continuous model for transport networks [16], where there is a feedback between gradients and transport capacity. We provide numerical simulations showing it is capable of generating a mid-vein pattern as well as simple branched patterns
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