Auxin transport model for leaf venation

Abstract

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 membranelocalised 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 midvein 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.

Leaf morphogenesis: The multifaceted roles of mechanics

“…Sachs proposed (20 years prior) a canalization model for venation patterning: positive feedback between the auxintransporting ability of a cell and its auxin flux, so that high-flux cells undergo differentiation into veins that further amplify their auxin transport abilities and eventually canalize into the vascular strands (Sachs, 1975;Sachs, 1981). Computational canalization models are able to demonstrate the patterning of auxin fluxes in venation development (Sawchuk et al, 2013;Cieslak et al, 2015;Feller et al, 2015;Haskovec et al, 2019) as well as the patterning of organ initiation in the SAM with typical phyllotactic patterns (Stoma et al, 2008). The guiding principles specifying the concentration-based auxin transport model and canalization model need to be further elucidated (Bayer et al, 2009;Bennett et al, 2014;Cieslak et al, 2015;Hayakawa et al, 2015;Hartmann et al, 2019).…”

Genome-wide identification and expression analysis of AUX/LAX family genes in Chinese hickory (Carya cathayensis Sarg.) Under various abiotic stresses and grafting

“…LAX2 was involved in leaf vein formation and xylem development ( Péret et al., 2012 ). The AUX/LAX gene family influences phyllotactic patterns and is required for the establishment of embryonic root cell organization and plant embryogenesis in Arabidopsis ( Bainbridge et al., 2008 ; Ugartechea-Chirino et al., 2010 ; Haskovec et al, 2019 ). In addition to the aforementioned functions, the auxin influx carrier is involved in processes such as seed germination, adventitious roots, and female gametophyte development ( Panoli et al., 2015 ; Chai et al., 2016 ; Lee et al., 2019 ).…”

Scaling relations for auxin waves

“…Haskovec and his coworkers derive local and global existence results together with an appropriate continuum limit for their graph-based diffusion model in Haskovec et al. ( 2019 ).

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