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.
| Original language | English |
|---|---|
| Article number | 20190015 |
| Journal | Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences |
| Volume | 475 |
| Issue number | 2231 |
| Early online date | 20 Nov 2019 |
| DOIs | |
| Publication status | Published - 29 Nov 2019 |
Bibliographical note
Funding Information:Data accessibility. All data are presented within this paper. Authors’ contributions. S.C.-S. carried out the sample design, manufacture and specimen preparation. P.M. conducted the sample analysis using SRAS. M.H. conducted the optical microscopy work and drafted the manuscript. S.C.-S. and R.P. supported the drafting of the manuscript. W.L. conducted the simulation of texture orientation. M.H. and A.T.C. coordinated the study. S.D.S. and C.T. gave feedback on the manuscript. All authors gave final approval for publication. Competing interests. The authors have no competing commercial or academic interests which may compromise the integrity of this work. Funding. This work was supported by the Engineering and Physical Sciences Research Council (grant number EP/L022125/1) through the ‘UK Research Centre in Nondestructive Evaluation’. Acknowledgements. The authors acknowledge Mark Hardy (Additive Manufacturing and 3D Printing Research Group, University of Nottingham) for his support in specimen production and Alistair Speidel (Advanced Component Engineering Laboratory, University of Nottingham) for his support in SEM micrograph generation. Furthermore, the authors acknowledge Paul Dryburgh for his support in advice for correction to the manuscript.
Funding
Data accessibility. All data are presented within this paper. Authors’ contributions. S.C.-S. carried out the sample design, manufacture and specimen preparation. P.M. conducted the sample analysis using SRAS. M.H. conducted the optical microscopy work and drafted the manuscript. S.C.-S. and R.P. supported the drafting of the manuscript. W.L. conducted the simulation of texture orientation. M.H. and A.T.C. coordinated the study. S.D.S. and C.T. gave feedback on the manuscript. All authors gave final approval for publication. Competing interests. The authors have no competing commercial or academic interests which may compromise the integrity of this work. Funding. This work was supported by the Engineering and Physical Sciences Research Council (grant number EP/L022125/1) through the ‘UK Research Centre in Nondestructive Evaluation’. Acknowledgements. The authors acknowledge Mark Hardy (Additive Manufacturing and 3D Printing Research Group, University of Nottingham) for his support in specimen production and Alistair Speidel (Advanced Component Engineering Laboratory, University of Nottingham) for his support in SEM micrograph generation. Furthermore, the authors acknowledge Paul Dryburgh for his support in advice for correction to the manuscript.
Keywords
- Continuum limit
- Mathematical modelling
- Numerical simulation
- Weak solutions
ASJC Scopus subject areas
- General Mathematics
- General Engineering
- General Physics and Astronomy