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Bulletin of Mathematical Biology

, Volume 76, Issue 11, pp 2834–2865 | Cite as

Mathematical Modelling of the Phloem: The Importance of Diffusion on Sugar Transport at Osmotic Equilibrium

  • S. PayvandiEmail author
  • K. R. Daly
  • K. C. Zygalakis
  • T. Roose
Original Article

Abstract

Plants rely on the conducting vessels of the phloem to transport the products of photosynthesis from the leaves to the roots, or to any other organs, for growth, metabolism, and storage. Transport within the phloem is due to an osmotically-generated pressure gradient and is hence inherently nonlinear. Since convection dominates over diffusion in the main bulk flow, the effects of diffusive transport have generally been neglected by previous authors. However, diffusion is important due to boundary layers that form at the ends of the phloem, and at the leaf-stem and stem-root boundaries. We present a mathematical model of transport which includes the effects of diffusion. We solve the system analytically in the limit of high Münch number which corresponds to osmotic equilibrium and numerically for all parameter values. We find that the bulk solution is dependent on the diffusion-dominated boundary layers. Hence, even for large Péclet number, it is not always correct to neglect diffusion. We consider the cases of passive and active sugar loading and unloading. We show that for active unloading, the solutions diverge with increasing Péclet. For passive unloading, the convergence of the solutions is dependent on the magnitude of loading. Diffusion also permits the modelling of an axial efflux of sugar in the root zone which may be important for the growing root tip and for promoting symbiotic biological interactions in the soil. Therefore, diffusion is an essential mechanism for transport in the phloem and must be included to accurately predict flow.

Keywords

Vascular transport Plant modelling Boundary layers  Matched asymptotic analysis 

Notes

Acknowledgments

This work was sponsored by Defra, BBSRC (BB/J000868/1), Scottish Government, AHDB, and other industry partners through Sustainable Arable LINK Project LK09136 and the BBSRC (BB/I024283/1). Tiina Roose is funded by a Royal Society University Research Fellowship.

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Copyright information

© Society for Mathematical Biology 2014

Authors and Affiliations

  • S. Payvandi
    • 1
    • 2
    Email author
  • K. R. Daly
    • 1
    • 2
  • K. C. Zygalakis
    • 2
    • 3
  • T. Roose
    • 1
    • 2
  1. 1.Engineering Sciences, Faculty of Engineering and the EnvironmentUniversity of SouthamptonSouthamptonUK
  2. 2.Crop Systems Engineering Group, Institute for Life SciencesUniversity of SouthamptonSouthamptonUK
  3. 3.Mathematical Sciences, Faculty of Social and Human SciencesUniversity of SouthamptonSouthamptonUK

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