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


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.


Vascular transport Plant modelling Boundary layers  Matched asymptotic analysis 



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.


  1. Barbaroux C, Bréda N, Dufrêne E (2003) Distribution of above-ground and below-ground carbohydrate reserves in adult trees of two contrasting broad-leaved species (Quercus petraea and Fagus sylvatica). New Phytol 157(3):605–615CrossRefGoogle Scholar
  2. Bingham IJ, Stevenson EA (1993) Control of root growth: effects of carbohydrates on the extension, branching and rate of respiration of different fractions of wheat roots. Physiol Plant 88(1):149–158CrossRefGoogle Scholar
  3. De Schepper V, De Swaef T, Bauwerarts I, Steppe K (2013) Phloem transport: a review of mechanisms and controls. J Exp Bot 64(16):4839–4850CrossRefGoogle Scholar
  4. Dilkes NB, Jones DL, Farrar J (2004) Temporal dynamics of carbon partitioning and rhizodeposition in wheat. Plant Physiol 134(2):706–715CrossRefGoogle Scholar
  5. Fernández V, Brown PH (2013) From plant surface to plant metabolism: the uncertain fate of foliar-applied nutrients. Front Plant Sci 4(289)Google Scholar
  6. Hall AJ, Minchin PEH (2013) A closed-form solution for steady-state coupled phloem/xylem flow using the Lambert-W function. Plant Cell Environ 36(12):2150–2162CrossRefGoogle Scholar
  7. Hinch EJ (1991) Perturbation methods, vol 6. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  8. Hölttä T, Vesala T, Sevanto S, Perämäki M, Nikinmaa E (2006) Modeling xylem and phloem water flows in trees according to cohesion theory and Münch hypothesis. Trees 20(1):67–78CrossRefGoogle Scholar
  9. Jensen KH, Rio E, Hansen R, Clanet C, Bohr T (2009) Osmotically driven pipe flows and their relation to sugar transport in plants. J Fluid Mech 636:371–396CrossRefzbMATHGoogle Scholar
  10. Jensen KH, Lee J, Bohr T, Bruus H, Holbrook NM, Zwieniecki MA (2011) Optimality of the Münch mechanism for translocation of sugars in plants. J R Soc Interface 8(61):1155–1165CrossRefGoogle Scholar
  11. Jensen KH, Berg-Sørensen K, Friis SM, Bohr T (2012) Analytic solutions and universal properties of sugar loading models in Münch phloem flow. J Theor Biol 304:286–296CrossRefGoogle Scholar
  12. Jensen KH, Savage JA, Holbrook NM (2013) Optimal concentration for sugar transport in plants. J R Soc Interface 10(83):0055Google Scholar
  13. Kramer PJ, Boyer JS (1995) Water relations of plants and soils, Chapter 2. Academic Press, San DiegoGoogle Scholar
  14. Kutschera L, Lichtenegger E, Sobotik M (2009) Wurzelatlas der Kulturpflanzen gemigter Gebiete mit Arten des Feldgemsebaues. DLG -Verlag, FrankfurtGoogle Scholar
  15. Lacointe A, Minchin PEH (2008) Modelling phloem and xylem transport within a complex architecture. Funct Plant Biol 35(10):772–780CrossRefGoogle Scholar
  16. Minchin PEH, Thorpe MR (1987) Measurement of unloading and reloading of photo-assimilate within the stem of bean. J Exp Bot 38(117):211–220CrossRefGoogle Scholar
  17. Minchin PEH, Thorpe MR (1996) What determines carbon partitioning between competing sinks? J Exp Bot 47:1293–1296CrossRefGoogle Scholar
  18. Münch E (1926) Über Dynamik der Saftströmungen. Deut Bot Ges 44:68–71Google Scholar
  19. Patrick JW (1997) Phloem unloading: sieve element unloading and post-sieve element transport. Annu Rev Plant Physiol 48(1):191–222CrossRefGoogle Scholar
  20. Payvandi S, Daly KR, Jones DL, Talboys P, Zygalakis KC, Roose T (2014) A mathematical model of water and nutrient transport in xylem vessels of a wheat plant. Bull Math Biol 76(3):566–596MathSciNetCrossRefzbMATHGoogle Scholar
  21. Phillips RJ, Dungan SR (1993) Asymptotic analysis of flow in sieve tubes with semi-permeable walls. J Theor Biol 162(4):465–485Google Scholar
  22. Pickard WF, Abraham-Shrauner B (2009) A simplest steady–state Münch-like model of phloem translocation, with source and pathway and sink. Funct Plant Biol 36(7):629–644CrossRefGoogle Scholar
  23. Rawson HM, Evans LT (1971) The contribution of stem reserves to grain development in a range of wheat cultivars of different height. Aust J Agric Res 22(6):851–863CrossRefGoogle Scholar
  24. Thompson MV, Holbrook NM (2003a) Application of a single-solute non-steady-state phloem model to the study of long-distance assimilate transport. J Theor Biol 220(4):419–455CrossRefGoogle Scholar
  25. Thompson MV, Holbrook NM (2003b) Scaling phloem transport: water potential equilibrium and osmoregulatory flow. Plant Cell Environ 26(9):1561–1577CrossRefGoogle Scholar
  26. Thompson MV, Holbrook NM (2004) Scaling phloem transport: information transmission. Plant Cell Environ 27(4):509–519CrossRefGoogle Scholar
  27. Thompson MV (2005) Scaling phloem transport: elasticity and pressure-concentration waves. J Theor Biol 236(3):229–241CrossRefGoogle Scholar
  28. Turgeon R (2010) The role of phloem loading reconsidered. Plant Physiol 152(4):1817–1823CrossRefGoogle Scholar
  29. van Bel AJE (2003) The phloem, a miracle of ingenuity. Plant Cell Environ 26(1):125–149CrossRefGoogle Scholar

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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