# American Institute of Mathematical Sciences

August  2017, 14(4): 1055-1069. doi: 10.3934/mbe.2017055

## A surface model of nonlinear, non-steady-state phloem transport

 1 Laboratoire Amiénois de Mathématique Fondamentale et Appliquée, CNRS UMR 7352, Université de Picardie Jules Verne, 80069 Amiens, France 2 SCION, New Zealand Forest Research Institute, Private bag 3020, Rotorua 3046, New Zealand

* Corresponding author: youcef.mammeri@u-picardie.fr

Received  April 2016 Accepted  February 2017 Published  April 2017

Phloem transport is the process by which carbohydrates produced by photosynthesis in the leaves get distributed in a plant. According to Münch, the osmotically generated hydrostatic phloem pressure is the force driving the long-distance transport of photoassimilates. Following Thompson and Holbrook[35]'s approach, we develop a mathematical model of coupled water-carbohydrate transport. It is first proven that the model presented here preserves the positivity. The model is then applied to simulate the flow of phloem sap for an organic tree shape, on a 3D surface and in a channel with orthotropic hydraulic properties. Those features represent an significant advance in modelling the pathway for carbohydrate transport in trees.

Citation: Youcef Mammeri, Damien Sellier. A surface model of nonlinear, non-steady-state phloem transport. Mathematical Biosciences & Engineering, 2017, 14 (4) : 1055-1069. doi: 10.3934/mbe.2017055
##### References:
 [1] P. Cabrita, M. Thorpe and G. Huber, Hydrodynamics of steady state phloem transport with radial leakage of solute, Frontiers Plant Sci., 4 (2013), 531-543.  doi: 10.3389/fpls.2013.00531. [2] A. L. Christy and J. M. Ferrier, A mathematical treatment of Münch's pressure-flow hypothesis of phloem translocation, Plant Physio., 52 (1973), 531-538.  doi: 10.1104/pp.52.6.531. [3] T. K. Dey and J. A. Levine, Delaunay meshing of isosurfaces, Visual Comput., 24 (2008), 411-422.  doi: 10.1109/SMI.2007.15. [4] J. M. Ferrier, Further theoretical analysis of concentration-pressure-flux waves in phloem transport systems, Can. J. Bot., 56 (1978), 1086-1090.  doi: 10.1139/b78-118. [5] F. G. Feugier and A. Satake, Dynamical feedback between circadian clock and sucrose availability explains adaptive response of starch metabolism to various photoperiods, Frontiers Plant Sci., 305 (2013), 1-11.  doi: 10.3389/fpls.2012.00305. [6] D. B. Fisher and C. Cash-Clark, Sieve tube unloading and post-phloemtransport of fluorescent tracers and proteins injected into sieve tubes via severed aphid stylets, Plant Physio., 123 (2000), 125-137. [7] J. D. Goeschl and C. E. Magnuson, Physiological implications of the Münch--Horwitz theory of phloem transport: effect of loading rates, Plant Cell Env., 9 (1986), 95-102.  doi: 10.1111/j.1365-3040.1986.tb01571.x. [8] J. Gričar, L. Krže and K. Čufar, Number of cells in xylem, phloem and dormant cambium in silver fir (Abies alba), in trees of different vitality, IAWA Journal, 30 (2009), 121-133. [9] J. Hansen and E. Beck, The fate and path of assimilation products in the stem of 8-year-old {Scots} pine (Pinus sylvestris {L}.) trees, Trees, 4 (1990), 16-21.  doi: 10.1007/BF00226235. [10] F. Hecht, New Developments in Freefem++, J. Num. Math., 20 (2012), 251-265. [11] L. Horwitz, Some simplified mathematical treatments of translocation in plants, Plant Physio., 33 (1958), 81-93. [12] T. Hölttä, M. Mencuccini and E. Nikinmaa, Linking phloem function to structure: Analysis with a coupled xylem-phloem transport model, J. Theo. Bio., 259 (2009), 325-337. [13] T. Hölttä, T. Vesala, S. Sevanto, M. Perämäki and E. Nikinmaa, Modeling xylem and phloem water flows in trees according to cohesion theory and Münch hypothesis, Trees, 20 (2006), 67-78. [14] W. Hundsdorfer and J. G. Verwer, Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations Springer Series in Comput. Math., 33, Springer, 2003. doi: 10.1007/978-3-662-09017-6. [15] K. H. Jensen, J. Lee, T. Bohr, H. Bruus, N. M. Holbrook and M. A. Zwieniecki, Optimality of the Münch mechanism for translocation of sugars in plants, J. R. Soc. Interface, 8 (2011), 1155-1165.  doi: 10.1098/rsif.2010.0578. [16] A. Kagawa, A. Sugimoto and T. C. Maximov, CO 2 pulse-labelling of photoassimilates reveals carbon allocation within and between tree rings, Plant Cell Env., 29 (2006), 1571-1584. [17] E. M. Kramer, Wood grain pattern formation: A brief review, J. Plant Growth Reg., 25 (2006), 290-301.  doi: 10.1007/s00344-006-0065-y. [18] H.-O. Kreiss and J. Lorenz, Initial-Boundary Value Problems and the Navier-Stokes Equations Classics in Applied Mathematics, SIAM, 2004. doi: 10.1137/1.9780898719130. [19] A. Lacointe and P. E. H. Minchin, Modelling phloem and xylem transport within a complex architecture, Funct. Plant Bio., 35 (2008), 772-780.  doi: 10.1071/FP08085. [20] A. Lang, A model of mass flow in the phloem, Funct. Plant Bio., 5 (1978), 535-546.  doi: 10.1071/PP9780535. [21] P. E. H. Minchin, M. R. Thorpe and J. F. Farrar, A simple mechanistic model of phloem transport which explains sink priority, Journal of Experimental Botany, 44 (1993), 947-955.  doi: 10.1093/jxb/44.5.947. [22] E. Münch, Die Stoffbewegungen in der Pflanze Jena, Gustav Fischer, 1930. [23] K. A. Nagel, B. Kastenholz, S. Jahnke, D. van Dusschoten, T. Aach, M. Mühlich, D. Truhn, H. Scharr, S. Terjung, A. Walter and U. Schurr, Temperature responses of roots: Impact on growth, root system architecture and implications for phenotyping, Funct. Plant Bio., 36 (2009), 947-959.  doi: 10.1071/FP09184. [24] E. M. Ouhabaz, Analysis of Heat Equations on Domains London Math. Soc. Monographs Series, Princeton University Press, 2005. [25] S. Payvandi, K. R. Daly, K. C. Zygalakis and T. Roose, Mathematical modelling of the phloem: The importance of diffusion on sugar transport at osmotic equilibrium, Bull. Math Biol., 76 (2014), 2834-2865.  doi: 10.1007/s11538-014-0035-7. [26] S. Pfautsch, J. Renard, M. G. Tjoelker and A. Salih, Phloem as capacitor: Radial transfer of water into xylem of tree stems occurs via symplastic transport in ray parenchyma, Plant Physio., 167 (2015), 963-971.  doi: 10.1104/pp.114.254581. [27] O. Pironneau and M. Tabata, Stability and convergence of a Galerkin-characteristics finite element scheme of lumped mass type, Int. J. Num. Meth. Fluids, 64 (2000), 1240-1253.  doi: 10.1002/fld.2459. [28] G. E. Phillips, J. Bodig and J. Goodman, Flow grain analogy, Wood Sci., 14 (1981), 55-64. [29] R. J. Phillips and S. R. Dungan, Asymptotic analysis of flow in sieve tubes with semi-permeable walls, J. Theor. Biol., 162 (1993), 465-485.  doi: 10.1006/jtbi.1993.1100. [30] D. Rotsch, T. Brossard, S. Bihmidine, W. Ying, V. Gaddam, M. Harmata, J. D. Robertson, M. Swyers, S. S. Jurisson and D. M. Braun, Radiosynthesis of 6'-Deoxy-6'[18F]Fluorosucrose via automated synthesis and its utility to study in vivo sucrose transport in maize (Zea mays) leaves PLoS ONE 10 (2015), e0128989. doi: 10.1371/journal.pone.0128989. [31] D. Sellier and J. J. Harrington, Phloem transport in trees: A generic surface model, Eco. Mod., 290 (2014), 102-109.  doi: 10.1016/j.ecolmodel.2013.11.021. [32] D. Sellier, M. J. Plank and J. J. Harrington, A mathematical framework for modelling cambial surface evolution using a level set method, Annals Bot., 108 (2011), 1001-1011.  doi: 10.1093/aob/mcr067. [33] R. Spicer, Symplasmic networks in secondary vascular tissues: Parenchyma distribution and activity supporting long-distance transport, J. Exp. Bot., 65 (2014), 1829-1848.  doi: 10.1093/jxb/ert459. [34] J. F. Swindells, C. F. Snyder, R. C. Hardy and P. E. Golden, Viscosities of sucrose solutions at various temperatures: Tables of recalculated values, NBS Circular 440 (1958). [35] M. V. Thompson and N. M. Holbrook, Application of a single-solute non-steady-state phloem model to the study of long-distance assimilate transport, J. Theo. Bio., 220 (2003), 419-455.

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##### References:
 [1] P. Cabrita, M. Thorpe and G. Huber, Hydrodynamics of steady state phloem transport with radial leakage of solute, Frontiers Plant Sci., 4 (2013), 531-543.  doi: 10.3389/fpls.2013.00531. [2] A. L. Christy and J. M. Ferrier, A mathematical treatment of Münch's pressure-flow hypothesis of phloem translocation, Plant Physio., 52 (1973), 531-538.  doi: 10.1104/pp.52.6.531. [3] T. K. Dey and J. A. Levine, Delaunay meshing of isosurfaces, Visual Comput., 24 (2008), 411-422.  doi: 10.1109/SMI.2007.15. [4] J. M. Ferrier, Further theoretical analysis of concentration-pressure-flux waves in phloem transport systems, Can. J. Bot., 56 (1978), 1086-1090.  doi: 10.1139/b78-118. [5] F. G. Feugier and A. Satake, Dynamical feedback between circadian clock and sucrose availability explains adaptive response of starch metabolism to various photoperiods, Frontiers Plant Sci., 305 (2013), 1-11.  doi: 10.3389/fpls.2012.00305. [6] D. B. Fisher and C. Cash-Clark, Sieve tube unloading and post-phloemtransport of fluorescent tracers and proteins injected into sieve tubes via severed aphid stylets, Plant Physio., 123 (2000), 125-137. [7] J. D. Goeschl and C. E. Magnuson, Physiological implications of the Münch--Horwitz theory of phloem transport: effect of loading rates, Plant Cell Env., 9 (1986), 95-102.  doi: 10.1111/j.1365-3040.1986.tb01571.x. [8] J. Gričar, L. Krže and K. Čufar, Number of cells in xylem, phloem and dormant cambium in silver fir (Abies alba), in trees of different vitality, IAWA Journal, 30 (2009), 121-133. [9] J. Hansen and E. Beck, The fate and path of assimilation products in the stem of 8-year-old {Scots} pine (Pinus sylvestris {L}.) trees, Trees, 4 (1990), 16-21.  doi: 10.1007/BF00226235. [10] F. Hecht, New Developments in Freefem++, J. Num. Math., 20 (2012), 251-265. [11] L. Horwitz, Some simplified mathematical treatments of translocation in plants, Plant Physio., 33 (1958), 81-93. [12] T. Hölttä, M. Mencuccini and E. Nikinmaa, Linking phloem function to structure: Analysis with a coupled xylem-phloem transport model, J. Theo. Bio., 259 (2009), 325-337. [13] T. Hölttä, T. Vesala, S. Sevanto, M. Perämäki and E. Nikinmaa, Modeling xylem and phloem water flows in trees according to cohesion theory and Münch hypothesis, Trees, 20 (2006), 67-78. [14] W. Hundsdorfer and J. G. Verwer, Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations Springer Series in Comput. Math., 33, Springer, 2003. doi: 10.1007/978-3-662-09017-6. [15] K. H. Jensen, J. Lee, T. Bohr, H. Bruus, N. M. Holbrook and M. A. Zwieniecki, Optimality of the Münch mechanism for translocation of sugars in plants, J. R. Soc. Interface, 8 (2011), 1155-1165.  doi: 10.1098/rsif.2010.0578. [16] A. Kagawa, A. Sugimoto and T. C. Maximov, CO 2 pulse-labelling of photoassimilates reveals carbon allocation within and between tree rings, Plant Cell Env., 29 (2006), 1571-1584. [17] E. M. Kramer, Wood grain pattern formation: A brief review, J. Plant Growth Reg., 25 (2006), 290-301.  doi: 10.1007/s00344-006-0065-y. [18] H.-O. Kreiss and J. Lorenz, Initial-Boundary Value Problems and the Navier-Stokes Equations Classics in Applied Mathematics, SIAM, 2004. doi: 10.1137/1.9780898719130. [19] A. Lacointe and P. E. H. Minchin, Modelling phloem and xylem transport within a complex architecture, Funct. Plant Bio., 35 (2008), 772-780.  doi: 10.1071/FP08085. [20] A. Lang, A model of mass flow in the phloem, Funct. Plant Bio., 5 (1978), 535-546.  doi: 10.1071/PP9780535. [21] P. E. H. Minchin, M. R. Thorpe and J. F. Farrar, A simple mechanistic model of phloem transport which explains sink priority, Journal of Experimental Botany, 44 (1993), 947-955.  doi: 10.1093/jxb/44.5.947. [22] E. Münch, Die Stoffbewegungen in der Pflanze Jena, Gustav Fischer, 1930. [23] K. A. Nagel, B. Kastenholz, S. Jahnke, D. van Dusschoten, T. Aach, M. Mühlich, D. Truhn, H. Scharr, S. Terjung, A. Walter and U. Schurr, Temperature responses of roots: Impact on growth, root system architecture and implications for phenotyping, Funct. Plant Bio., 36 (2009), 947-959.  doi: 10.1071/FP09184. [24] E. M. Ouhabaz, Analysis of Heat Equations on Domains London Math. Soc. Monographs Series, Princeton University Press, 2005. [25] S. Payvandi, K. R. Daly, K. C. Zygalakis and T. Roose, Mathematical modelling of the phloem: The importance of diffusion on sugar transport at osmotic equilibrium, Bull. Math Biol., 76 (2014), 2834-2865.  doi: 10.1007/s11538-014-0035-7. [26] S. Pfautsch, J. Renard, M. G. Tjoelker and A. Salih, Phloem as capacitor: Radial transfer of water into xylem of tree stems occurs via symplastic transport in ray parenchyma, Plant Physio., 167 (2015), 963-971.  doi: 10.1104/pp.114.254581. [27] O. Pironneau and M. Tabata, Stability and convergence of a Galerkin-characteristics finite element scheme of lumped mass type, Int. J. Num. Meth. Fluids, 64 (2000), 1240-1253.  doi: 10.1002/fld.2459. [28] G. E. Phillips, J. Bodig and J. Goodman, Flow grain analogy, Wood Sci., 14 (1981), 55-64. [29] R. J. Phillips and S. R. Dungan, Asymptotic analysis of flow in sieve tubes with semi-permeable walls, J. Theor. Biol., 162 (1993), 465-485.  doi: 10.1006/jtbi.1993.1100. [30] D. Rotsch, T. Brossard, S. Bihmidine, W. Ying, V. Gaddam, M. Harmata, J. D. Robertson, M. Swyers, S. S. Jurisson and D. M. Braun, Radiosynthesis of 6'-Deoxy-6'[18F]Fluorosucrose via automated synthesis and its utility to study in vivo sucrose transport in maize (Zea mays) leaves PLoS ONE 10 (2015), e0128989. doi: 10.1371/journal.pone.0128989. [31] D. Sellier and J. J. Harrington, Phloem transport in trees: A generic surface model, Eco. Mod., 290 (2014), 102-109.  doi: 10.1016/j.ecolmodel.2013.11.021. [32] D. Sellier, M. J. Plank and J. J. Harrington, A mathematical framework for modelling cambial surface evolution using a level set method, Annals Bot., 108 (2011), 1001-1011.  doi: 10.1093/aob/mcr067. [33] R. Spicer, Symplasmic networks in secondary vascular tissues: Parenchyma distribution and activity supporting long-distance transport, J. Exp. Bot., 65 (2014), 1829-1848.  doi: 10.1093/jxb/ert459. [34] J. F. Swindells, C. F. Snyder, R. C. Hardy and P. E. Golden, Viscosities of sucrose solutions at various temperatures: Tables of recalculated values, NBS Circular 440 (1958). [35] M. V. Thompson and N. M. Holbrook, Application of a single-solute non-steady-state phloem model to the study of long-distance assimilate transport, J. Theo. Bio., 220 (2003), 419-455.
Schematic representation of a tree and the layered organisation of its secondary tissues: phloem, vascular cambium and xylem (bark not shown). Flows of water/solute in a phloem element also shown.
Sucrose concentration (C), hydrostatic pressure (P) and water flux (J) in the phloem as simulated with the model of [35] (solid line) and with the proposed model (dashed line)
Maximum of the relative error obtained by comparison with the approximated solution for $\Delta t = 0.1$s, $\Delta x = 0.1$mm after 12 hours.
Effects of model simplications on the axial water flux at an early (10 minutes), intermediate (1h10) and late stage (24h). The considered simplifications are a constant viscosity ($\mu = \mu_0$, dashed line); neglecting the partial molal volume of sucrose ($V_s =0$, dash-dot line); initial geometry ($e = e_0$, dotted line); all simplifications combined (grey, solid line); no simplifications (black, solid line).
Mesh reconstruction of Te Matua Ngahere from a photograph (credit: D. Sellier).
Phloem pressure (P), sucrose concentration (C) and water flux (J) distributions on a Te Matua Ngahere after 12 hours.
Phloem pressure (P), sucrose concentration (C) and water flux (J) distributions on an orthotropic plate after 12 hours.
Phloem pressure (P), sucrose concentration (C) and water flux (J) distributions in a 3D fork after 12 hours.
 Algorithm 1 Semi-implicit scheme Given $P_0, C_0$ and $N\in \mathbb{N}^*$. For $n=1$ to $N$do $\langle \frac{C^{n+1/2}-C^n}{\Delta t}, \varphi_1 \rangle - \langle \left( \frac{k}{\mu} \nabla P^n \right) \cdot \nabla C^n, \varphi_1 \rangle - \langle \frac{U^n}{e}, \varphi_1 \rangle = 0 \\ \breve{C}^{n+1/2}=\log(C^{n+1/2}) \\ \langle \frac{\breve{C}^{n+1}-\breve{C}^{n+1/2}}{\Delta t}, \varphi_2 \rangle + \langle \frac{k}{\mu} \nabla P^{n+1}, \nabla \varphi_2 \rangle + \int_{\partial \Omega} \frac{k}{\mu} \partial_n P^{n+1} \varphi_2 = 0 \\ \langle \frac{P^{n+1}-P^n}{\Delta t}, \varphi_3 \rangle + \langle \frac{E}{\mu} k \nabla P^{n+1}, \nabla \varphi_3 \rangle - \int_{\partial \Omega} \frac{E}{\mu} k \partial_n P^{n+1} \varphi_3 \\ - \langle \frac{E L_R}{e} (\psi - P^n + RT C^n, \varphi_3 \rangle - \langle \frac{V_s U^n}{e}, \varphi_3 \rangle = 0 \\ C^{n+1} = \exp(\breve{C}^{n+1}).$
 Algorithm 1 Semi-implicit scheme Given $P_0, C_0$ and $N\in \mathbb{N}^*$. For $n=1$ to $N$do $\langle \frac{C^{n+1/2}-C^n}{\Delta t}, \varphi_1 \rangle - \langle \left( \frac{k}{\mu} \nabla P^n \right) \cdot \nabla C^n, \varphi_1 \rangle - \langle \frac{U^n}{e}, \varphi_1 \rangle = 0 \\ \breve{C}^{n+1/2}=\log(C^{n+1/2}) \\ \langle \frac{\breve{C}^{n+1}-\breve{C}^{n+1/2}}{\Delta t}, \varphi_2 \rangle + \langle \frac{k}{\mu} \nabla P^{n+1}, \nabla \varphi_2 \rangle + \int_{\partial \Omega} \frac{k}{\mu} \partial_n P^{n+1} \varphi_2 = 0 \\ \langle \frac{P^{n+1}-P^n}{\Delta t}, \varphi_3 \rangle + \langle \frac{E}{\mu} k \nabla P^{n+1}, \nabla \varphi_3 \rangle - \int_{\partial \Omega} \frac{E}{\mu} k \partial_n P^{n+1} \varphi_3 \\ - \langle \frac{E L_R}{e} (\psi - P^n + RT C^n, \varphi_3 \rangle - \langle \frac{V_s U^n}{e}, \varphi_3 \rangle = 0 \\ C^{n+1} = \exp(\breve{C}^{n+1}).$
Description of parameters employed in the model. Numerical values correspond to the initial values used in section 4.1.
 Symbol Description Value Units $L_R$ Radial hydraulic conductivity $1.57\times 10^{-13}$ m Pa$^{-1}$ s$^{-1}$ $\psi$ Xylem hydrostatic pressure $0$ Pa $R$ Gas constant $8.31$ J mol$^{-1}$ K$^{-1}$ $T$ Temperature $293$ K $V_s$ Partial molal volume of sucrose $2.15\times 10^{-4}$ m$^{3}$ mol$^{-1}$ $k_L$ Longitudinal permeability $9.28\times 10^{-12}$ m$^{2}$ $k_T$ Tangential permeability $9.28\times 10^{-13}$ m$^{2}$ $e$ Phloem thickness $7.5\times 10^{-6}$ m $E$ Phloem Young's modulus $1.7\times 10^{7}$ Pa $\widetilde{U}$ Loading rate $3.375\times 10^{-6}$ mol m$^{-2}$ s$^{-1}$ $C^*$ Reference sucrose concentration $500$ mol m$^{-3}$ $\mu$ Viscosity $10^{-3}$ at $C=0$ Pa s
 Symbol Description Value Units $L_R$ Radial hydraulic conductivity $1.57\times 10^{-13}$ m Pa$^{-1}$ s$^{-1}$ $\psi$ Xylem hydrostatic pressure $0$ Pa $R$ Gas constant $8.31$ J mol$^{-1}$ K$^{-1}$ $T$ Temperature $293$ K $V_s$ Partial molal volume of sucrose $2.15\times 10^{-4}$ m$^{3}$ mol$^{-1}$ $k_L$ Longitudinal permeability $9.28\times 10^{-12}$ m$^{2}$ $k_T$ Tangential permeability $9.28\times 10^{-13}$ m$^{2}$ $e$ Phloem thickness $7.5\times 10^{-6}$ m $E$ Phloem Young's modulus $1.7\times 10^{7}$ Pa $\widetilde{U}$ Loading rate $3.375\times 10^{-6}$ mol m$^{-2}$ s$^{-1}$ $C^*$ Reference sucrose concentration $500$ mol m$^{-3}$ $\mu$ Viscosity $10^{-3}$ at $C=0$ Pa s
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