# 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
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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.
] (solid line) and with the proposed model (dashed line)">Figure 2.  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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