Model-based approach

2. Model-based approach • Purpose • Improve understanding • Optimization • Control • Macroscale approach • Mannapperuma et al. (1991); Lammertyn et al. (2003) • Geometry: intact fruit • Gas transport coupled with respiration kinetics

3. Gas transport properties Effective parameters Deff Liquid phase 1-ε Ci,l ε Ci,g Gas phase • Macroscale approach • Volume-averaged parameter • Measurement • Biological variability • DCO2 > DO2 • Anisotropic diffusivity • Apparant values

4. Microscopic overview of tissue • Parenchyma tissue structure • Grouped cells • Random distribution of cells and pores • Cell wall • Plasma membranes • Transport phenomena • Geometry required • Two phases • Gas • Liquid • Cell membrane • Passive transport • Active transport • Intra-cellular enzymatic reactions Plant Cell wall (Albert et al., 1994)

5. Objective • To verify the applicability of a microscale modelling approach to the gas transport at tissue level in a multiscale framework • To quantify the pathways of gas transport in relation to the microstructure of fruit tissue

6. Microscale modelling of gas transport in pears Q. Tri Ho, Hibru K. Mebatsion, Bert E. Verlinden, Pieter Verboven, Stefan Vandewalle and Bart M. Nicolaï

7. Geometry model • Light microscopy images • Parenchyma tissues of ‘Conference’ pear • Resolution 1pixel~0.735µm • Digitization of image • Geometry model generation (Mebatsion et al, 2006) • Ellipse tessellation algorithm

8. TEM image of Conference pear Cells • Ellipse tesselation Cell Intercellular space Cell wall

9. Concept of gas transport Liquid Pore O2,g ADP +Pi O2,l ATP synthase Mitochondrion ATP Cytosol CO2,l Work ATP CO2,g HCO3- Gas exchange of fruit Air filled intercellular space At the interface Intra-cell

10. Model of O2 transport in tissue • Assumption • Cell wall was assumed gas phase with effective diffusivity DO2,w • Passive gas transport through cell membrane • Henry’s law at the inter-phase • Model equation (Fick’s second law of diffusion) • Pore, cell, cell wall (i= pore, cell, cellwall) • O2 consumption at intra-cell • Michaelis-Menten reaction • Flux through cell membrane • with C*O2,l the equilibrated O2 concentration outside of the membrane

11. Model of CO2 transport in tissue CO2,l CO32- HCO3- Fraction of spieces H2CO3 pH • Fraction of CO2 spieces in liquid phase

12. Model of CO2 transport in tissue O2 Mitochondrion k1 H2CO3 CO2,l + H2O k2 ka1 HCO3- +H+ CO2,g • In pore, cell wall: the same as Eq of O2 • In cellular liquid phase

13. Physical parameters of microscale model (1)Lide (1996), (2) Gunning and Steer (1996), (3) Uchida et al. (1992), (4)Geers and Gros (2000), (5) Jolly (1991)

14. Numerical solution C2 Ltissue C1 • Meshing • 125050 elements • Solution • Finite element method • Comsol 3.3 (Comsol AB, Stockholm) • Estimation of Dtissue, eff • Steady state • Boundary condition • Side 1: C1 ; Side 2: C2 Isolated boundary

15. Results • Simulation of O2 transport

16. Simulation of CO2 transport

17. Effect of vacuole in the model • Occupy 30-90% of cell volume • Storage function • Maintain internal acidic pH • Lumped pH intra-cell model • Constant pH in the cell (DH+=9.3×10-9 m2/s, Geers and Gros, 2000) • pHintra-cell=5 • Model with vacuole • Constant pH in the cytoplasm and vacuole • Regulation of pH in the cytoplasm and vacuole • pHcytoplasm=7, pHvacuole=4.82

18. Lumped pH intra-cell versus intra-cell including vacuole

19. Comparison of lumped pH intracellular model and intra-cell including vacuole

20. Estimated O2 diffusivity of pear tissue DO2,cell wall =5×10-9 m2/s Cell wall thickness= 0.73 µm (TEM, Mebatsion 2006, unpublished data) 9 different geometries

21. Estimated CO2 diffusivity of pear tissue Dw, CO2cell= 5×10-9m2/s Cell wall thickness= 0.73 µm (TEM, Mebatsion 2006, unpublished data) 9 different geometries

22. Current work • Toward 3D model • Digitization vs. mapping X-ray image Digitial geometry Mapping parameters 13853 elements 16896 elements

23. Solution using mapping parameters Solution based on digitial geometry DO2=1.29e-9 m2/s Solution using mapping parameters DO2=1.33e-9 m2/s

24. 3D geometry based on mapping • Geometry information • Synchrotron X-ray tomography • Resolution 1 pixel~1.4µm • Pore distribution

25. 3D geometry based on mapping • Geometry meshing • Lagrange-linear cube • Edge dim ~ 2.8µm • Nr of elements: 3968750 (3968750 DOFs) • Solving • Comsol script 1.1 (Comsol 3.3) • GMRES • HPC Clusters • CPU freq: 2000-2400 MHz • Used Mem: 5792736kb • Processing time: 12.25h

26. 3D geometry based on mapping • Postprocessing • Scripts written in Matlab • Running on cluster • CPU freq: 2000-2400 MHz • Used Mem: 6.5 Mb

27. Primary 3D simulation CO2,gas (mol/m3) CO2,l (mol/m3) • O2 diffusion in tissue O2 gas concentration O2 liquid concentration

28. Primary 3D simulation CO2,gas (mol/m3) • O2 diffusion in different tissues Cortex Epidermis Subepidermis

29. 3D diffusivity results

30. Conclusions • A model was presented to study gas transport at the microscale • O2 mainly transports in the gas phase of intercellular space and cell wall networks • CO2 transfer in both gas and liquid phase • Macroscopic diffusivity was estimated using microscale simulations • Future: 3D simulations based on synchrotron X-ray tomography 500 micron

31. Thank you