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The spatial dynamics of a zebrafish model with cross-diffusions
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 |
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[
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.
|
show all references
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.
|







Algorithm 1 Semi-implicit scheme |
Given |
For |
Algorithm 1 Semi-implicit scheme |
Given |
For |
Symbol | Description | Value | Units |
| Radial hydraulic conductivity | | m Pa |
| Xylem hydrostatic pressure | | Pa |
| Gas constant | | J mol |
| Temperature | | K |
| Partial molal volume of sucrose | | m |
| Longitudinal permeability | | m |
| Tangential permeability | | m |
| Phloem thickness | | m |
| Phloem Young's modulus | | Pa |
| Loading rate | | mol m |
| Reference sucrose concentration | | mol m |
| Viscosity | | Pa s |
Symbol | Description | Value | Units |
| Radial hydraulic conductivity | | m Pa |
| Xylem hydrostatic pressure | | Pa |
| Gas constant | | J mol |
| Temperature | | K |
| Partial molal volume of sucrose | | m |
| Longitudinal permeability | | m |
| Tangential permeability | | m |
| Phloem thickness | | m |
| Phloem Young's modulus | | Pa |
| Loading rate | | mol m |
| Reference sucrose concentration | | mol m |
| Viscosity | | Pa s |
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