In this paper we present a derivation and multiscale analysis of a mathematical model for plant cell wall biomechanics that takes into account both the microscopic structure of a cell wall coming from the cellulose microfibrils and the chemical reactions between the cell wall’s constituents. Particular attention is paid to the role of pectin and the impact of calcium-pectin cross-linking chemistry on the mechanical properties of the cell wall. We prove the existence and uniqueness of the strongly coupled microscopic problem consisting of the equations of linear elasticity and a system of reaction-diffusion and ordinary differential equations. Using homogenization techniques (two-scale convergence and periodic unfolding methods) we derive a macroscopic model for plant cell wall biomechanics.
Mots clés : Homogenization, two-scale convergence, periodic unfolding method, elasticity, reaction-diffusion equations, plant modelling
@article{M2AN_2016__50_2_593_0, author = {Ptashnyk, Mariya and Seguin, Brian}, title = {Homogenization of a system of elastic and reaction-diffusion equations modelling plant cell wall biomechanics}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {593--631}, publisher = {EDP-Sciences}, volume = {50}, number = {2}, year = {2016}, doi = {10.1051/m2an/2015073}, mrnumber = {3482556}, zbl = {1342.35031}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2015073/} }
TY - JOUR AU - Ptashnyk, Mariya AU - Seguin, Brian TI - Homogenization of a system of elastic and reaction-diffusion equations modelling plant cell wall biomechanics JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2016 SP - 593 EP - 631 VL - 50 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2015073/ DO - 10.1051/m2an/2015073 LA - en ID - M2AN_2016__50_2_593_0 ER -
%0 Journal Article %A Ptashnyk, Mariya %A Seguin, Brian %T Homogenization of a system of elastic and reaction-diffusion equations modelling plant cell wall biomechanics %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2016 %P 593-631 %V 50 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2015073/ %R 10.1051/m2an/2015073 %G en %F M2AN_2016__50_2_593_0
Ptashnyk, Mariya; Seguin, Brian. Homogenization of a system of elastic and reaction-diffusion equations modelling plant cell wall biomechanics. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 2, pp. 593-631. doi : 10.1051/m2an/2015073. http://www.numdam.org/articles/10.1051/m2an/2015073/
An extension theorem from connected sets, and homogenization in general periodic domains. Nonlin. Anal. Theory Methods Appl. 18 (1992) 481–496. | DOI | MR | Zbl
, , and ,Lp bounds of solutions of reaction-diffusion equations. Commun. Partial Differ. Eq. 4 (1976) 827–868. | DOI | MR | Zbl
,Homogenization and two-scale convergence. SIAM J. Math. Anal. 23 (1992) 1482–1518. | DOI | MR | Zbl
,G. Allaire, Shape Optimization by the Homogenization Method. Springer (2002). | MR | Zbl
A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures. North Holland (1978). | MR | Zbl
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer (2010). | MR | Zbl
The strain energy function of an ideal plant cell wall. J. Theoret. Biol. 163 (1993) 77–97. | DOI
,Homogenization approach to water transport in plant tissues with periodic microstructures. Math. Model. Nat. Phenom. 8 (2013) 80–111. | DOI | MR | Zbl
and ,P.-G. Ciarlet, Mathematical elasticity. Volume I: Three-dimensional elasticity. North-Holland (1988). | MR
Another approach to linear elasticity and Korn’s inequality. C.R. Acad. Sci. Paris Ser. I 339 (2004) 307–312. | DOI | MR | Zbl
and ,D. Cioranescu and J. Saint Jean Paulin, Homogenization of reticulated structures. Springer (1999). | MR | Zbl
The periodic unfolding method in homogenization. SIAM J. Math. Anal. 40 (2008) 1585–1620. | DOI | MR | Zbl
, and ,The periodic unfolding method in domains with holes. SIAM J. Math. Anal. 44 (2012) 718–760. | DOI | MR | Zbl
, , , and ,The size of the cellulose microfibril. J. Cell Biol. 17 (1963) 105–109. | DOI
,Growth of the plant cell wall. Nat. Rev. Molec. Cell Biol. 6 (2005) 850–86. | DOI
,Anisotropic elastic properties of cellulose measured using inelastic X-ray scattering. Macromolecules 41 (2008) 9755–9759. | DOI
, , and ,An anisotropic-viscoplastic model of plant cell morphogenesis by tip growth. Int. J. Developmental Biol. 50 (2006) 209–222. | DOI
, , , and ,Identification and characterization of stretch-activated ion channels in pollen protoplasts. Plant Physiol. 135 (2004) 1398–1406. | DOI
and ,A fibre-reinforced fluid model of anisotropic plant cell growth. J. Fluid Mech. 655 (2010) 472–503. | DOI | MR | Zbl
, ,A model of crosslink kinetics in the expanding plant cell wall: Yield stress and enzyme action. J. Theoret. Biol. 307 (2012) 125–136. | DOI | MR | Zbl
, and ,Error estimate and unfolding method for homogenization of a reaction-diffusion system modeling sulfate corrosion. Appl. Anal. 91 (2012) 1129–1154. | MR | Zbl
, and ,Y.C. Fung, Biomechanics: mechanical properties of living tissues. Springer (1993).
Homogenizing the acoustic properties of the seabed: Part I. Nonlin. Anal. 40 (2000) 185–212. | DOI | MR | Zbl
and ,M.-E. Gurtin, E. Fried and L. Anand, The Mechanics and Thermodynamics of Continua. Cambridge University Press (2010). | MR
Diffusion of polymers through polyacrylamide gels. Polymer 29 (1988) 1058–1063. | DOI
, , ,Diffusion, convection, adsorption, and reaction of chemicals in porous media. J. Differ. Eq. 92 (1991) 199–225. | DOI | MR | Zbl
and ,Homogenization limit of a model system for interaction of flow, chemical reactions, and mechanics in cell tissues. SIAM J. Math. Anal. 43 (2011) 1390–1435. | DOI | MR | Zbl
, and ,V.-V. Jikov, S.-M. Kozlov and O.-A. Oleinik, Homogenization of Differential Operators and Integral Functionals. Springer (1994). | MR | Zbl
Hydration effects on spacing of primary-wall cellulose microfibrils: a small angle X-ray scattering study. Cellulose 14 (2007) 401–408. | DOI
, , and ,A. Korn, Über einige ungleichungen, welche in der theorie del elastichen und elektrishen schwingungen eine rolle spielen. Bullettin Internationale, Cracovie Akademie Umiejet, Classe des sciences mathématiques et naturelles (1909) 705–724. | JFM
Regulator or driving force? The role of turgor pressure in oscillatory plant cell growth. PLoS One 6 (2011) e18549. | DOI
, and ,O. Ladyzhenskaya, V. Solonnikov and N. Ural’ceva, Linear and quasilinear equations of parabolic type. American Mathematical Society (1968). | MR
J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires. Dunod (1969). | MR | Zbl
DOLFIN: automated finite element computing. ACM Trans. Math. Software 37 (2010). | DOI | MR | Zbl
and ,A. Logg, K.-A. Mardal, G.N. Wells et al., Automated solution of differential equations by the finite element method. Springer (2012). | Zbl
A.J. Majda and A.L. Bertozzi, Vorticity and Incompressible Flow. Cambridge Texts Appl. Math. (2001). | MR | Zbl
Derivation of a macroscopic receptor-based model using homogenisation techniques. SIAM J. Math. Anal. 40 (2008) 215–237. | DOI | MR | Zbl
and ,Homogenization of a viscoelastic equations with non-periodic coefficients. Proc. Roy. Soc. Edinburgh: Sect. A Math. 106 (1987) 143–160. | DOI | MR | Zbl
,On the interface law between a deformable porous medium containing a viscous fluid and an elastic body. Math. Models Methods Appl. Sci. 22 (2012) 1250031. | DOI | MR | Zbl
and ,F. Murat and L. Tartar, H-convergence, in Topics in the Mathematical Modelling of Composite Materials. Vol. 31 of Progr. Nonlin. Differ. Equ. Appl. Birkhäuser Boston, Boston, MA (1997) 21–43. | MR | Zbl
J. Necas, Les méthodes directes en théorie des équations elliptiques. Academie, Prague (1967). | MR | Zbl
A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20 (1989) 608–623. | DOI | MR | Zbl
,O. Oleinik, A.-S. Shamaev and G.-A. Yosifian, Mathematical problems in Elasticity and Homogenization. North Holland (1992). | MR | Zbl
Optimisations for quadrature representations of finite element tensors through automated code generation. ACM Trans. Math. Software 37 (2010). | DOI | MR
and ,Turgor and cell expansion: beyond the Lockhart equation. Aust. J. Plant Physiol. 19 (1992) 565–576.
and ,Pectin-induced changes in cell wall mechanics underlie organ initiation in Arabidopsis. Curr. Biol. 21 (2011) 1720–1726. | DOI
, , , , and ,A role for pectin de-methylesterification in a developmentally regulated growth acceleration in dark-grown Arabidopsis hypocotyls. New Phytol. 188 (2010) 726–739. | DOI
, , , , , , , , , , , and ,Calcium deprivation disrupts enlargement of Chara corallina cells: further evidence for the calcium pectate cycle. J. Exp. Bot. 63 (2012) 1–6.
and ,Derivation of a macroscopic model for nutrient uptake by a single branch of hairy-roots. Nonlin. Anal.: Real World Appl. 11 (2010) 4586–4596. | DOI | MR | Zbl
,M. Ptashnyk and B. Seguin, Periodic homogenization and material symmetry in linear elasticity, (2015). | arXiv | MR
Invariant sets for strongly coupled reaction-diffusion systems under general boundary conditions. Arch. Rational Mech. Anal. 108 (1989) 281–291. | DOI | MR | Zbl
,Chemically mediated mechanical expansion of the pollen tube cell wall. Biophys. J. 101 (2011) 1844–1853. | DOI
, and ,E. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory. Springer (1980). | MR | Zbl
J. Simon, Compact sets in the space . Ann. Mat. Pure Appl. (IV) CXLVI (1987) 65–96. | MR | Zbl
J. Smoller, Shocke Waves and Reaction-Diffusion Equations. Springer (1994). | MR | Zbl
Toward a systems approach to understanding plant cell walls. Science 306 (2004) 2206. | DOI
, , , , , , , , , , and ,Structure of cellulose microfibrils in primary cell walls from collenchyma. Plant Physiol. 161 (2013) 465–476. | DOI
, , , , , , , and ,A model of cell wall expansion based on thermodynamics of polymer networks. Biophys. J. 75 (1998) 2240–2250. | DOI
and ,The pathways of calcium movement to the xylem. J. Exp. Bot. 52 (2001) 891–899. | DOI
,Growth control by cell wall pectins. Protoplasma 249 (2012) 169–175. | DOI
and ,Growth control and cell wall signaling in plants. Ann. Rev. Plant Biol. 63 (2012) 381–407. | DOI
, and ,Plant cell wall homeostasis is mediated by Brassinosteroid feedback signaling. Curr. Biol. 22 (2012) 1732–1737. | DOI
, , , and ,Direction turgor pressure measurements in individual leave cells of Tradescantia virginiana. Planta 148 (1980) 445–453. | DOI
, and ,Material properties of concentrated pectin networks. Carbohyd. Res. 339 (2004) 1317–1322. | DOI
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