Homogenization of a system of elastic and reaction-diffusion equations modelling plant cell wall biomechanics

Abstract

Abstract. 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 reactiondiffusion and ordinary differential equations. Using homogenization techniques (two-scale convergence and periodic unfolding methods) we derive a macroscopic model for plant cell wall biomechanics.

Homogenization of Biomechanical Models for Plant Tissues

“…For the analysis presented here the diffusion term in the equations for calciumpectin cross-link density is important. However, the same results can be obtained if one assumes that calcium-pectin cross-links do not diffuse and that the reaction terms in equations for pectin, calcium, and calcium-pectin cross-links depend on a local average of the deformation gradient, reflecting the fact that in a dense pectin network mechanical forces have a nonlocal effect on the calcium-pectin chemistry; see [39].…”

“…For simplicity we assume that there is a constant concentration of PME enzyme in the cell wall. By simple modifications of the analysis considered here, the same results can be obtained for a generalized model including the dynamics of PME and chemical reactions between PME and pectin; see [39] for the derivation of the corresponding system of equations.…”

“…Homogenization of the microscopic model for plant cell wall biomechanics, composed of equations of linear elasticity and reaction-diffusion equations for chemical processes, has been studied in [39].…”

“…We assume that as the result of the breakage of a calcium-pectin crosslink by mechanical stresses we obtain one calcium molecule and two galacturonic acid monomers of demethylesterified pectin. A detailed derivation of the chemical reaction term g e is given in [39]. See also Remark 2.3 for the detailed form of the reaction terms.…”

“…Thus assuming a constant temperature, the calcium-pectin chemistry can be described as a reaction between calcium molecules and pectins, where the breakage of cross-links depends on the deformation gradient of the cell walls. Hence we assume that the cross-links are disturbed by the mechanical stresses in cell walls and middle lamella; see [39] for a detailed description of the modelling of the calcium-pectin chemistry. A similar approach was used in [41] to model a chemically mediated mechanical expansion of the cell wall of a pollen tube.…”