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

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.


Introduction
The main feature of plant cells are their walls, which must be strong to resist high internal hydrostatic pressure (turgor pressure) and flexible to permit growth.The primary wall of a plant cell consists mainly of oriented cellulose microfibrils, pectin, hemicellulose, structural proteins, and water, Fig. 1.The cross-linked pectin network is the main composite of the middle lamella which joins individual cells together and ensures the continuity of symplast by building plasmodesmata.The main force for cell elongation (turgor pressure) acts isotropically, and so it is the microscopic structure of the cell wall which determines the anisotropic growth of plant cells and tissue.The orientation of microfibrils, their length, high tensile strength, and interaction with wall matrix macromolecules strongly influences the wall's stiffness.Hemicelluloses form hydrogen bonds with the surface of cellulose microfibrils which may strengthen the cell wall by creating a microfibril-hemicelluluse network, but also weaken the mechanical strength of cell walls by preventing cellulose aggregation [52].Pectin can be modified by the enzyme pectin methylesterase (PME), which removes methyl groups by breaking ester bonds.The de-esterified pectin is able to form calcium-pectin cross-links, and so stiffen the cell wall and reduce its expansion, see e.g.[56,60].For irreversible deformation, the deposition of new wall materials and the loosening of the cell wall through the breaking of the load-bearing cross-links between microfibrils, pectin, and hemicellulose are required, see e.g.[14,51].
There are a number of models of a plant cell wall, each of which focuses on different aspects of its structure.Mathematical models of the cellulose-hemicellulose network were proposed in [18,43].The account of the microstructure of a cell wall has been addressed by considering the anisotropic yield stresses or by distinguishing between the free energies related to the elasticity of (i) macromolecules and hydrogen bonds or (ii) the matrix and microfibrils [8,16,54].The influence of the microfibril orientation and the external torque on the expansion process has been considered in [19].The effect of changes in the chemical configurations of pectins (methylesterified and demethylesterified) and the calcium concentration on the viscous behavior of a cell wall in a pollen tube has been analyzed in [32,48].
The biomechanics of plant cell walls is determined by the cell wall microstructure, given by microfibrils, and the physical properties of the cell wall matrix.It is supposed that calciumpectin cross-linking chemistry is one of the main regulators of cell wall elasticity and extension [57].It has been shown that the modification of pectin by PME and the control of the amount of calcium-pectin cross-links greatly influence the mechanical deformations of plant cell walls [44,45], and the interference with PME activity causes dramatic changes in growth behavior of plant cells and tissues [58].
In this paper we focus on two aspects which have not been considered together before: the influence of the microstructure, associated with the cellulose microfibrils, and the calcium-pectin cross-links on the mechanical properties of plant cell walls.In the microscopic model of cell wall biomechanics derived in this paper, the cell wall microstructure and the dynamics of the formation and dissociation of calcium-pectin cross-links are considered explicitly.
We model the cell wall as a three-dimensional continuum consisting of a polysaccharide matrix embedded with cellulose microfibrils.Within the matrix, we consider the dynamics of the enzyme PME, methylesterfied pectin, demethylesterfied pectin, calcium ions, and calcium-pectin cross-links.The cell wall matrix is assumed to be either a linearly elastic or viscoelastic Kelvin-Voigt material, whereas microfibrils are modelled as an anisotropic linearly elastic material.The interplay between the mechanics and the cross-link dynamics comes in by assuming that the elastic and viscous properties of the matrix depend on the density of the cross-links and that strain or stress within the cell wall can break calcium-pectin cross-links.The strain-or stressdependent opening of calcium channels in the cell plasma membrane is addressed in the flux boundary conditions for calcium ions.The resulting microscopic model is a system of strongly coupled four diffusion-reaction equations, one ordinary differential equation, and the equations of linear elasticity or linear viscoelasticity, depending on which constitutive assumption is used.
In our model we focus on the interactions between the chemical reactions within the cell wall and its deformation and, hence, do not consider the growth of the cell wall.
To analyse the macroscopic behaviour of plant cell walls, comprising a complex microscopic structure and a dynamic cell wall matrix, we rigorously derive a macroscopic biomechanical model for plant cell walls.As there are thousands of microfibrils in a plant cell wall, the derivation of macroscopic equations is also important for effective numerical simulations.The two-scale convergence, e.g.[4,41], and the periodic unfolding method, e.g.[10,11], are applied to obtain the macroscopic equations.In the case of an elastic constitutive law, the resulting macroscopic equations have a similar structure as the microscope ones, whereas for the elastic-viscoelastic model the macroscopic momentum balance equation contains a term that depends on the history of the strain represented by an integral term (fading memory effect).Due to the coupling between the viscoelastic properties and the biochemistry of a plant cell wall, the elastic and viscous tensors depend on space and time.This fact introduces additional complexity in the derivation and in the structure of the macroscopic equations.
A multiscale analysis of the viscoelastic equations with time-independent coefficients was considered previously in [21,23,36,49].Macroscopic equations for scalar elastic-viscoelastic equations with time-independent coefficients were derived in [20] by applying the H-convergence method [38].A microscopic viscoelastic Kelvin-Voigt model with time-dependent coefficients in the context of thermo-viscoelasticity was analyzed in [1].Macroscopic equations were derived by applying the method of asymptotic expansion.A multiscale analysis of microscopic models comprising the equations of linear elasticity for a solid matrix or cells combined with the Stokes equations for the fluid part was considered in [24,28,37].Some previous results on homogenization of problems in linear elasticity can be found in [5,6,29,42,49] (and the references therein).
The main novelty of this paper is twofold: (i) we derive a new model for plant cell wall biomechanics where the mechanical properties and biochemical processes in a cell wall are considered on the scale of its structural elements (on the scale of the microfibrils) and (ii) using homogenisation techniques we obtain a macroscopic model for plant cell wall biomechanics from a microscopic description of the mechanical and chemical processes.This approach allows us to take into account the complex microscopic structure of a plant cell wall and to analyze the impact of the heterogeneous distribution of cell wall structural elements on the mechanical properties and development of plants.The main mathematical difficulty arises from the strong coupling between the equations of linear elasticity or viscoelasticity for cell wall mechanics and the system of reaction-diffusion and ordinary differential equations for the chemical processes in the wall matrix.
The paper is organised as follows.In Section 1 we derive the model for plant cell wall biomechanics.In Section 2 we give the general setting of the microscopic model and show the existence and uniqueness of a weak solution of the model equations by using a fixed point argument.The theory of positively invariant regions [47,50] and the Moser [33] and Alikakos [3] iteration techniques are applied to show the non-negativity and uniform boundedness of solutions of the microscopic model.The multiscale analysis of the model involving elastic deformations of plant cell walls are conducted in Sections 3. The viscoelastic case is analysed in Section 4. Since we assume that only the cell wall matrix exhibits viscoelastic behaviour and microfibrils are elastic, the viscous tensor is zero in the domain occupied by the microfibrils.This fact causes some technical difficulties in the multiscale analysis of the microscopic model.To derive the macroscopic equations for the elastic-viscoelastic model for cell wall biomechanics we first consider perturbed equations by introducing an inertial term.Then, letting the perturbation parameter in the macroscopic model tend to zero, we obtain the effective homogenized equations for the original elastic-viscoelastic model.

Derivation of the mathematical model
In this section a mathematical model for plant cell wall biomechanics is derived.We consider interactions between the mechanical properties of the plant cell wall and the chemical modifications of pectin.We consider the most abundant subclass of pectin, homogalacturonan, which is important for the regulation of plant biomechanics and growth.Homogalacturonan consists of a long linear chain of galacturonic acids.Pectin is deposited into the cell wall in a highly methylestrified state and is modified by the wall enzyme pectin-methylesterase (PME), which removes methyl groups [56].The demethylesterified pectin interacts with calcium ions to produce load bearing cross-links, which reduce cell wall expansion, see e.g.[57].
In the mathematical model a flat section of a cell wall composed of a polysaccharide matrix and cellulose microfibrils is considered.Let Ω be a reference configuration of the plant cell wall.
The domains Ω M and Ω F denote the parts of Ω occupied by the cell wall matrix and microfibrils, respectively.The microscopic structure of Ω is specified in Section 2. We consider five species within the plant cell wall matrix: methylestrified pectin, the enzyme PME, demethylestrified pectin, calcium ions, and calcium-pectin cross-links.The methylestrified pectin consists of a chain of galacturonic acids that are esterified with a methyl group.To form cross-links with calcium ions Ca 2+ , pectin molecules need to have only some of their constituent acids de-estrified, see e.g.[14,56].Thus, when describing the density of pectin in the different states, we refer to the density of the galacturonic acid groups in the different states.
Let n e , n E , n d , n c , and n b denote the number densities of methylestrified pectin acid groups, PME, demethylestrified pectin acid groups, calcium ions, and calcium-pectin cross-links in the reference configuration Ω M , respectively.Let S = {e, E, d, c, b} be an index set and n S will denote all five of the densities.
We assume that the densities n α , with α ∈ S, are changing due to spacial movement, reactions between the species, and external agencies.Thus, the balance equation for n α is given by where r α models the chemical reactions between the species, j α is the flux, and h α is the species supply due to external agencies.The momentum balance for the cell wall reads where T R is the Piola stress and b denote the external body forces, including inertial forces.We consider elastic and viscoelastic behavior of the wall material and assume that the chemical processes in the wall matrix influence the mechanical properties of the cell walls, see e.g.[14,56].First, the constitutive law of linear elasticity, see e.g.[9], for the stress is assumed: where E M (n S ) and E F are elasticity tensors, e(u) = 1 2 (∇u + ∇u ) is the symmetric part of the displacement gradient, and χ A is the characteristic function of a domain A.
It was observed that in addition to elastic deformations, the cell wall matrix exhibits viscous behavior, see e.g.[26].Thus, we also consider a viscoelastic Kelvin-Voigt type constitutive law, see e.g.[22], for the matrix and a linear elastic one for the microfibrils where V M is a viscosity tensor.The interactions between the mechanical properties of the cell wall and the biochemistry of the wall matrix are also reflected in the reactions terms where We consider two scenarios by assuming that the strain or the elastic stress influence the chemical reactions and the dynamics of calciumpectin cross-links [46].Since pectin are long molecules, we assume the nonlocal impact of cell wall mechanics on chemical processes.Thus, strains or stresses within a neighborhood of a point affect the rate of the chemical reactions.The length scale δ is associated with the length of the pectin molecules.The flux of species α is assumed to be determined by Fick's law: where D α is the diffusion coefficient of the species α.
Next, we specify assumptions on the constitutive laws introduced in (3)-( 6) that reflect the physics of the plant cell wall.
The cell wall matrix has the same properties in all directions and, hence, is isotropic, see e.g.Zsivanovits et al. [60].This is expressed mathematically by requiring Using standard representation theorems for isotropic functions, see e.g.Gurtin, Fried, and Anand [25], (7) implies that We assume that calcium-pectin cross-links do not diffuse, i.e.D b = 0. Unlike the matrix, the microfibrils have different elastic properties in different directions, see e.g.[15].For a plant cell wall, the amount of calcium-pectin cross-links plays a decisive role in determining the elastic and viscous properties of the wall matrix, [52,57].Thus, we assume that E M and V M , or, equivalently, µ, λ, η 1 , and η 2 , only depend on n b .
We consider the following four interactions between the species in the matrix: 1.The enzyme PME interacts with methylestrified pectin to form demethylestrified pectin.
3. Demethylestrified pectin and calcium ions bind together to form calcium-pectin cross-links.
4. Under the presence of strain or stress, calcium-pectin cross-links break to yield demethylestrified pectin and calcium ions.
For a detailed discussion of Interactions 1-4, see e.g.[46,56,60].We begin by discussing the reaction term r d , which is decomposed into the sum of three terms: where r eE is the rate of change of the density of demethylestrified pectin acid groups, n d , associated with Interaction 1, r dd is the rate of decay of n d mentioned in Interaction 2, and r f b is the rate of change of n d associated with the formation and breakage of calcium-pectin bonds specified in Interactions 3 and 4.
From Interaction 1, we have r eE = −r e .
We assume that the binding of PME to and dissociation from a pectin acid group are very fast, and that the enzyme PME is not used up during the demethyl-esterification process so that r E = 0. From Interactions 3 and 4 it follows that The factor of a half in front of r f b reflects the fact that two demethylestirified galacturonic acids are needed to form a calcium-pectin cross-link.We assume that where R eE defines the demethyl-esterification reaction between methylestrified galacturonic acid groups and PME and R d > 0 is a decay constant of the demethylestrified pectin.The form of r b is more complicated since there are two different mechanisms associated with it.Interactions between demethylestirified pectin and calcium ions increase the number of cross-links, while strain or stress can break the cross-links.Here, r b is assumed to take the form where R dc models the formation of cross-links through the interactions between demethylesterified pectin and calcium ions, (N δ (e(u))) + = max{N δ (e(u)), 0}, and N δ (e(u) is defined as or for all x ∈ Ω.Having r b depend on the positive part of N δ (e(u)) does not follow from ( 5), but it is consistent with the isotropy assumption.The reason for the choice ( 9) is based on the idea that stretching, rather than compressing, the cross-links will cause them to break.Possible choices for the functions R eE , R dc , and R b are where k eE , k dc,1 , k dc,2 , and k b are positive constants.Due to the high calcium concentration in plant cell walls, we assume saturation kinetics for the density of calcium ions in the reaction term R dc .
Remark.The constitutive laws we consider are consistent with the Second Law of Thermodynamics in that, in the elastic case, Maxwell's relation holds, where µ α is the chemical potential for species α, which depends on n S , and e(u), see [25].
The chemical potential is related to the flux j α through the relation To obtain the flux used in this section, set In the viscoelastic case, Maxwell's relation does not need to hold.
The environment can effect the cell wall in two different ways: through external influences and boundary conditions.The effects of the supply of species h α and external body forces b, including inertial terms, are neglected so The boundary ∂Ω of Ω is decomposed into three disjoint surfaces: Γ I , Γ E , and ∂Ω \ (Γ I ∪ Γ E ).Γ I is the part of ∂Ω in contact with the interior of the cell and Γ E is the part of ∂Ω in contact with the middle lamella.For the rest of the paper, ν denotes the exterior unit-normal to whatever surface is under discussion.On Γ, ν points away from Ω M .PME, produced in the Golgi apparatus of a plant cell, is deposited into the cell wall, and diffuses through the cell wall into the middle lamella.PME can also diffuse back into the cell to degrade.Thus, we assume that the enzyme PME can enter or leave the cell wall through Γ I but can only leave the wall through Γ E .To account for mechanisms controlling the amount of PME in a cell wall, [57] we assume that the inflow of PME into the cell wall depends on the total amount of methylestrified pectin within the wall, which leads to the boundary fluxes where ζ E and γ E are non-negative constants.Methylesterified pectin is produced by the cell and then transported into the cell wall through Γ I , e.g.[56].To account for mechanisms controlling the amount of pectin in the cell wall, we assume that the inflow of new methylestrified pectin decreases with an increasing amount of methylestrified pectin in the wall.Methylestrified pectin can leave the wall through Γ E and enter the middle lamella.Thus, on Γ I , where γ e is a non-negative constant.We assume an outflow of demethylesterified pectin from the cell wall into the middle lamella: Calcium ions may enter or leave the cell wall through both Γ I and Γ E , but the flow of calcium through Γ I is controlled by stretch activated calcium channels in plasma membrane, see e.g.Dutta and Robinson [17] and White [55].Thus, the flow of calcium through Γ I is assumed to depend on the local average of the strain or elastic stress and on the density of calcium, so that where with non-negative constants γ c,i and ζ c,i , where i = 1, 2, and N δ (e(u)) is given in (10) or (11).Similar to (9), we assume that the right-hand side of ( 14) 2 depends on the positive part of N δ (e(u)).
The traction boundary-conditions come from the constant, positive turgor pressure p I within the cell and the traction force f E , caused by surrounding cells.We consider zero-flux boundary conditions on the surface of the microfibrils: E, d, c.
On ∂Ω \ (Γ I ∪ Γ E ), consisting of four flat surfaces in two sets of parallel surfaces, periodic boundary conditions for the densities and displacement are imposed.
Possible choices for the functions that determine the boundary conditions are where β E , β e , ζ e , and p E are positive constants.

Microscopic model
In this section we summarize the model equations derived in Section 1 and prove the existence and uniqueness of a weak solution of the microscopic model.Here we assume that the stress in the plant cell wall matrix is determined by the equations of linear elasticity.The viscoelastic behavior of the cell wall matrix will be analyzed in Section 4.
We choose a coordinate system (x 1 , x 2 , x 3 ) so that Ω = (0, a 1 ) × (0, a 2 ) × (0, a 3 ), where a i , i = 1, 2, 3, are positive numbers and the microfibrils are oriented in the x 3 -direction, see Fig. 2(a).The part of ∂Ω on the exterior of the cell wall, which is in contact with the middle lamella, is given by Γ E = {a 1 } × (0, a 2 ) × (0, a 3 ), and the interior boundary Γ I of the cell wall is given by To determine the microscopic structure of the cell wall, we consider the cuboid Y = (0, 1) 2 × (0, a 3 ) with Y = Y M ∪ Y F , where Y M represents the cell wall matrix and Y F corresponds to a microfibril.We define Y F ⊂ Y as a cylinder with axis parallel to the x 3 -axis, whose ends lie in the boundary of Y , and the centers of the ends of Y F are centrally located in the faces they lie in, see Fig. 2(b).We also denote We assume that the microfibrils in the cell wall are distributed periodically and have a diameter on the order of ε, where the small parameter ε is defined as the ratio of the size of the microstructure to the thickness of the cell wall.The microfibrils of a plant cell wall are about 3 nm in diameter and are separated by a distance of about 6 nm, see e.g.[13,30,53], whereas the thickness of a plant cell wall is of the order of a few micrometers.The domain denotes the part of Ω occupied by the microfibrils, while corresponds to the part of Ω occupied by the cell wall matrix.The boundary between the matrix and the microfibrils is denoted by Summarizing the balance equations and boundary conditions introduced in Section 1, the microscopic model for the number densities of all species reads where N δ (e(u ε )) is defined in (10) or (11).The boundary conditions are given by and where ζ E and γ α , for α = e, E, d, are non-negative constants.We consider zero Neumann boundary conditions on Γ ε and assume periodicity in x 2 and x 3 The initial densities are given by The displacement satisfies the equations of linear elasticity The elasticity tensor is defined as We adopt the following notations: By Korn's second inequality, the L 2 -norm of the strain defines a norm on W(Ω), see [31,42].For a given measurable set A we consider the notation where the product of φ 1 and φ 2 is the scalar-product if they are vector valued, and by ψ 1 , ψ 2 V,V we denote the dual product between 1.The diffusion constants D α are positive, for α = e, E, d, c, and the constants ζ E , R d , γ c,j , ζ c,j , and γ α are non-negative, for α = e, E, d and j = 1, 2.
Remark.Assumption 1.9 is only required in the case Definition 2.1.A weak solution of the microscopic model ( 15)-( 20) are functions E, d, c, and satisfy the equations and for all ψ ∈ L 2 (0, T ; W(Ω)).Furthermore, n ε α satisfy the initial conditions and the estimate where the constant C 1 is independent of ε and n ε b,i , i = 1, 2. Additionally, we have where the constant C 2 is independent of ε.
Proof.Due to the assumptions on E ε , see Assumprion 1.8, the solutions u ε i of ( 25) exist by the Lax-Milgram Theorem.Taking u ε i as a test function in the weak formulation of (25) and using the properties of E ε and the non-negativity of n ε b,i we obtain for a.a.t ∈ (0, T ), where σ > 0 is arbitrary and C σ is independent of ε.Applying the second Korn inequality for u ε i ∈ L ∞ (0, T ; W(Ω)) and the trace estimate in H 1 (Ω), and choosing σ > 0 sufficiently small yield the estimate (26).
Taking u ε 1 − u ε 2 as a test function in the weak formulation of (25) for i = 1, 2 and subtracting the results imply Using the coercivity and regularity of E(n ε b,1 , x) and the boundedness of , yields the inequality (27).
Using classical extension results [2,12], we obtain the following lemma.

Lemma 2.3. There exists an extension n
, where α = e, E, d, c, and the constant µ 1 depends only on Y and Y M .
Remark.Notice that the microfibrils do not intersect the boundaries Γ I and Γ E , and near the boundaries (∂Ω \ (Γ I ∪ Γ E )) ∩ {x 2 = const} it is sufficient to extend n ε α by reflection in the direction normal to the microfibrils and parallel to the boundary.Thus, classical extension results [2,12] apply. For ) almost everywhere in time.Since the extension operator is linear and bounded and Ω ε M does not depend on t, we have nε α ∈ L p (0, T ; W 1,p (Ω)) ∩ W 1,p (0, T ; L p (Ω)) and In the sequel, we identify n ε α with its extension, for α = e, E, d, c.Theorem 2.4.Under Assumption 1 and for where the constant C is independent of ε, there exists a unique weak solution and the a priori estimates where α = e, E, d, c and the constant C is independent of ε.
Proof.To show the existence of a solution of the microscopic model ( 15)-( 19) for a given u ε ∈ L ∞ (0, T ; W(Ω)), we shall apply the Schauder fixed point theorem and the Galerkin method.
For ñε Applying the Galerkin method and using estimates similar to those shown below, we obtain the existence of a unique weak solution of the problem (30).
First we show the non-negativity of the solutions of (30).The assumptions on R eE and J e ensure Thus, the Theorem on positive invariant regions [47,50], with G(n ε e , n ε E ) = −n ε e , and the nonnegativity of n e0 , implies n ε e (t, x) ≥ 0 for a.a.(t, x) ∈ Ω ε M,T .Integrating (30) 1 over Ω ε M and using the assumptions on J e and n e0 we obtain where the constant C is independent of ε.From the assumptions on J E and estimate (31) we have for g ≥ 0 for all n ε e ≥ 0 and g is Lipschitz continuous.
Applying the Theorem on positive invariant regions, where G(n ε e , n ε E ) = −n ε E , and the nonnegativity of the initial condition yields n ε E (t, x) ≥ 0 for a.a.(t, x) ∈ Ω ε M,T .Considering n ε e and n ε E as test functions in (30) 1 and (30) 2 , respectively, and using the nonnegativity of n ε e and n ε E , along with the assumptions on J e , J E , and R eE , ensure the estimates , where α = e, E. To estimate the boundary terms we use the extension of n ε α from Ω ε M to Ω, for α = e, E, see Lemma 2.3, and the trace inequality In the same way we also obtain the uniform estimates for , where α = e, E, with n ε e instead of ñε e in (30).Next, we show that n ε e and n ε E are bounded.We define Φ ε β as the solution of the linear problem where D > 0 and β ≥ 0. In the same way as in [35] using the extension of The properties of J e imply that the boundary term in the last inequality is non-positive.Thus, we conclude n ε e (t, x) Since A in the assumptions on ñε E is an arbitrary constant, it can be chosen so that From the weak formulation of the equations in (30) and the estimates for n ε e and n ε E in L 2 (0, T ; V(Ω ε M )) shown above, we obtain the boundedness of ∂ t n ε e and ∂ t n ε E in L 2 (0, T ; V(Ω ε M ) ) for every fixed ε.
To show the existence of a solution (n ε e , n ε E ) of the first two equations in (15) with the corresponding boundary and initial conditions, we consider and define an operator K 1 : X → X, where n ε E = K 1 (ñ ε E ) is given as a solution of the problem (30).The continuity of the functions R eE , J e , and J E , along with the a priori estimates for n ε e and n ε E and the compact embedding of L 2 (0, T ; ensures the continuity of K 1 .Utilizing the a priori estimates and the compact embedding of ) again, and applying the Schauder fixed point theorem yield the existence of a non-negative, bounded weak solution (n ε e , n ε E ) of the first two equations in (15) with boundary and initial conditions in ( 16)-( 18) and ( 19), for every fixed ε.
To show the existence of a weak solution of the equations for Applying the Galerkin method and estimates similar to those shown below, we obtain the existence of a unique weak solution of the problem (35).
First we show the non-negativity of n ε d and n ε b .We define the reaction terms in the equations for n ε d and n ε b in (35) by Using the properties of the functions R eE , R dc , and R b and the non-negativity of n ε e , n ε E , and ñε c , we obtain Thus, applying the Theorem on positively invariant regions [47,50], with , and using the non-negativity of the initial data yield n ε d (t, x) ≥ 0 and n ε b (t, x) ≥ 0 for a.a.(t, x) ∈ Ω ε M,T .Next, we derive estimates for the solutions of (35).Adding (35) 1 and (35) 3 , multiplied by 2, using the non-negativity of n ε d and n ε b , and integrating over Ω ε M give . Then, the boundedness of n ε e , n ε E , n d0 , and n b0 implies sup (0,T ) Testing (35) 3 with n ε b and applying the assumptions on R dc and the Gronwall lemma ensure Taking n ε d as a test function in the weak formulation of (35) 1 yields Notice that the estimate (28) for u ε , and Assumption 1.9 in the case of N δ (e(u ε )) given by ( 21), ensure for a.a.t ∈ [0, T ], where the constant C is independent of ε.Using (37) and (38) and applying the Gagliardo-Nirenberg inequality yields ) ) for any τ ∈ (0, T ].Using the boundedness of n ε e and n ε E , the estimates ( 36) and (37), and the regularity of initial data, and applying the Gronwall inequality imply We show the boundedness of n ε d by using Lemma 3.2 in Alikakos [3].Taking |n ε d | p−1 , with p = 2 κ , κ = 1, 2, 3, . .., as a test function in the weak formulation of (35) 1 , we obtain Applying the extension Lemma 2.3 to |n ε d | p 2 and using the Gagliardo-Nirenberg inequality imply .
Then, using similar recursive iterations as in Lemma 3.2 in Alikakos [3], we obtain for a.a.τ ∈ [0, T ] and C ≥ 1. Applying the pth root, and taking p → ∞, we obtain for all τ ∈ (0, T ] and C 1 is independent of ε.Multiplying (35) 3 with n ε b , integrating over (0, τ ), and considering the supremum over Ω ε M give (39) and iterating over time intervals of length 1/(2C 2 (C 1 + 1)) yield the estimates for n ε b and, hence, for Using the definitions of J c,E and J c,I and estimate (38) we obtain that the boundary terms J c,E and J c,I are Lipschitz continuous for n ε c = 0.Moreover, J c,E (0) ≥ 0 and J c,I (0) ≥ 0. Thus, the Theorem on positive invariant regions, with where σ > 0 is arbitrary fixed.Here we used the properties of the extension of n ε c , see Lemma 2.3.In the same way as for n ε d we show the boundedness of n ε c by using Lemma 3.2 in Alikakos [3].Taking |n ε c | p−1 , with p = 2 κ , κ = 1, 2, 3, . .., as a test function in the weak formulation of (35) 2 , using the boundedness of n ε d and n ε b , and applying the Gagliardo-Nirenberg [7] inequality yield Here, the boundary terms are estimated by applying the extension Lemma 2.3 to |n ε c | p/2 and the trace inequality for H 1 -functions: Applying Lemma 2.3 and the Gagliardo-Nirenberg inequality we also have .
Using similar recursive iterations as in Lemma 3.2 in Alikakos [3], we obtain Applying the pth root, and taking p → ∞, we obtain the estimate Since A in the assumptions on ñε c is an arbitrary constant, we can choose A ≥ C.
The boundedness of (N δ (e(u ε ))) + and n ε α , for α = d, b, ensures the estimate for ) and the weak formulation of equations (35) we obtain the boundedness of ∂ t n ε α in L 2 (0, T ; V(Ω ε M ) ), for α = d, c.To show the existence of a weak solution (n ε d , n ε c , n ε b ) of the corresponding equations in ( 15)-( 19), we consider the operator K 2 : X → X defined by n ε c = K 2 (ñ ε c ), where n ε c solves the problem (35) and X is defined in (34).The continuity of K 2 is ensured by the continuity of R eE , R dc , R b , J c,I , and J c,E , the a priori estimates for n ε α , with α = d, c, b, the compact embedding of Estimate ( 40) is obtained by considering the difference of equation ( 35) 3 for n ε b,1 and n ε b,2 , testing by n ε b,2 − n ε b,2 , and using the properties of R dc and R b .Applying the Schauder fixed point theorem and the compact embedding of L 2 (0, T ; ) yields the existence of a fixed point of K 2 .Hence, combining this result with the existence result for (n ε e , n ε E ), ensures the existence of a weak solution of the microscopic model ( 15)- (19).Considering the equations for the difference of two solutions n ε α,1 − n ε α,2 , for α = e, E, d, c, b, and using the uniform boundedness of n ε α,1 and n ε α,2 , we obtain the uniqueness of a weak solution of the problem ( 15)-( 19) for a given u ε ∈ L ∞ (0, T ; W(Ω)).
Remark.If we assume that demethylesterified pectin does not diffuse (D d = 0), then considering the sum of the equation for n ε d with twice the equation for n ε b yields Thus, we could show the boundedness of n ε d and n ε b under weaker assumptions on the functions R dc and R b , i.e. the sublinearity and global Lipschitz continuity are not required.Lemma 2.5.Under Assumption 1, weak solutions of (15)-( 20) satisfy where the constant C is independent of ε.
Proof.To show the estimates for time derivatives, we first consider a regularisation with respect to t of the weak solutions.Then, applying the lower semicontinuity of the L 2 -norm we obtain the corresponding estimates for ∂ t n ε α and ∂ t u ε .Differentiating the equations of linear elasticity (20) with respect to time t, testing it with ∂ t u ε , and using the uniform boundedness of ∂ t n ε b imply the estimate for ∂ t e(u ε ) L 2 (Ω T ) .Applying the second Korn inequality we obtain the estimate for ∂ t u ε in L 2 (0, T ; W(Ω)).
Taking ∂ t n ε α as a test function in (22), for α = e, E, d, c, using the properties of R eE , R dc , R b , J α , for α = e, E, J c,E , and J c,I , and applying the extension Lemma 2.3, we obtain the estimates for the L 2 -norm of the time-derivatives, independent of ε.Here, the boundary integrals are estimated as for arbitrary fixed σ > 0 and τ ∈ (0, T ].Thus, the estimates for u ε and n ε b ensure and for a.a.τ ∈ [0, T ]. Theorem 2.6.Under Assumption 1 there exists a unique weak solution of the microscopic model ( 15)- (20).
Proof.For a given ñε b ∈ L ∞ (Ω ε M,T ), by Lemma 2.2, there exists a unique u ε ∈ L ∞ (0, T ; W(Ω)) satisfying (20), with n ε b replaced by ñε b .Then for u ε ∈ L ∞ (0, T ; W(Ω)), by Theorem 2.4 and Lemma 2.5, there are unique E, d, c, and n and show that for sufficiently small T ∈ (0, T ], the operator K is a contraction.First, we consider ñε b,1 , ñε b,2 ∈ L ∞ (Ω ε M,T ) and the cor- E, d, c, and , where p = 2 κ and κ = 1, 2, 3, . .., as test functions in the differences of the equations for n ε α,1 and n ε α,2 , where α = d, c.For the boundary integrals in the equation for n ε c,1 − n ε c,2 we have and, using the extension Lemma 2.3 and the trace inequality, Then, the uniform boundedness of n ε α,j , for α = d, c, b and j = 1, 2, the Gagliardo-Nirenberg estimate and the estimates for the extension of |n Applying Lemma 3.2 in Alikakos [3] with p = 2 κ and κ = 2, 3, . . ., we obtain for all τ ∈ (0, T ] and C δ ≥ 1.Here we also used the estimate . Taking the pth root, and considering p → ∞ yield for all τ ∈ (0, T ].We consider the difference of equations ( 23) for n ε b,i , where i = 1, 2, and multiply by Using the estimate (42) yields for any τ ∈ (0, T ].Then, iterating over time intervals of length 1/(2C δ ), ensures for all T ∈ (0, T ] and the constant C independent of ε.Combining the last inequality together with (27) ensures that for fixed δ and T sufficiently small, K is a contraction.Thus, by the Banach fixed point theorem, K has a unique fixed point.Hence, there exists a unique weak solution of ( 15)-( 20) in (0, T ) × Ω.Since T depends only on model parameters, iterating over time intervals yields the existence of a unique weak solution in (0, T ) × Ω.

Homogenization for the elastic model of plant cell walls
In this section we derive macroscopic equations for the microscopic model ( 15)-( 20) by applying the two-scale convergence and the periodic unfolding methods, see e.g.[4,10,11,40,41].
Definition 3.1.For a measurable function φ on Ω ε M , the unfolding operator We define the space E, d, c, and , such that, up to a subsequence, E, d, c, Proof.The a priori estimates in Lemma 2.2, Theorem 2.4, and Lemma 2.5, together with the extension Lemma 2.3 and the compactness theorems for the two-scale convergence, see e.g.[4,41], ensure the weak and two-scale convergence of n ε α and u ε , stated in Lemma.The strong convergence in L 2 (Ω T ) follows from the Lions-Aubin Lemma [34].The embedding W β,2 (Ω) ⊂ L 2 (∂Ω), with β ∈ (1/2, 1), and the compact embedding H 1 (Ω) ⊂ W β,2 (Ω) ensure the strong convergence on the boundary ∂Ω.The a priori estimate for ∂ t e(u ε ) yields the last strong convergence stated in Lemma.
Remark.Notice that the two-scale convergence of u ε and the estimate for ∂ t e(u ε ), together with the Kolmogorov theorem [7], imply where êy,33 (v) = 0, êy,3j (v) = êy,j3 (v) = 1 2 ∂ y j v 3 for j = 1, 2, and êy,ij (v) = 1 2 (∂ y i v j + ∂ y j v i ) for i, j = 1, 2, and as ε → 0, for all x ∈ Ω.Then, Lebesgue's dominant convergence theorem ensures In the same way we also obtain ∩Ω tr e(u ε (t, x))dx for x ∈ Ω and t ∈ (0, T ), we have, up to a subsequence, where Using the strong convergence of n ε d and n ε c in L 2 (Ω T ) and the boundedness of n ε α , where α = d, c, for the first term we have where Using the properties of the unfolding operator T ε , i.e.T ε (φ Applying the weak convergence of e(u ε ) and the strong convergence of B δ (x)∩Ω e(u ε )dx yields where σ 3 (ε j ) → 0 as ε j → 0, with j = m, k.We estimate I 3 as Combining all of the estimates from above and applying the Gronwall inequality we obtain sup (0,T ) To proof the strong convergence of T ε (n ε b ) in the case N δ depends on stress E ε (n ε b , x)e(u ε (t, x)) we have to consider a different approach.Lemma 3.4.In the case )dx, we have, up to a subsequence The construction of the extension for n ε α and the uniform boundedness of for α = d, c, with a constant C independent of ε.Then, from equation ( 43), we also obtain the uniform boundedness of n ε b and ∂ t n ε b in L ∞ (Ω T ).We show the strong convergence of n ε b by applying the Kolmogorov theorem [7,39].Considering equation ( 43) at (t, x+h j ) and (t, x), where x) as a test function and using the Lipschitz continuity of R dc and R b yield where , and the constants C 1 , C 2 are independent of ε and h.Using the regularity of the initial condition n b0 ∈ H 1 (Ω), the a priori estimates for ∇n ε α , with α = d, c, and e(u ε ), the fact that |B δ,h (x) ∩ Ω| ≤ Cδ 2 h for all x ∈ Ω, and applying the Gronwall inequality we obtain Extending n ε b by zero from Ω T into R + × R 3 and using the uniform boundedness of where Ω2h = {x ∈ R 3 | dist(x, ∂Ω) ≤ 2h} and the constant C is independent of ε and h.The estimates for ∂ t n ε b ensure where C 1 and C 2 are independent of ε and h.Combining ( 44)-( 46) and applying the Kolmogorov theorem, yield the strong convergence of n ε b in L 2 (Ω T ).Using the properties of the unfolding operator [10,11], we obtain the strong convergence of Theorem 3.5.A sequence of solutions of the microscopic model ( 15)-( 20) converges to a solution of the macroscopic equations in Ω T together with the boundary conditions where the functions v j α and w ij are solutions of the unit cell problems with Γ = Γ ∩ {y 3 = const}, for a.a.(t, x) ∈ Ω T and w 33 = 0. Here, b jk = 1 2 (b j ⊗b k +b k ⊗b j ), where (b j ) 1≤j≤3 is the canonical basis of R 3 , for v ∈ R 3 we denote div y v = ∂ y 1 v 1 + ∂ y 2 v 2 , and b1 = (1, 0) T , b2 = (0, 1) T .
Proof.We consider φ α (t, x) = ϕ α (t, x) + εψ α (t, x, x/ε) as a test function in ( 22)-( 23 The strong convergence of T ε (n ε b ), the two-scale convergence of u ε and ∂ t u ε ensure Choosing ϕ α ≡ 0 yields for α = e, E, d, c.The linearity of (51) implies that n α,1 has the form where v j α are solutions of the unit cell problems (49).Considering ψ α ≡ 0, we obtain the macroscopic equations for n α in (47), where α = e, E, d, c.The two-scale convergence of ∂ t n ε α , see Lemma 3.2, ensures that n α satisfies the initial condition, for α = e, E, d, c, b.
In the same manner as for the microscopic model, we show the uniqueness of solutions of the macroscopic equations and obtain that the whole sequence of microscopic solutions converges to a solution of the macroscopic problem.

Homogenization for the elastic-viscoelastic model
In this section we consider the equations ( 15)- (19) for the dynamics of pectin and calcium-pectin cross-links together with the equations of linear viscoelasticity for the mechanics of the plant cell wall: Here, V ε (ξ, x) = V(ξ, x/ε), where V(ξ, y) = V M (ξ)χ ŶM (y) for y ∈ Ŷ and extended Ŷ -periodically into R 3 .
In addition to Assumption 1, we consider the following assumptions.
Hence, the bounds for n ε b and (∂ t n ε b ) + are independent of ũε .Using the estimates for u ε and ∂ t u ε , similar to those shown below, and applying the Galerkin method, yield the existence of a weak solution of the equations (53).
Considering ∂ t u ε as a test function in (54) and using the non-negativity of n ε b , as well as the uniform boundedness of n ε b and Choosing σ sufficiently small and using Gronwall's inequality imply with a constant C independent of ε.Applying the second Korn inequality yields (55).Notice that (56) implies the estimate for , independent of ε.Considering the difference of the equations ( 54) for n ε b,i , with i = 1, 2, and taking . By the assumptions on E ε (n ε b,1 , x) and V ε (n ε b,1 , x), we have Applying the Gronwall inequality and the estimates for ∂ t n ε b,1 and e(u ε 2 ) implies for all T ∈ (0, T ].In the same way as in Theorem 2.6, we derive the estimate for all T ∈ (0, T ], where n ε b,i is a weak solution of the problem ( 15)-( 19) and (53).Thus, we obtain that the operator K : L ∞ (0, T ; W(Ω)) → L ∞ (0, T ; W(Ω)), defined by K(ũ ε ) = u ε , where u ε is a weak solution of (53), is a contraction for sufficiently small T .Hence, using the Banach fixed point theorem and iterating over time intervals, we obtain the existence of a unique weak solution of the microscopic model ( 15)-( 19) and (53).
for all x ∈ Ω, and the equations div E hom e(u) + V hom ∂ t e(u) where where the constant C is independent of ε, ϑ, and h.Here we used the extension property e , with a constant C 1 independent of ε, see e.g.[42], and the estimates (64).
Considering the difference of (63) for t and t+h and taking δ h u ε (t, x) = u ε (t+h, x)−u ε (t, x) as a test function yield In the same way we also have where C is independent of ε, ϑ, and h.In the estimates for the first two terms on the lefthand side of (67) we use the uniform boundedness of n ε b and ∂ t n ε b , the equality δ h e(u ε (t, x)) = h 1 0 ∂ t e(u ε (t + hs, x))ds, and estimates (64): with a constant C independent of ε, ϑ, and h.Then, the assumptions on E, f E , and p I , estimates (64) and (66), and the boundedness of with a constant C independent of ε, ϑ, and h.
In the same way as in Lemma 3.2, using (64), we obtain the weak and two-scale convergences as ε → 0 for subsequences of n ε α to limit functions n ϑ α , where α = e, E, d, c, b.
In a similar way as in the proof of Theorem 4.2, considering the assumptions on E and V, together with the boundedness of n ϑ b and ∂ t n ϑ b , uniformly in ϑ, we obtain the existence of weak solutions of (72) satisfying where the constant C is independent of ϑ.The estimates (76) ensure the existence of a weak solution of (73) such that  (79) Using the assumptions on V M , we obtain the symmetry and strong ellipticity of V ϑ hom , see e.g.[49,42].The assumptions on E and V M , the uniform boundedness of n ϑ b , and estimates (76)-( 77) ensure with a constant C independent of ϑ.Taking u ϑ t as a test function in the weak formulation of (78), using the strong ellipticity of V ϑ hom and estimates (74) and (80), and applying the second Korn inequality for u ϑ (t) ∈ W(Ω) yield ϑ ∂ t u ϑ 2 L ∞ (0,T ;L 2 (Ω)) + e(u ϑ ) 2 H 1 (0,T ;W(Ω)) ≤ C, with a constant C independent of ϑ.
To pass to the limit as ϑ → 0 in the macroscopic equations (78) we have to show the strong convergence of E ϑ hom , V ϑ hom , and K ϑ as ϑ → 0. First, we show the strong convergence of Ŷ êy (w ij ϑ )dy and ŶM ∂ t êy (w ij ϑ )dy in L 2 (Ω T ).Considering (72) 1 for t + h and t, with h > 0, taking δ h w ij ϑ (t, x, y) = w ij ϑ (t + h, x, y) − w ij ϑ (t, x, y) as a test function, and using δ h êy (w ij ϑ (t)) = h for a.a.x ∈ Ω and the constants C 1 , C 2 , and C 3 are independent of ϑ and h.Here, we used the fact that due to the periodicity of w ij ϑ and the Korn inequality we have for a.a.x ∈ Ω, and êy (δ h ∂ t w ij ϑ ) L 2 ((0,T −h)× Ŷ ) ≤ C êy (δ h ∂ t w ij ϑ ) L 2 ((0,T −h)× ŶM ) .Considering (72) 1 for x + h j and x, with h j = hb j , and using (75) imply δ h j êy (w ij ϑ ) 2 L ∞ (0,T ;L 2 (Ω× Ŷ )) + δ h j êy (∂ t w ij ϑ ) 2 L 2 (Ω T × ŶM ) ≤ Ch, where δ h j w ij ϑ (t, x, y) = w ij ϑ (t, x+h j , y)−w ij ϑ (t, x, y), n ϑ b is extended by zero from Ω T in R + ×R 3 , and C is independent of ϑ.In the same manner we obtain In this paper we developed a mathematical model for plant cell walls which considers the microscopic structure of the cell wall and the biochemical processes that take place within the wall matrix explicitly.The microscopic model defined on the scale of the cell wall's structural elements describes the interconnections between the calcium-pectin cross-links dynamics and the changes in the mechanical properties of the cell wall.We assume a biologically relevant non-local effect of strain or elastic stress on the calcium-pectin cross-link dynamics.The analysis of the microscopic model and the derivation of the macroscopic equations in the case where the reaction terms in the equations for n ε d , n ε c and n ε b depend on the elastic stress or strain pointwise, instead of their local average, appears to require uniform in ε estimates for e(u ε ) L ∞ (Ω T ) .However, in our situation the regularity of the elasticity tensor is not sufficient to derive the estimate for e(u ε ) in L ∞ (Ω T ).
The macroscopic equations, rigorously obtained by applying homogenization techniques, allow us to develop efficient multiscale numerical simulations for the biomechanical model of the cell wall, which takes into account its complex microscopic processes and microstructure.The numerical analysis of the macroscopic model will be the subject of future research.

Figure 1 :
Figure 1: Schematic diagram of a plant cell wall.MF denotes cell wall microfibrils, CMT denotes cortical microtubules in a plant cell.

Figure 2 :
Figure 2: (a) A depiction of the domain Ω with the subsets representing the matrix Ω ε M and the microfibrils Ω ε F .The surface Γ I is in contact with the interior of the cell, and the (hidden) surface Γ E is facing the outside of the cell.(b) A depiction of the unit cell Y .
where Φ ε β denotes the extension of Φ ε β into Ω T and the constant C is independent of ε.Considering nε e = n ε e − Φ ε βe , where Φ ε βe is the solution of the problem (33) with β = βe = β e (1 + n ε e L ∞ (0,T ;L 1 (Ω ε M )) ) and D = D e , and taking (n ε e − A e ) + as a test function, where A e ≥ n e0 L ∞ (Ω) , imply and the non-negativity of the initial condition n c0 implies n ε c (t, x) ≥ 0 for a.a.(t, x) ∈ Ω ε M,T .Taking n ε c as a test function for (35) 2 and using the non-negativity and boundedness of n ε d and n ε b yield the estimates for n ε c L 2 (0,T ;H 1 (Ω ε M )) and n ε c L ∞ (0,T ;L 2 (Ω ε M )) , independent of ε.The boundary integrals are estimated as Applying the unfolding operator T ε to the equation (15) 5 and taking T εm (n εm b ) − T ε k (n ε k b ) as a test function in the difference of the equations for T εm (n εm b ) and

Remark. 4 . 0 K
If we assume that n ε d does not diffuse, i.e.D d = 0, then Assumption 2.3 on R dc is not required and the uniform boundedness of n ε d , n ε b and (∂ t n ε b ) + follows from the boundedness of n ε e and n ε E .See also the Remark after Theorem 2.Theorem 4.3.A sequence of solutions of the microscopic model (15)-(19) and (53) converges to a solution of the macroscopic equations (47) with the boundary conditions (48) and N δ (e(u)) = − B δ (x)∩Ω tr e(u) dx or N δ (e(u)) = − B δ (x)∩Ω tr E hom (n b )e(u) + t (t − s, s, n b )∂ s e(u)ds dx,