Homogenization of a viscoelastic model for plant cell wall biomechanics

The microscopic structure of a plant cell wall is given by cellulose microfibrils embedded in a cell wall matrix. In this paper we consider a microscopic model for interactions between viscoelastic deformations of a plant cell wall and chemical processes in the cell wall matrix. We consider elastic deformations of the cell wall microfibrils and viscoelastic Kelvin--Voigt type deformations of the cell wall matrix. Using homogenization techniques (two-scale convergence and periodic unfolding methods) we derive macroscopic equations from the microscopic model for cell wall biomechanics consisting of strongly coupled equations of linear viscoelasticity and a system of reaction-diffusion and ordinary differential equations. As is typical for microscopic viscoelastic problems, the macroscopic equations for viscoelastic deformations of plant cell walls contain memory terms. The derivation of the macroscopic problem for degenerate viscoelastic equations is conducted using a perturbation argument.


Introduction
To obtain a better understanding of the mechanical properties and development of plant tissues it is important to model and analyse the interactions between the chemical processes and mechanical deformations of plant cells. 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 biomechanics of plant cell walls is determined by the cell wall microstructure, given by microfibrils, and the physical properties of the cell wall matrix. The orientation of microfibrils, their length, high tensile strength, and interaction with wall matrix macromolecules strongly influences the wall's stiffness. It is also supposed that calcium-pectin cross-linking chemistry is one of the main regulators of cell wall elasticity and extension [30]. 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. [29]. 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 [23,24], and the interference with PME activity causes dramatic changes in growth behavior of plant cells and tissues [31].
To address the interactions between chemistry and mechanics, in the microscopic model for plant cell wall biomechanics we consider the influence of the microstructure, associated with the cellulose microfibrils, and the calcium-pectin cross-links on the mechanical properties of plant cell walls. We model the cell wall as a threedimensional 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. It was observed experimentally that plant cell wall microfibrils are anisotropic, see e.g. [10], and the cell wall matrix in addition to elastic deformations exhibits viscous behaviour, see e.g. [14]. Hence we model the cell wall matrix as a linearly viscoelastic Kelvin-Voigt material, whereas microfibrils are modelled as an anisotropic linearly elastic material. The model for plant cell wall biomechanics in which the cell wall matrix was assumed to be a linearly elastic was derived and analysed in [26]. The interplay between the mechanics and the cross-link dynamics comes in by assuming that the elastic and viscous properties of the cell wall matrix depend on the density of the cross-links and that stress within the cell wall can break calcium-pectin cross-links. The stress-dependent 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 viscoelasticity. Since only the cell wall matrix is viscoelastic we obtain degenerate elastic-viscoelastic equations.
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 mechanical properties of the plant cell wall we rigorously derive macroscopic equations from the microscopic description of plant cell wall biomechanics. The two-scale convergence, e.g. [4,21], and the periodic unfolding method, e.g. [7,8], are applied to obtain the macroscopic equations. For the viscoelastic equations 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 variables. This fact introduces additional complexity in the derivation and in the structure of the macroscopic equations, compered to classical viscoelastic equations.
The main novelty of this paper is the multiscale analysis and derivation of the macroscopic problem 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 viscoelasticity for cell wall mechanics and the system of reaction-diffusion and ordinary differential equations for the chemical processes in the wall matrix. Also the degeneracy of the viscoelastic equations, due to the fact that only the cell wall matrix is assumed to be viscoelastic and microfibres are assumed to be elastic, induces additional technical diffuculties.
A multiscale analysis of the viscoelastic equations with time-independent coefficients was considered previously in [12,13,18,27]. Macroscopic equations for scalar elastic-viscoelastic equations with time-independent coefficients were derived in [11] by applying the H-convergence method [19]. 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.
The paper is organised as follows. In Section 1 we formulate a mathematical model for plant cell wall biomechanics in which the cell wall matrix is assumed to be viscoelastic. In Section 2 we summarise the main results of the paper. The well-possednes of the microscopic model is shown in Section 3. The multiscale analysis of the microscopic model is conducted 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 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.

Microscopic model for viscoelastic deformations of plant cell walls
It was observed experimentally that in addition to elastic deformations the plant cell wall matrix exhibit viscoelastic behaviour [14]. Hence, in contrast to the problem considered in [26], here we assume that the deformation in the plant cell wall matrix is determined by the equations of linear viscoelasticity.
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 characterise the size of the microstructure. The domains and Ω ε M = Ω \ Ω ε F denote the part of Ω occupied by the microfibrils and by the cell wall matrix, respectively. The boundary between the matrix and the microfibrils is denoted by Γ ε = ∂Ω ε M ∩ ∂Ω ε F . We adopt the following notation: for all u ∈ W(Ω) defines a norm on W(Ω), see [6,17,22]. For more details see also [26]. The microscopic model for elastic-viscoelastic deformations u ε of cell walls and the densities of enzyme and pectins: esterified pectin p ε 1 , PME enzyme p ε 2 , de-esterified pectin n ε 1 , calcium ions n ε 2 , and calcium-pectin and in the cell wall matrix Ω ε M,T we consider ∂ t n ε = div(D n ∇n ε ) + F n (p ε , n ε ) + R n (n ε , b ε , N δ (e(u ε ))) ) T , and div(D n ∇n ε ) = (div(D 1 n ∇n ε 1 ), div(D 2 n ∇n ε 2 )) T , together with the initial and boundary conditions Here N δ (e(u ε )), defined as in Ω T , for δ > 0, represent the nonlocal impact of mechanical stresses on the calcium-pectin cross-links chemistry. From a biological point of view the non-local dependence of the chemical reactions on the displacement gradient is motivated by the fact that pectins are very long molecules and hence cell wall mechanics has a nonlocal impact on the chemical processes. The positive part in the definition of N δ (e(u ε )) reflects the fact that extension rather than compression causes the breakage of cross-links. In the boundary conditions (3) we assumed that the flow of calcium ions between the interior of the cell and the cell wall depends on the displacement gradient, which corresponds to the stress-dependent opening of calcium channels in the plasma membrane [28]. The elasticity and viscosity tensors are defined as E ε (ξ, x) = E(ξ,x/ε) and V ε (ξ, x) = V(ξ,x/ε), where thê Y -periodic in y functions E and V are given by E(ξ, y) = E M (ξ)χŶ M (y)+E F χŶ F (y) and V(ξ, y) = V M (ξ)χŶ M (y). For a given measurable set A we use the notation φ 1 , φ 2 A = A φ 1 φ 2 dx, 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 ψ 1 ∈ L 2 (0, T ; V(Ω ε M )) and ψ 2 ∈ L 2 (0, T ; V(Ω ε M ) ). We also denote I k µ = (−µ, +∞) k for µ > 0 and k ∈ N.

Remark.
Notice that Assumption 1.9 is not restrictive from a physical point of view, since every biological material will have a maximal possible stiffness. Also, in contrast to [26], we assume that (R b (ξ, η, ζ)) + is bounded. This is required to show a priori estimates for solutions of equations of linear viscoelasticity independent of b ε .

Main results
The main result of the paper is the derivation of the macroscopic equations for the microscopic viscoelastic model for plant cell wall biomechanics. The main difference between the homogenization results presented here and those in [26] is due to the presence of degenerate viscose term in the equation for mechanical deformations of a cell wall. The fact that only the cell wall matrix is viscoelastic and the dependence of the viscosity tensor on the time variable, via the dependence on the cross-links density b ε , make the multiscale analysis nonclassical and complex.
First we formulate the well-posedness result for the model (1) where the constant C 1 is independent of ε and δ, M,T −h and the constant C 3 is independent of ε and h. The proof of Thorem 2.1 follows similar lines as the proof of the corresponding existence and uniqueness results in [26]. Thus here we will only sketch the main ideas of the proof and emphasise the steps that are different from those of the proof in [26].
To formulate the macroscopic equations for the microscopic model (1)-(3), first we define the macroscopic coefficients which will be obtained by the derivation of the limit equations. The macroscopic coefficients coming from the elasticity tensor are given by and the macroscopic elasticity and viscosity tensors and memory kernel read: for a.a. x ∈ Ω and s ∈ The macroscopic diffusion coefficients are defined by where∇ y v j α,l = (∂ y1 v j α,l , ∂ y2 v j α,l , 0) T and the functions v j α,l , for l = 1, 2 and j = 1, 2, 3, are solutions of the unit cell problems in Ω T together with the initial and boundary conditions where θ M = |Ŷ M |/|Ŷ |, and the macroscopic equations of linear viscoelasticity Here 3. Existence of a unique weak solution of the microscopic problem (1)-(3). A priori estimates.
In the derivation of a priori estimates for solutions of the microscopic problem (1)-(3) we shall use an extension of a function defined on a connected perforated domain Ω ε M to Ω. Applying classical extension results [2,9,25], we obtain the following lemma.
Remark. Notice that the microfibrils do not intersect the boundaries Γ I , Γ U , and Γ E , and near the boundaries ∂Ω \ (Γ I ∪ Γ E ∪ Γ U ) it is sufficient to extend v ε by reflection in the direction normal to the microfibrils and parallel to the boundary. Thus, classical extension results [2,9,15,25] apply to Ω ε M . In the sequel, we identify p ε and n ε with their extensions.
First we show the well-possedness and a priori estimates for equations (2)-(3) for a given u ε ∈ L ∞ (0, T ; W(Ω)). Next for a given b ε we show the existence of a unique solution of the viscoelastic problem (1). Then using the fact that the estimates for b ε can be obtain independently of u ε and applying a fixed point argument we show the well-possedness of the coupled system. Lemma 3.2. Under Assumption 1 and for u ε ∈ L ∞ (0, T ; W(Ω)) such that (21) u ε L ∞ (0,T ;W(Ω)) ≤ C, where the constant C is independent of ε, there exists a unique weak solution (p ε , n ε , b ε ) of the microscopic model (2) × Ω ε M , j = 1, 2, and the a priori estimates (8) and (10).
Proof. The proof of this lemma follows the same lines as the proof of Theorem 3.3 in [26]. The only difference is in the derivation of the estimates for b ε . Using the non-negativity of n ε 1 , n ε 2 , b ε , and Assumptions 1.4 and 1.5 we obtain from the equation for b ε for a.a. (t, x) ∈ Ω ε M,T . Hence, the bounds for b ε and (∂ t b ε ) + are independent of the bound for u ε L ∞ (0,T ;W(Ω)) . This fact is important for the derivation of a priori estimates for u ε and the fixed point argument for the proof of the existence of a solution for the coupled system.
Using the equation for b ε , the definition of N δ and the estimates for n ε (5), respectively, we obtain the last estimate in (10).
Next we prove the existence, uniqueness and a priori estimates for a solution of viscoelastic equations for a given b ε ∈ L ∞ (0, T ; L ∞ (Ω ε M )). Proof. Using the estimates for u ε and ∂ t u ε , similar to those in (23), along with the positive definiteness of E and V, and applying the Galerkin method, yield the existence of a weak solution of the problem (1).
Considering ∂ t u ε as a test function in (7) and using the non-negativity of b ε and the assumptions on E and V, we obtain e(u ε )(τ ) 2 Choosing σ sufficiently small, using the boundedness of b ε and (∂ t b ε ) + , independent of ε and u ε , and applying Gronwall's inequality imply (23) e(u ε ) 2 L ∞ (0,T ;L 2 (Ω)) + ∂ t e(u ε ) 2 L 2 (Ω ε M,T ) ≤ C, with a constant C independent of ε. Then using the second Korn inequality yields (9). Now applying a fixed point argument and using the results in Lemmas 3.2 and 3.3 we obtain the wellpossedness result for the coupled system (1)-(3).

Derivation of the macroscopic equations of the problem (1)-(3): Proof of Theorem 2.2.
Due to the fact that viscous term is defined only in the cell wall matrix and is zero for cell wall microfibrils, to conduct the multiscale analysis of the viscoelastic problem (1) we first consider a perturbed problem by adding the inertial term ϑ∂ 2 t u ε , where ϑ > 0 is a small perturbation parameter: (25) ϑχ on Ω T , and the additional initial condition We split the proof of Theorem 2.2 into two steps. First we derive the macroscopic equations for the perturbed system. Then letting the perturbation parameter ϑ go to zero we obtain the macroscopic equations (19) for the original degenerate viscoelastic problem.  (1) and (26), satisfying the a priori estimates with a constant C independent of ε and ϑ, and where θ h v(t, x) = v(t + h, x) for a.e. (t, x) ∈ Ω ε M,T −h , and the constant C is independent of ε and ϑ. Proof. For a given u ε ∈ L ∞ (0, T ; W(Ω)), with u ε L ∞ (0,T ;W(Ω)) ≤ C, in the same way as in Lemma 3.2 we obtain the existence of a unique solution of the problem (2)-(3), satisfying the a priori estimates (28). Notice that the estimates for b ε and (∂ t b ε ) + are independent of u ε , ε, and ϑ.  (1) and (26), satisfying the a priori estimates (27).
Similar to the proof of Theorem 2.1, considering the difference of the equations (25) for b ε,j , with j = 1, 2, and taking ∂ t (u ε,1 − u ε,2 ) as a test function yield for τ ∈ (0, T ]. By the assumptions on E ε (b ε,1 , x) and V ε (b ε,1 , x), and applying the Gronwall inequality and the estimates for ∂ t b ε,1 and e(u ε,2 ) we obtain for all T ∈ (0, T ]. Then, using the estimates (24), (27) and (30), and the a priori estimates for p ε , n ε , and b ε in the same way as in the proof of Theorem 2.1 we obtain the existence of a unique weak solution of the perturbed problem (2), (3), and (25) with initial and boundary conditions in (1) and (26).
Proof. A priori estimates in (8) and (10) imply weak and two-scale convergences of p ε , n ε , b ε , and ∂ t b ε . Using the estimates for p ε (t + h, x) − p ε (t, x) and n ε (t + h, x) − n ε (t, x) together with the estimates for ∇n ε and ∇p ε in (10) and the properties of the extension of n ε and p ε from Ω ε M to Ω, see Lemma 3.1, and applying the Kolmogorov theorem [5,20] we obtain the strong convergence of n ε and p ε in L 2 (Ω T ).
In the same way as in [26] we show that, up to a subsequence, Here we present only the sketch of the calculations. Using the extension of n ε from Ω ε M to Ω, see Lemma 3.1, we define the extension of b ε from Ω ε M to Ω as a solution of the ordinary differential equation The construction of the extension for n ε and the uniform boundedness of n ε 1 , n ε 2 in Ω ε M,T , see (10), ensure n ε L ∞ (0,T ;L ∞ (Ω)) ≤ C n ε L ∞ (0,T ;L ∞ (Ω ε M )) , with the constant C independent of ε. Hence from (32) we obtain also the boundedness of b ε and ∂ t b ε . We show the strong convergence of b ε by applying the Kolmogorov theorem [5,20]. Considering equation (32) at x) as a test function and using the Lipschitz continuity of , and the constants C 1 , C 2 are independent of ε and h. Using the regularity of the initial condition b 0 ∈ H 1 (Ω), the a priori estimates for e(u ε ) and ∇n ε , along with the fact that |B δ,h (x) ∩ Ω| ≤ Cδ 2 h for all x ∈ Ω, and applying the Gronwall inequality we obtain Extending b ε by zero from Ω T into R + × R 3 and using the uniform boundedness of b ε in L ∞ (0, T ; L ∞ (Ω)) imply where C 1 and C 2 are independent of ε and h. Combining (33)-(35) and applying the Kolmogorov theorem yield the strong convergence of b ε to b ϑ in L 2 (Ω T ). The definition of the two-scale convergence yields that b ϑ = b ϑ and hence the two-scale limit of b ε is independent of y. Then using the properties of the unfolding operator, see e.g. [7,8], we obtain the strong convergence of T * ε (b ε ). Considering an extension ∂ t u ε of ∂ t u ε from Ω ε M into Ω and applying the Korn inequality [22] yield where the constant C 3 is independent of ε and ϑ. Estimates (27) and (36) ensure the existence of u ϑ ∈ L 2 (0, T ; W(Ω)), u ϑ 1 ∈ L 2 (Ω T ; H 1 per (Ŷ )), ξ ϑ ∈ L 2 (0, T ; H 1 (Ω)) and ξ ϑ 1 ∈ L 2 (Ω T ; H 1 per (Ŷ M )) such that two-scale, see e.g. [4]. Considering the two-scale convergence of u ε and ∂ t u ε , we obtain . Hence, ∂ t u ϑ ∈ L 2 (Ω T ), and ξ ϑ = ∂ t u ϑ a.e. in Ω T ×Ŷ . The two-scale convergence of ∇u ε and ∂ t ∇u ε implies ). Thus, ∂ t∇y u ϑ 1 ∈ L 2 (Ω T ×Ŷ M ) and∇ y ξ ϑ 1 = ∂ t∇y u ϑ 1 a.e. in Ω T ×Ŷ M . Therefore, u ϑ ∈ H 1 (0, T ; W(Ω)), ∂ t u ϑ 1 ∈ L 2 (Ω T ; H 1 per (Ŷ M )) and χ Ω ε M ∂ t e(u ε ) → χŶ M (∂ t e(u ϑ ) + ∂ têy (u ϑ 1 )) two-scale.
To derive macroscopic equations for the microscopic problem (1)-(3), we first derive the macroscopic equations for the perturbed system (2), (3), (25). Then letting the perturbation parameter to go to zero we obtain the macroscopic equations for (1)-(3). Theorem 4.3. A sequence of solutions (u ε , p ε , n ε , b ε ), of the microscopic problem (2), (3), (25), converges to a solution (u ϑ , p ϑ , n ϑ , b ϑ ) of the macroscopic perturbed equations and (38) in Ω T together with the initial and boundary conditions and w ij ϑ (t, x, y), χ ij V,ϑ (t, x, y), and v ij ϑ are solutions of the unit cell problems (13) and (14) with b ϑ instead of b. The macroscopic diffusion matrices D l α , with α = n, p and l = 1, 2, are defined as in (15) and N eff δ is defined in (20).
Proof. To pass to the limit in the equations for n ε and b ε , we shall prove the strong convergence of Ω e(u ε )dx in L 2 (0, T ) using the Kolmogorov compactness theorem [5,20]. Considering the difference of (25) 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 To estimate the first term on the right-hand side we consider δ h u ε x) as a test function, with u ε t being an extension of u ε t from Ω ε M to Ω as in Lemma 3.1, where the constant C is independent of ε, ϑ, and h ∈ (0, T ). Here we used estimates (27) and the property of the extension, i.e. e(u ε t ) 2 with a constant C 1 independent of ε, see e.g. [22].
In the same way as for the macroscopic elasticity tensor for the equations of linear elasticity, see e.g. [16,22], we obtain that V ϑ hom is positive-definite and possesses major and minor symmetries, as in Assumption 1.8. The assumptions on E and V M and the uniform boundedness of b ϑ ensure the boundedness of E ϑ hom and K ϑ . Notice that the positive-definiteness and symmetry properties of V ϑ hom together with the boundedness of E ϑ hom and K ϑ ensure the well-possedness of the viscoelastic equations (37). Now we can complete the proof of the main result of the paper.
Proof of Theorem 2.2. To complete the proof of Theorem 2.2, we have to show that {p ϑ }, {n ϑ }, {b ϑ }, and {u ϑ } converge to solutions of the macroscopic model (17)- (20). Using the fact that the estimates (27) and (46) for u ε are independent of ϑ and ε and applying the weak and two-scale convergence of u ε together with the lower semicontinuity of a norm yield with a constant C independent of ϑ and h. Similar to the proof of Lemma 3.2, using the estimates (50) we obtain the estimates for p ϑ and n ϑ in L 2 (0, T ; V(Ω)) ∩ L ∞ (0, T ; L ∞ (Ω)), and b ϑ in W 1,∞ (0, T ; L ∞ (Ω)) uniformly in ϑ. In a similar way as in the proof of Lemma 4.2, we show where b ϑ is extended by zero from Ω T into R 3 × R + and h j = hb j , with h ∈ (0, T ). Then, applying the Kolmogorov theorem we obtain the strong convergence of a subsequence of b ϑ in L 2 (Ω T ) as ϑ → 0.
In a similar way as in the proof of Lemma 3.3, considering the assumptions on E and V, together with the boundedness of b ϑ and ∂ t b ϑ , uniformly in ϑ, we obtain the existence of weak solutions of the unit cell problems (13), with b ϑ instead of b, satisfying w ij ϑ 2 L ∞ (0,T ;H 1 per (Ŷ )) + ∂ têy (w ij ϑ ) 2 L 2 (0,T ;L 2 (Ŷ M )) ≤ C for a.a. x ∈ Ω, where the constant C is independent of ϑ. The estimates (52) and boundedness of b ϑ and ∂ t b ϑ ensure the existence of a weak solution of the unit cell problem (14) for a.a. x ∈ Ω and s ∈ [0, T ].