Homogenization of biomechanical models of plant tissues with randomly distributed cells

In this paper homogenization of a mathematical model for biomechanics of a plant tissue with randomly distributed cells is considered. Mechanical properties of a plant tissue are modelled by a strongly coupled system of reaction-diffusion-convection equations for chemical processes in plant cells and cell walls, the equations of poroelasticity for elastic deformations of plant cell walls and middle lamella, and the Stokes equations for fluid flow inside the cells. The nonlinear coupling between the mechanics and chemistry is given by the dependence of elastic properties of plant tissue on densities of chemical substances as well as by the dependence of chemical reactions on mechanical stresses present in a tissue. Using techniques of stochastic homogenization we derive rigorously macroscopic model for plant tissue biomechanics with random distribution of cells. Strong stochastic two-scale convergence is shown to pass to the limit in the non-linear reaction terms. Appropriate meaning of the boundary terms is introduced to define the macroscopic equations with flux boundary conditions and transmission conditions on the microscopic scale.


Introduction
To better understand and to improve plant growth and development, we need to model and analyse interactions between chemical processes in and mechanical properties of plant tissues. A plant tissue is composed of cells surrounded by cell walls and connected by cross-linked pectin network of middle lamella. Plant cell walls mainly consist of cellulose microfibrils, pectin, hemicellulose, macromolecules, and water. It is supposed that calcium-pectin cross-linking chemistry strongly influences elastic properties of plant cell walls [47]. Pectin is produced in Golgi apparatus inside the cells and is deposited to a cell wall in a methyl-esterified form, where it can be de-methylesterified by the enzyme pectin methylesterase (PME), which removes methyl groups by breaking ester bonds. The de-methylesterified pectin is able to form calcium-pectin cross-links, and so stiffen the cell wall and reduce its expansion, see e.g. [46], whereas mechanical stresses can breaking calcium-pectin cross-links and hence increase the extensibility of plant cell walls and middle lamella.
In [39] we derived and analysed a mathematical model for plant tissue biomechanics, which describes the interactions between calcium-pectin dynamics and deformations of a plant tissue. The microscopic model, at the length scale of plant cells, comprises a strongly coupled system of the Stokes equations modelling water flow inside plant cells, the equations of poro-elasticity defining elastic deformations of plant cell walls and middle lamella, and reaction-diffusion-convection equations describing the dynamics of the methyl-esterfied pectin, de-methyl-esterfied pectin, calcium ions, and calciumpectin cross-links. The interplay between the mechanics and the chemistry comes in by assuming that the elastic properties of cell walls and middle lamella depend on the density of the calcium-pectin cross-links and the stress within cell walls and middle lamella can break the cross-links. Assuming periodic distribution of cells in a plant tissue in [39] we derived rigorously macroscopic model for plant tissue biomechanics. In this paper we generalise the results obtain in [39] by considering random distribution of cells in a plant tissue.
To analyse macroscopic mechanical properties of plant tissues with a random distribution of cells, we derive rigorously a macroscopic model for plant biomechanics using techniques of stochastic homogenization. The stochastic two-scale convergence [51] is applied to obtain the macroscopic equations. The main mathematical difficulties in the derivation of the macroscopic equations arise from the strong coupling between the equations of poro-elasticity and the system of reaction-diffusion-convection equations, as well as due to transmission conditions between the free fluid and poro-elastic material. The strong stochastic two-scale convergence for the deformation gradient and flow velocity is proven to pass to the limit in the nonlinear reactions terms. Extension arguments and formulations of surface integrals as volume integrals are used to pass to the stochastic two-scale limit in the equations with non-homogeneous Neumann boundary conditions and transmission conditions. To pass to the limit in the flux boundary conditions defined on the surfaces of the microstructure, Palm measure and the proven here trace inequality for H 1 -function in the probability space, see Lemma 8.1, are used.
Some of the first results on the stochastic homogenization of linear second-order elliptic equations were obtained in [28,38,49]. The homogenization of quasi-linear elliptic and parabolic equations with stochastic coefficients and convex integral operators was considered in [6,17,21,22]. Subadditive ergodic theory and the method of viscosity solutions were applied to homogenize Hamilton-Jacobi, viscous Hamilton-Jacobi equations, and fully nonlinear elliptic and parabolic equations in stationary ergodic media [2,16,27,30,31] (see also references therein). The stochastic two-scale convergence introduced in [51] has been extended to Riemannian manifolds and has been applied to analyze heat transfer through composite and polycrystalline materials with nonlinear conductivities [24,25]. The two-scale convergence in the mean [12] has been applied to derive macroscopic equations for singleand two-phase fluid flows in randomly fissured media [10,48].
The poro-elastic equations, modelling interactions between fluid flow and elastic deformations of a porous medium, has been first obtained by Biot using a phenomenological approach [7,8,9] and subsequently derived by applying formal asymptotic expansion [3,14,29,40] or the two-scale convergence method [19,23,33,36]. Along many results for poroelastic equations, only few studies of interactions between a free fluid and a deformable porous medium can be found. In [43] nonlinear semigroup method was used for mathematical analysis of a system of poroelastic equations coupled with the Stokes equations for free fluid flow. A rigorous derivation of interface conditions between a poroelastic medium and an elastic body was considered in [34]. Numerical methods for coupled system of poroelastic and Navier-Stokes equations were studied in [4,15].
The paper is organised as follows. In Section 2 we formulate the microscopic model for plant tissue biomechanics. The main results of the paper are summarised in Section 3. The a priori estimates and convergence results are given in Sections 4 and 5. In Section 6 we derive macroscopic equations for the coupled poro-elastic and Stokes problem. The strong stochastic two-scale convergence for deformation gradient and flow velocity is proven in Section 7. The macroscopic equations for the system of reaction-diffusion-convection equations are derived in Section 8.

Microscopic model
We consider a probability space (Ω, F, P) with probability measure P. We define a 3-dimensional dynamical system T x : Ω → Ω, i.e. a family {T x : x ∈ R 3 } of invertible maps, such that for each x ∈ R 3 , T x is measurable and satisfy the following conditions: (i) T 0 is the identity map on Ω, and for all x 1 , x 2 ∈ R 3 the semigroup property holds: (ii) P is an invariant measure for T x , i.e. for each x ∈ R 3 and F ∈ F we have that We consider a fixed measurable set Ω f such that P(Ω f ) > 0 and P(Ω \ Ω f ) > 0 and denote Ω e = Ω \ Ω f . We also consider Ω Γ ⊂ Ω, with P(Ω Γ ) > 0 and P(Ω Γ ∩ Ω j ) > 0, for j = e, f .
2. The distance between two connected components of G f (ω) is uniformly bounded from above and below.
3. The diameter of connected components of G f (ω) is bounded from below and above by some positive constants.
is open on Γ(ω) and Lipschitz continuous.
Consider a bounded C 1,α -domain G ⊂ R 3 , with α > 0, representing a part of a plant tissue. In a plant tissue individual cells, consisting of cell inside and cell walls, are connected by the pectin network of middle lamella. Then the microscopic structure of a plant tissue with a random distribution of cells is defined as where G ε e represent the subdomains occupied by cell walls and middle lamella, G ε f denotes the cell inside, and Γ ε defines a part of cell membrane which is impermeable to calcium ions. Assumption 1.2 states that the thickness of cell walls and middle lamella is uniformly bounded from above and below and Assumption 1.3 postulates that the diameter of cells is bounded from above and below.
We define statistically homogeneous random fields E(x, ω, ξ) = E(T x ω, ξ) and K p (x, ω) = K p (T x ω), where E(·, ξ) and K p (·) are given measurable functions from Ω to R 3 4 and R 3×3 , respectively, for ξ ∈ R. Then for each ω ∈ Ω and ξ ∈ R the microscopic elasticity tensor E ε and permeability tensor K ε p are defined as . In the mathematical model for biomechanics of a plant tissue we consider concentration of calcium c ε e and c ε f in cell walls and middle lamella G ε e and in cell cytoplasm G ε f (cell inside), respectively. In addition, in the domain of cell walls and middle lamella G ε e densities of methylesterified and demethylesterified pectins n ε e and n ε d and of calcium-pectin cross-links n ε b are considered. We shall use the notation b ε e = (n ε e , n ε d , n ε b ) and ) denotes the diagonal matrix of diffusion coefficients for n ε e , n ε d , and n ε b respectively. We assume that the inflow of new calcium is facilitated only on parts of the cell membrane Γ ε \ Γ ε . Here we consider a passive flow of calcium between cell wall and cell inside. The regulatory mechanism for calcium inflow by mechanical properties of cell walls will be considered in further studies. For elastic deformations of plant cell walls and middle lamella we consider homogenized equations of poro-elasticity reflecting the microscopic structure of cell walls composed of elastic cellulose microfibrils and cell wall matrix permeable for the fluid flow. The differences in the elastic properties of cell walls and middle lamella are reflected in the elasticity tensor E ε , which depends on the microscopic variable x/ε. Here we consider diffusion coefficients depending on calcium-pectin cross-links density. The analysis in the case of diffusion coefficients depending additionally on microscopic and macroscopic variables will follow the same steps.
For P-a.a. realisations ω ∈ Ω the microscopic model for the concentration of calcium and densities of pectins and calcium-pectin cross-links reads The water flow inside the cells and elastic deformations of plant cell walls and middle lamella are modelled by a coupled system of poro-elastic and Stokes equations The dependence of the elastic properties of the cell wall matrix and middle lamella on calcium-pectin cross-links is reflected in the dependence of the elasticity tensor E ε on b ε e,3 . On the external boundaries we consider some given forces applied to plant tissues and flux conditions for pectins and calcium: A detailed derivation of the model equations (1) and (2) can be found in [39]. System (1)-(3) is studied under the following assumption on the coefficients and nonlinear functions: ≤d e for all ξ ∈ R, with some d j , d e ,d j ,d e > 0 and j = 1, 2, 3.
A5. For functions g b , g e , g f , R, F b , and F c we assume and F c and g f are Lipschitz continuous. Moreover, the following estimates hold: where s ∈ R + , r ∈ R 3 + , and A is a symmetric 3 × 3 matrix.
Here and in what follows we identify the space of symmetric 3 × 3 matrices with R 6 .
It is also assumed that for any symmetric 3 × 3 matrix A we have that g b,j (s, r, A), F b,j (r), R j (r) are non-negative for r j = 0, s ≥ 0, and r i ≥ 0, with i = 1, 2, 3 and j = i, and g e (s, r, A), g f (s), and F c (s) are non-negative for s = 0 and r j ≥ 0, with j = 1, 2, 3.
Notice that in the equation for calcium c ε f inside plant cells we consider a bounded function of the water velocity u ε f . This technical assumption is biologically justified, since only bounded velocities are possible inside plant cells.
for P-a.a. ω ∈ Ω, that satisfy the integral relation that satisfy the integral relations and for all ϕ 1 ∈ L 2 (0, T ; H 1 (G ε e )) 3 and ϕ 2 ∈ L 2 (0, T ; H 1 (G \ Γ ε )), and for P-a.a. ω ∈ Ω. Moreover the initial conditions are satisfied in • Let P be a stationary ergodic point process in R 3 such that (i) for any two points x j and x k the inequality holds |x j − x k | ≥ c > 0 with a deterministic constant c; (ii) There exists r > 0 such that the intersection of the process with any ball of radius r is a.s. non-empty. We then set

Main results
The main result of the paper is the derivation of the macroscopic equations for the microscopic problem (1)-(3) using methods of stochastic homogenization. First we shall introduce the following notations. Denote by ∂ j ω the generator of a strongly continuous group of unitary operators in L 2 (Ω) associated with T x along e j -direction, i.e.
we denote the space of functions with continuous realisations and C 1 T (Ω) defines the set of functions from C T (Ω) such that (∂ j ω u) ∈ C T (Ω), for j = 1, 2, 3. First we introduce the spaces of potential and solenoidal vector fields: where the closure in the definition of L 2 pot (Ω) is with respect to the L 2 (Ω)-norm, see [50]. To introduce correctors we also need the space of functions whose realisations are discontinuous along the surface Γ(ω). We define We also denote We start with the definition of effective coefficients for macrosocpic poro-elastic equations, which are obtained by deriving the macroscopic equations for the microscopic problem (2)-(3). The macroscopic elasticity tensor E hom = (E hom ijkl ) and permeability tensor K hom where W kl e,sym denotes the symmetric part of the matrix W kl e , and W kl for k, l = 1, 2, 3, with b kl = 1 2 (e k ⊗ e l + e l ⊗ e k ) and {e j } 3 j=1 is the canonical basis of R 3 .
We also define Q(∂ t u f ) as where Then the macroscopic equations for the microscopic problem (2)-(3) are formulated as follows.
with boundary and initial conditions and the equations for the flow velocity for all ϕ ∈ L 2 (G T ; Remark. Notice that the equations for correctors (8) and (10), as well as problem (13) for ∂ t u f are formulated in the weak form as integral identities. This is due to the fact that the equations are define on Ω e ⊂ Ω and Ω f ⊂ Ω, respectively, and have strong formulation only for P-a.a. realisations ω ∈ Ω.
The homogenized coefficients in reaction-diffusion-convection equations that will be obtained by deriving macroscopic equations for microscopic problem (1), (3), are defined as where D(b e,3 , ω) = D e (b e,3 )χ Ωe (ω) + D f χ Ω f (ω) for ω ∈ Ω, with w j b ∈ L 2 pot (Ω) and w j ∈ L 2 pot,Γ (Ω) are solutions of the cell problems and The effective velocity is defined by where ϑ j = Ω χ Ω j (ω) dP(ω), for j = e, f , and Here µ is the Palm measure of the random measure µ ω of surfaces Γ(ω), see e.g. [20] for the definition of Palm measure.

A priori estimates
Considering assumptions on G ε j , with j = e, f , in the same way as in the periodic case [39], for Pa.a. realisations ω ∈ Ω, we obtain the existence, uniqueness and a priori estimates, uniform in ε, for solutions of microscopic problem (1)-(3).
and for the concentration of calcium c ε e and c ε f and densities of pectins and calcium-pectin cross-links b ε e we obtain b ε e L 2 (0,T ; and forT ∈ (0, T − h] and for P-a.a. ω ∈ Ω, where the constant C is independent of ε and For P-a.a. realisations ω ∈ Ω the proof of the a priori estimates follows the same lines as in [39,Lemma 6].
Using the assumptions on the random microscopic structure of G ε e and G ε f we obtain the following extension results for functions defined on G ε e and on a subdomain G ε ef ⊂ G, which will be specified below.
(ii) There exists an extension c ε of c ε from L 2 (0, T ; Here is a εσ-neighbourhood of Γ ε for P-a.a. realisations ω ∈ Ω and 0 < σ < d dim /4, with d min being the minimal distance between connected components of G f (ω).
Proof. The uniform boundedness of the diameter of cell walls and cell insider, independent on realisations ω ∈ Ω, implies the existence of the corresponding extension operators, see [1] Lemma 4.4. For extensions of b ε e , c ε e , u ε e , p ε e from G ε e,T to G T and c ε from G ε ef,T to G T (denoted again by b ε e , c ε e , u ε e , p ε e , and c ε ) we have the following estimates where the constant C is independent of ε. An extension of ∂ t u ε f from G ε f,T to G T , constructed below and denoted again by ∂ t u ε f satisfies the following estimates where the constant C is independent of ε. Also we have that and the constant C does not depend on ε.
Proof. The estimates for b ε e , c ε e , c ε , u ε e , and p ε e follow directly from estimates (20)-(22), Lemma 4.3, and the linearity of extension considered in Lemma 4.3.
Using geometrical assumptions on G f (ω), for P-a.a. ω ∈ Ω, we can extend ∂ t u ε f from G ε f to G in the following way. For each connected component G f,j (ω) of G f (ω), with j ∈ N, we can consider a σ-neighbourhood G σ f,j (ω) of G f,j (ω), where σ = d min /4 and d min is the minimal distance between G f,j (ω) for j ∈ N. Then since ∂ t u ε f ∈ L 2 (0, T ; H 1 (G ε f )), i.e. ∂ t u ε f ∈ L 2 (0, T ; H 1/2 (Γ ε )), there exists for P-a.a. realisations ω ∈ Ω and j ∈ N, see e.g. [45, Theorem 2.4, Lemma 2.4]. Each ∂ t u j f we extend by zero to G e (ω) \ G σ f,j (ω). Considering a scaling y = x/ε in ∂ t u j f and collecting all ∂ t u j f for j ∈ N we obtain an extension where the constant C is independent of ε. Similar to the periodic case to show the a priori estimates for p ε f we consider the first and third equations in (2) and use the a priori estimates for u ε e , p ε e , and ∂ t u ε f to obtain with φ ∈ L 2 (0, T ; H 1 (G)) 3 . Here we used the extension of p ε e from G ε e to G, see Lemma 4.3, and the trace estimate p ε e L 2 ((0,T )×∂G) ≤ C 1 p ε e L 2 (0,T ;H 1 (G)) ≤ C 2 p ε e L 2 (0,T ;H 1 (G ε e )) . For any q ∈ L 2 (G T ) there exists φ ∈ L 2 (0, T ; H 1 (G)) 3 satisfying div φ = q in G, φ · n = 1 |∂G| G q(·, x)dx on ∂G and φ L 2 (0,T ;H 1 (G)) 3 ≤ C q L 2 (G T ) . Thus using (28), the definition of the L 2 -norm, and the a priori estimates for p ε e we obtain where the constant C is independent of ε. Definition 5.1. Let G be a domain in R 3 , T x be an ergodic dynamical system, and ω be a "typical realisation". Then, we say that a sequence {v ε } ⊂ L 2 (0, T ; L 2 (G)) converges stochastically two-scale to v ∈ L 2 (G T ; L 2 (Ω, dP)) if

Convergence results
and As a "typical realisation" we denote such realisation ω ∈ Ω that Birkhoff's theorem is satisfied for P-a.s. for all bounded Borel sets A, |A| > 0, and all g(ω) ∈ C(Ω). Let us note that realisations are typical P-a.s., see e.g. [51]. Using compactness properties of stochastic two-scale convergence, see [51], we obtain the following result.
and for fluid velocity and pressure we have Proof. The estimates (26), the compactness of the embedding of H 1 (0, T ; L 2 (G)) ∩ L 2 (0, T ; H 1 (G)) in L 2 (G T ), and the compactness theorem for stochastic two-scale convergence, see e.g. [51], yield the convergence results in (34).
, respectively. Additionally we have that U 1 e χ Ωe , P 1 e χ Ωe , ∂ t u f χ Ω f , and ∇ ω ∂ t u f χ Ω f do not depend on the extension of u ε e , p ε e from G ε e to G and of ∂ t u ε f from G ε f to G. The estimate and definition of p ε in (28) and (31) ensure the stochastic two-scale convergence of χ G ε f p ε f .
In the following lemma, we shall use the same notation for b ε e , c ε e and their extensions from G ε e to G, whereas the extension for c ε from G ε ef to G will be denoted by c ε .
6 Derivation of macroscopic equations for flow velocity and elastic deformations.
To show the convergence of boundary terms we shall prove the relation between convergence with respect to P in G and Palm measure µ on the oscillating surfaces Γ ε .
Definition 6.1. [20] The Palm measure of the random stationary measure µ ω is the measure µ on (Ω, F) defined as Lemma 6.2. For u ∈ H 1 (Ω, P) we have that u ∈ L 2 (Ω, µ), where µ is the Palm measure of the random stationary measure µ ω of surfaces Γ(ω) for realisations ω ∈ Ω, and the embedding is continuous.
Proof. Consider u ∈ H 1 (Ω, P) and a random stationary measure µ ω given by dµ where dσ(x) is the standard surface measure. By µ we denote the Palm measure of the random stationary measure µ ω . Let Q ρ be the ball in R 3 of radius ρ centered at the origin. Since u ∈ H 1 (Ω, P), then a.s. u(T x ω) ∈ H 1 loc (R 3 ). Under our assumptions by the trace theorem there exist δ > 0 and C > 0 such that P-a.s. in Ω. We divide the left-and the right-hand sides of this relation by ρ 3 and pass to the limit, as ρ → ∞. By the Birkhoff theorem we obtain This yields the desired statement.
Proof of Theorem 3.1. To derive macroscopic equations for the system of poro-elastic and Stokes equations, first we consider as test functions in (4) the following functions , and P-a.a. realisations ω ∈ Ω. To apply stochastic two-scale convergence of u ε e , p ε e , and ∂ t u ε f , we rewrite the boundary integrals over Γ ε in the weak formulation (4) as volume integrals Here we have used the relation div∂ t u ε f = 0 in G ε f,T and the fact that χ G ε j (x, ω) = χ Ω j (T x/ε ω) P-a.s.
in Ω, where j = e, f . Using the convergence results in Lemma 5.2 and passing to the limit ε → 0 we obtain Letting first ψ 1 ≡ 0 and η 1 ≡ 0 and then φ 1 ≡ 0 and η 1 ≡ 0 we obtain the equations for the correctors U 1 e and P 1 e , i.e. and From (39) considering φ 1 ≡ 0 and ψ 1 ≡ 0 also yields Next, choosing in (4) test functions of the form (φ(t, x), ψ(t, x), η(t, x, x/ε)), where Letting ε → 0 and using the stochastic two-scale and strong convergences of u ε e and p ε e , the strong convergence of b ε e , and the stochastic two-scale convergence of ∂ t u ε f we obtain ∂ 2 t u e , φ χ Ωe G T ,Ω + E(ω, b e,3 ) e(u e ) + U 1 e,sym , e(φ) χ Ωe G T ,Ω + ∇p e + P 1 e , φ χ Ωe G T ,Ω + ∂ t p e , ψ χ Ωe G T ,Ω + K p (ω)(∇p e + P 1 Here we used the fact that χ Ω f p f = χ Ω f p e in G T × Ω. Since P 1 e ∈ L 2 (G T ; L 2 pot (Ω)) and η 1 ∈ C(G T ), η 2 ∈ L 2 sol (Ω) we obtain that P 1 e , η 1 η 2 G T ,Ω = 0. The stochastic two-scale convergence of ∂ t u ε f and the fact that ∂ t u ε f is divergence-free in G T (we identify here ∂ t u ε f with its extension constructed in Lemma 4.4) imply Thus div ω ∂ t u f = 0 a.e. in G T and P-a.s. in Ω.
Choosing φ ≡ 0 and ψ ≡ 0, and taking η = η 1 η 2 , where η 1 ∈ C 1 0 (G T ) and η 2 ∈ C 1 T (Ω) 3 , with div ω η 2 = 0 and Π τ η 2 (T x ω) = 0 on Γ(ω) P-a.s. in Ω, we conclude that ∂ t u f is a solution to problem (13). Taking η = η 1 η 2 , with η 2 = const and η 1 ∈ C 1 0 (0, T ; C 1 (G)) 3 as a test function in (13) yields Next we have to determine the boundary conditions for tangential components of ∂ t u f on Γ(ω) for P-a.a. ω ∈ Ω. From a priori estimates for ∂ t u ε e and ∂ t u ε f we have that where the constants C j , with j = 1, 2, 3, 4, are independent of ε. Thus using Lemmata 6.2 and 8.1 and the fact that ∂ t u f ∈ L 2 (G T ; H 1 (Ω)) and ∂ t u e ∈ L 2 (0, T ; H 1 (G)) we obtain (Ω) and typical realisations ω ∈ Ω. Thus for each typical realisation ω ∈ Ω we have , and using equality (44) together with where W kl e are solutions of the first equations in (8), yield the macroscopic equations for u e : where E hom is defined by (7), as well as the equation together with problem (41) for P 1 e . The structure of the problem (41) suggests that P 1 e should be of the form where W k p and W k u are solutions of cell problems (8), and Q f is a solution of problem (10). Substituting the right-hand side of (48) for P 1 e in (47) we obtain the macroscopic equations for p e in (11), where K hom p and K u are defined in (7). 7 Strong stochastic two-scale convergence of e(u ε e ), ∇p ε e , and ∂ t u ε f .
Due to the presence of nonlinear functions depending on e(u ε e ) and ∂ t u ε f in equations for b ε e , c ε e , and c ε f , in order to derive the macroscopic equations for b e and c we have to show that e(u ε e ) and ∂ t u ε f converge stochastically two-scale strongly. χ G ε e e(u ε e ) → χ Ωe (e(u e ) + U 1 e,sym ) strongly stochastic two-scale, χ G ε e ∇p ε e → χ Ωe (∇p e + P 1 e ) strongly stochastic two-scale, Proof. Similar to the periodic case [39], to show the strong stochastic two-scale convergence of e(u ε e ), p ε e , and ∂ t u ε f we prove the convergence of the energy related to the equations for u ε e , p ε e , and ∂ t u ε f . We consider a monotone decreasing function ζ : R + → R + , e.g. ζ(t) = e −γt for t ∈ R + , and define the energy functional for the microscopic problem (2)-(3) as for s ∈ (0, T ) and P-a.a. ω ∈ Ω. Considering ∂ t u ε e ζ 2 , p ε e ζ 2 , and ∂ t u ε f ζ 2 as test functions in (4) yields the equality Due to assumptions on E and ∂ t E there exists such γ > 0 that 2γ E 1 (ω, K(ξ))−∂ t E 1 (ω, K(ξ)) A·A ≥ 0 for all symmetric matrices A and ξ ∈ R, and P-a.a. ω ∈ Ω.
Considering equation (40) for the corrector U 1 e and taking ∂ t U 1 e ζ 2 as a test function imply Thus we obtain that E(u e , p e , ∂ t u f ) ≤ lim inf E ε (u ε e , p ε e , ∂ t u ε f ) ≤ lim sup E ε (u ε e , p ε e , ∂ t u ε f ) = E(u e , p e , ∂ t u f ), and, hence the strong stochastic two-scale convergence sated in Lemma.
8 Derivation of macroscopic equations for b e and c.
Using strong stochastic two-scale convergence of e(u ε e ) and ∂ t u ε f we derive macroscopic equations for concentrations of pectins b e and calcium c. First we shall prove convergence of sequences defined on the boundaries of the random microstructure Γ ε . Lemma 8.1. Consider the random measure µ ω denoting the surface measure of Γ(ω) and define dµ ε ω (x) = ε 3 dµ ω (x/ε). (1, ∞), then for any φ ∈ C ∞ (0, T ; C ∞ 0 (R 3 )) and any ψ ∈ C(Ω) we have and (ii) If b ε L p (G ε e,T ) + ε ∇b ε L p (G ε e,T ) ≤ C and b ε → b stochastic two-scale, b ∈ L p (G T , W 1,p (Ω, dP)), with p ∈ (1, ∞), then convergence (56) holds, and Proof. For P-a.a. realisations ω ∈ Ω, using the assumptions on the geometry of G ε e and the trace inequality in each G σ e,j = G σ f,j (ω) \ G f,j (ω), see proof of Lemma 4.4 for the definition of G σ f,j (ω), applying the scaling x/ε and summing up over j we obtain Moreover, in the case (i) the limit function b does not depend on ω, its trace on Γ ε e,T is well defined, and This yields (56).
In the case (ii), for b ∈ L p (G T ; W 1,p (Ω, dP)) using the same arguments as in the proof of Lemma 6.2 one can show that b ∈ L p (G T ; L p (Ω, µ)). This yields (58).
To justify (56) we regularize measures µ ω as follows. Let k = k(x) be a non-negative symmetric It is easy to check that a.s. for any test functions φ ∈ C ∞ (0, T ; C ∞ 0 (R d )) and ψ ∈ C(Ω) we have The Palm measure of dµ ω,δ (x) is dµ δ (ω) = ρ δ (ω)dP. Since for each δ > 0 the measure µ δ is absolutely continuous with respect to dP and the density ρ δ is bounded, the two-scale limit of b ε with respect to dµ ε ω,δ is b that is By the trace theorem a.s. lim sup ε→0 b ε L p (G T ,dµ ε ω ) ≤ C. Therefore, for a subsequence b ε stochastically two-scale converge in L p (G T , dµ ε ω ) to some function B ∈ L p (G T ; L p (Ω, dµ)). As was proved in [51], the measures dµ δ converge weakly to the measure dµ. Using one more time the same arguments as in the proof of Lemma 6.2 we obtain Passing to the limit δ → 0 and combining the above relations, we conclude that In view of arbitrariness of φ and ψ this implies that B = b in L p (G T ; L p (Ω, dµ)).
Using the convergence on the oscillating boundary Γ ε proved in Lemma 8.1 we can now derive macroscopic equations for b e and c.

Proof of Theorem 3.2. We can rewrite the microscopic equation for
T (Ω). From the a priori estimates for b ε and assumptions on R we have that where the constant C is independent of ε. Thus considering the stochastic two-scale convergence we obtain that there exists where µ ω is the random measure of Γ(ω). Using the assumptions on the geometry and on the function R together with the strong convergence of b ε e in L 2 (G T ) we have that Then using the strong convergence of b e , the continuity of R and the convergence result in Lemma 8.1 we obtain that R = R(b e ) P-a.s. in G T × Ω.
Taking the stochastic two-scale limit and using the strong convergence of b ε e and c ε e and the strong stochastic two-scale convergence of e(u ε e ), shown in Lemma 7.1, we obtain To show the convergence of g b (c ε e , b ε e , e(u ε e )) we considered an approximation of U 1 e,sym ∈ L 2 (G T ×Ω) by U δ ∈ C(G T ; C T (Ω)), such that U δ → U 1 e,sym in L 2 (G T × Ω) as δ → 0. For P-a.a. ω ∈ Ω we define U ε δ (t, x) = U δ (t, x, T x/ε ω). Using the strong stochastic two-scale convergence of U ε δ to U δ we obtain lim δ→0 lim ε→0 U ε δ L 2 (G T ) = lim δ→0 U δ L 2 (G T ×Ω) = U 1 e,sym L 2 (G T ×Ω) .
Considering φ 1 = 0 and using the linearity of the resulted equation we obtain B 1 e (t, x, ω) = 9 Well-posedness of the macroscopic problem In the same way as in the case of periodic microstructure [39], using fixed point iteration we show existence of an unique solution of the limit problem.
Proof. First we show estimates for two iterations (u j−1 e , ∂ t p j−1 e , ∂ t u j−1 f ), (b j−1 e , c j−1 ) and (u j e , ∂ t p j e , ∂ t u j f ), (b j e , c j ) for limit problem (11)-(13), (19). We begin with the equations for fluid flow velocity ∂ t u f and for elastic displacement u e . Taking ∂ t u f − ∂ t u e as a test function in the equation for the difference ∂ t u j f and ∂ t u e as a test function in the equations for the difference u j e we obtain ∂ t u j e (s) 2 L 2 (G) + E hom (b j−1 e,3 )e( u j e (s)), e( u j e (s)) Ω − ∂ t E hom (b j−1 e,3 )e( u j e ), e( u j e ) Gs + 2 (E hom (b j e,3 ) − E hom (b j−1 e,3 )) e(u j e (s)), e( u j e (s)) G − 2 ∂ t (E hom (b j e,3 ) − E hom (b j−1 e,3 )) e(u j e ) + (E hom (b j e,3 ) − E hom (b j−1 e,3 )) ∂ t e(u j e ), e( u j e ) Gs + ∂ t u j f (s) χ Ω f 2 L 2 (G×Ω) + 2µ e ω (∂ t u j f ) χ Ω f 2 L 2 (Gs×Ω) + 2 ∇ p j e , ∂ t u j e χ Ωe + ∂ t u j f χ Ω f Gs,Ω = 2 P 1,j e , ∂ t u j f χ Ωe − ∂ t u j e χ Ωe Gs,Ω + ∂ t u j f (0) χ Ω f 2 L 2 (G×Ω) + ∂ t u j e (0) 2 3 ) e( u j e (0)), e( u j e (0)) G + 2 (E hom (b j e,3 ) − E hom (b j−1 e,3 )) e(u j e (0)), e( u j e (0)) G , where u j e = u j e − u j−1 e , p j e = p j e − p j−1 e , u j f = u j f − u j−1 f , and P 1,j e = P 1,j e − P 1,j−1 e . The equation (11) for p j e and p j−1 e yields p j e (s) 2 Due to the assumptions in A1 on E, the definition of the macroscopic elasticity tensor E hom and the properties of a solution W kl e , with k, l = 1, 2, 3, of the corresponding cell problems in (8) (8) and (10) yield P 1,j e L 2 (Gs×Ω) ≤ C ∇ p j e L 2 (Gs) + ∂ t u j e L 2 (Gs) + ∂ t u j f χ Ω f L 2 (Gs×Ω) .
Adding (66) and (67), and applying the Hölder and Gronwall inequalities yield ∂ t u j e L ∞ (0,s;L 2 (G)) + e( u j e ) L ∞ (0,s;L 2 (G)) + p j e L ∞ (0,s;L 2 (G)) + ∇ p j e L 2 (Gs) for all s ∈ (0, T ] and the constant C does not depend on s and solutions of the macroscopic problem.