Acta Mathematica Scientia 2009, 29B(1):7482 [610095]

Acta Mathematica Scientia 2009, 29B(1):74–82
http://actams.wipm.ac.cn
HOMOGENIZATION RESULTS FOR ENZYME
CATALYZED REACTIONS THROUGH
POROUS MEDIA∗
Claudia Timofte
Department of Mathematics, Faculty of Physics, University of Bucharest,
P.O. Box MG-11, Bucharest-Magurele, Romania
E-mail: [anonimizat]
Abstract The aim of this article is to study the effective behavior of the solution of
a nonlinear problem arising in the modelling of enzyme catalyzed reactions through the
exterior of a domain containing periodically distributed reactive solid obstacles, with period
ε. The asymptotic behavior of the solution of such a problem is governed by a new elliptic
boundary-value problem, with an extra zero-order term that captures the effect of theenzymatic reactions.
Key words Homogenization, enzyme, reactive flows, Michaelis-Menten mechanism
2000 MR Subject Classification 35B27, 35B40, 35Q35
1 Introduction
The goal of this article is to analyze the effective behavior of some nonlinear reactive
flows through a domain containing periodically distributed reactive solid grains. Such problems
are very natural in the study of enzymatic reactions through periodic perforated media and,
more precisely, in the study of the so-called Michaelis-Menten model (for the chemical aspects
involved in such a model, see [8, 9] and the references therein). Enzymes are proteins that
speed up the rate of a chemical reaction without being used up. They are specific to particular
substrates. The substrates in the reaction bind to active sites on the surface of the enzyme.
The enzyme-substrate complex then undergoes a reaction to form a product along with the
original enzyme. The rate of chemical reactions increases with the substrate concentration.
However, enzymes become saturated when the substrate concentration is high. Additionally,
the reaction rate depends on the properties of the enzyme and the enzyme concentration.We can describe the reaction rate with a simple equation to understand how enzymes affect
chemical reactions. Michaelis-Menten equation remains the most generally applicable equation
for describing enzymatic reactions.
∗Received April 21, 2006

No.1 Claudia: HOMOGENIZATION RESULTS FOR ENZYME CATALYZED REACTIONS 75
Let Ω be an open bounded set in Rn,n≥3. We shall consider that Ω is an ε-periodic
structure, consisting of two parts: a fluid phase Ωεand a solid skeleton (grains), Ω \
Ωε;ε
represents a small parameter related to the characteristic size of the grains.
The nonlinear problem studied in this article concerns the stationary flow of a fluid confined
in Ωε, of concentration uε, reacting inside Ωεand, also, on the boundary of the grains:
⎧
⎪⎪⎪⎨
⎪⎪⎪⎩−D
fΔuε+β(uε)=fin Ωε,
−Df∂uε
∂ν=aεg(uε)o n Sε,
uε=0 o n∂Ω.(1.1)
Here, νis the exterior unit normal to Ωε,a>0,f∈L2(Ω) and Sεis the boundary of the porous
medium Ω \
Ωε. The fluid is assumed to be homogeneous and isotropic, with a constant diffusion
coefficient Df>0. We shall take the function βin (1.1) to be a continuously differentiable
function, monotonously nondecreasing, and such that β(0) = 0. For example, we can take βto
be a linear function, that is, β(v)=λv, or we can consider the nonlinear case, in which
β(v)=a1v
1+a2v,a1,a2>0 (Langmuir kinetics) .
Also, in the boundary condition on Sεin problem (1.1), the function gis assumed to
be given. We shall deal here with the case of a single-valued maximal monotone graph with
g(0) = 0, that is, the case in which gis the subdifferential of a convex lower semicontinuous
function G. More precisely, we shall consider an important practical example, arising in the
diffusion of enzymes (the Michaelis-Menten model):
g(v)=⎧
⎨
⎩δv
v+γ,v ≥0,
0,v < 0,
forδ, γ > 0.
The existence and uniqueness of a weak solution of (1.1) is ensured by the classical theory
of monotone problems (see [1] and [6]). Therefore, we know that there exists a unique weak
solution uε∈Vε/intersectiontextH2(Ωε), where
Vε={v∈H1(Ωε)|v=0o n ∂Ω}.
If we consider the following nonempty convex subset of Vε:
Kε={v∈Vε|G(v)|
Sε∈L1(Sε)}, (1.2)
then, uεis also the unique solution of the following variational problem:
⎧
⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩Find u
ε∈Kεsuch that
Df/integraldisplay
ΩεDuεD(vε−uε)dx+/integraldisplay
Ωεβ(uε)(vε−uε)dx
−/integraldisplay
Ωεf(vε−uε)dx+a/angbracketleftμε,G(vε)−G(uε)/angbracketright≥0,∀vε∈Kε,(1.3)

76 ACTA MATHEMATICA SCIENTIA Vol.29 Ser.B
where μεis the linear form on W1,1
0(Ω) defined by
/angbracketleftμε,ϕ/angbracketright=ε/integraldisplay
Sεϕdσ,∀ϕ∈W1,1
0(Ω).
We shall consider, here, only periodic structures obtained by removing periodically from
Ω, with period εY(where Yis a given hyper-rectangle in Rn), an elementary obstacle Twhich
has been appropriately rescaled and which is strictly included in Y,t h a ti s ,
T⊂Y.
We shall prove that the solution uε, properly extended to the whole Ω, converges to the
unique solution of the following variational inequality:
⎧
⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩u∈H
1
0(Ω),/integraldisplay
ΩQDuD (v−u)dx+/integraldisplay
Ωβ(u)(v−u)dx≥/integraldisplay
Ωf(v−u)dx
−a|∂T|
/vextendsingle/vextendsingleY\
T/vextendsingle/vextendsingle/integraldisplay
Ω(G(v)−G(u))dx,∀v∈H1
0(Ω).(1.4)
Here, Q=(qij) is the homogenized matrix, whose entries are defined by:
qij=Df/parenleftBigg
δij+1
/vextendsingle/vextendsingleY\
T/vextendsingle/vextendsingle/integraldisplay
Y\
T∂χj
∂yidy/parenrightBigg
, (1.5)
in terms of the functions χ
i,i=1,···,n, solutions of the following cell problems:
⎧
⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩−Δχ
i=0 i n Y\
T,
∂(χi+yi)
∂ν=0 o n ∂T,
χiY−periodic.(1.6)
Notice that ualso satisfies
⎧
⎪⎪⎨
⎪⎪⎩−n/summationdisplay
i,j=1qij∂2u
∂xi∂xj+β(u)+a|∂T|
/vextendsingle/vextendsingleY\
T/vextendsingle/vextendsingleg(u)=fin Ω,
u=0 o n∂Ω.(1.7)
Let us also remark that the effect of the enzymatic reactions initially situated on the
boundaries of the grains spreaded out in the limit all over the domain, giving the extra zero-
order term which captures this boundary effect. In fact, one could obtain a similar result
by considering interior enzymatic nonlinear chemical reactions given by the same well-known
nonlinear function g. The only difference in the limit equation would be the coefficient appearing
in front of this extra zero-order term. So, in fact, we can control the effective behavior of suchreactive flows by choosing different locations for the involved chemical reactions. Moreover,
we could obtain similar effects by considering transmission problems, with an unknown flux
on the boundary of each grain, that is, we can consider the case in which we have chemical
reactions in Ω
ε, but also inside the grains, instead on their boundaries. The difference in the
limit equation would be the coefficient appearing in front of this extra zero-order term. Hence,
we can control the effective behavior of such reactive flows by choosing different locations for
the involved chemical reactions.

No.1 Claudia: HOMOGENIZATION RESULTS FOR ENZYME CATALYZED REACTIONS 77
To illustrate this situation, we can focus on the case of a function gwhich satisfies (2.1),
it is continuous and monotone increasing, with g(0) = 0; our previous example is still covered
by this class of functions.
If we denote by vεthe concentration inside the grains, one can consider a transmission
problem described by the following system:
⎧
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩−D
fΔuε+β(uε)=fin Ωε,
−DpΔvε+ag(vε)=0 i nΩ \
Ωε,
−Df∂uε
∂ν=Dp∂vε
∂νonSε,
uε=vεonSε,
uε=0 o n∂Ω,(1.8)
where Dpis the diffusion coefficient characterizing the granular material filling the obstacles.
As in the previous case, the classical semilinear theory guarantees the well-posedness of this
problem.
If we define θεas
θε(x)=⎧
⎨
⎩uε(x),x ∈Ωε,
vε(x),x ∈Ω\
Ωε,
and introduce
A=⎧
⎨
⎩DfId, inY\
T,
DpId, inT,
then, it is not difficult to prove, reasoning exactly like in this article, that θεconverges weakly
inH1
0(Ω) to the unique solution of the following homogenized problem:
⎧
⎪⎪⎨
⎪⎪⎩−n/summationdisplay
i,j=1a0
ij∂2u
∂xi∂xj+a|T|
/vextendsingle/vextendsingleY\
T/vextendsingle/vextendsingleg(u)+β(u)=fin Ω,
u=0 o n∂Ω.(1.9)
Here, A0=(a0
ij) is the homogenized matrix, whose entries are defined by:
a0
ij=1
|Y|/integraldisplay
Y/parenleftbigg
aij+aik∂χj
∂yk/parenrightbigg
dy, (1.10)
in terms of the functions χ
j,j=1,···,n,solutions of the following cell problems:
⎧
⎪⎨
⎪⎩−div(AD(yj+χ
j)) = 0 in Y,
χ
j−Yperiodic .(1.11)
So, at this time, as already mentioned, we get a similar extra term capturing the enzymatic
reaction effects given by the nonlinear function g, but with a different coefficient in front of it
(compare (1.7) with (1.9)).
We can treat in a similar manner the case of a multi-valued maximal monotone graph,
which includes various semilinear boundary-value problems, such as Dirichlet, Neumann or
Robin problems, Signorini’s problems, and problems arising in chemistry (see [2], [4] and [5]).

78 ACTA MATHEMATICA SCIENTIA Vol.29 Ser.B
The structure of the article is as follows: Section 2 is devoted to the setting of our problem
and to the formulation of the main result of this article. Section 3 contains some necessary
preliminary results. In the last section, we give the proof of our main result.
2 Setting of the Problem and the Main Result
Let Ω be a smooth (of class C2) bounded connected open subset of Rn(n≥3) and let
Y=[ 0,l1[×···[0,ln[ be the representative cell in Rn.D e n o t eb y Tan open subset of Y,w i t h
boundary ∂Tof class C2, such that
T⊂Y.
Letεbe a real parameter taking values in a sequence of positive numbers convergent to
zero. For each εand for any integer vector k∈Zn,s e tTε
k=ε(kl+T) the translated image of
εTby the vector εkl=ε(k1l1,···,knln) and denote by Tεthe set of all the obstacles contained
in Ω, that is,
Tε=/uniondisplay/braceleftbig
Tε
k|
Tε
k⊂Ω,k∈Zn/bracerightbig
.
Set Ωε=Ω\
TεandSε=∪{∂Tε
k|
Tε
k⊂Ω,k∈Zn}. Also, let Y∗=Y\
Tandρ=|Y∗|
|Y|.
Moreover, for an arbitrary function ψ∈L2(Ωε),we shall denote by /tildewideψits extension by zero
inside the obstacles.
As already mentioned, we are interested in studying the asymptotic behavior of the solution
of problem (1.1).
We consider that the function βin (1.1) is a continuously differentiable function, monoto-
nously nondecreasing with β(0) = 0, and we take B(v)=/integraltextv
0β(s)dy. Also, we address here
the case in which the function gappearing in (1.1) is a single-valued maximal monotone
graph in R×R,w i t h g(0) = 0. Moreover, if we denote by D(g) the domain of g,t h a ti s ,
D(g)={ξ∈R|g(ξ)/negationslash=∅}, then suppose that D(g)=R. Finally, we assume that gis
continuous and there exist C≥0, and an exponent q,w i t h0 ≤q<n / (n−2), such that
⎧
⎪⎨
⎪⎩/vextendsingle/vextendsingle/vextendsingle/vextendsingledβ
dv/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(1 +|v|
q),
|g(v)|≤C(1 +|v|q).(2.1)
We know that in this case there exists a lower semicontinuous convex function Gfrom Rto
]−∞,+∞],Gbeing proper, that is, G/negationslash≡+∞, such that gis the subdifferential of G:g=
∂G.L e t G(v)=/integraltextv
0g(s)dy.This general situation is well illustrated by the above mentioned
important practical examples.
By classical results (see [1] and [6]), we know that there exists a unique weak solution
uε∈Vε/intersectiontextH2(Ωε).
If the convex set Kεis defined by (1.2), then, for a given function f∈L2(Ω), the weak
solution of the variational problem (1.3) is also the unique solution of the minimization problem:
⎧
⎨
⎩uε∈Kε,
Jε(uε)= i n f
v∈KεJε(v),(2.2)
where
Jε(v)=1
2Df/integraldisplay
Ωε|Dv|2dx+/integraldisplay
ΩεB(v)dx+a/angbracketleftμε,G(v)/angbracketright−/integraldisplay
Ωεfvdx.

No.1 Claudia: HOMOGENIZATION RESULTS FOR ENZYME CATALYZED REACTIONS 79
Let us introduce the following functional defined on H1
0(Ω):
J0(v)=1
2/integraldisplay
ΩQDvDv dx+a|∂T|
|Y∗|/integraldisplay
ΩG(v)dx+/integraldisplay
ΩB(v)dx−/integraldisplay
Ωfvdx.
The main result of this article is the following
Theorem 2.1 One can construct an extension Pεuεof the solution uεof the variational
inequality (1.3), such that
Pεuε/arrowrighttophalfu weakly in H1
0(Ω),
where uis the unique solution of the following minimization problem:
⎧
⎨
⎩Findu∈H1
0(Ω) such that
J0(u)= i n f
v∈H1
0(Ω)J0(v).(2.3)
Moreover, G(u)∈L1(Ω). Here, Q=(qij) is the classical homogenized matrix whose entries are
defined by (1.5)–(1.6).
3 Preliminary Results
To extend the solution uεof problem (1.3) to the whole Ω, let us recall the following
well-known result (see [3]):
Lemma 3.1 There exist a linear continuous extension operator L(L2(Ωε);L2(Ω))∩
L(Vε;H1
0(Ω)) and a positive constant C, independent of ε, such that, for any v∈Vε,
/bardblPεv/bardblL2(Ω)≤C/bardblv/bardblL2(Ωε)
and
/bardbl∇Pεv/bardblL2(Ω)≤C/bardbl∇v/bardblL2(Ωε).
For getting the effective behavior of our solution uε, we have to pass to limit in (1.3). To
do this, let us introduce, for any h∈Ls/prime(∂T), 1≤s/prime≤∞, the linear form με
honW1,s
0(Ω)
defined by
/angbracketleftμε
h,ϕ/angbracketright=ε/integraldisplay
Sεh(x
ε)ϕdσ∀ϕ∈W1,s
0(Ω),
with 1 /s+1/s/prime=1.It was proved in [2] that
μεh→μhstrongly in ( W1,s
0(Ω))/prime, (3.1)
where /angbracketleftμh,ϕ/angbracketright=μh/integraltext
Ωϕdx,w i t h
μh=1
|Y|/integraldisplay
∂Th(y)dσ.
In the particular case h∈L∞(∂T)o re v e n his constant, we have
μεh→μhstrongly in W−1,∞(Ω). (3.2)
We denote by μεthe above introduced measure in the case h=1 .

80 ACTA MATHEMATICA SCIENTIA Vol.29 Ser.B
LetFbe a continuously differentiable function, monotonously nondecreasing, and such
thatF(v) = 0 if and only if v= 0. We suppose that there exist a positive constant Cand an
exponent q,w i t h0 ≤q<n / (n−2), such that/vextendsingle/vextendsingle∂F
∂v/vextendsingle/vextendsingle≤C(1 +|v|q). It is not difficult to prove
(see [4]) that for any ϕ∈D(Ω) = C∞
0(Ω) and for any zε/arrowrighttophalfzweakly in H1
0(Ω), we get
ϕF(zε)/arrowrighttophalfϕ F (z)w e a k l y i n W1,
q
0(Ω), (3.3)
where
q=2n
q(n−2)+n.
4 Proof of the Main Result
Proof of Theorem 2.1 Letuεbe the solution of the variational inequality (1.3) and let
Pεuεbe the extension of uεgiven in Lemma 3.1. One can easily see that Pεuεis bounded in
H1
0(Ω). So, by passing to a subsequence, we have
Pεuε/arrowrighttophalfu weakly in H1
0(Ω).
Letϕ∈D(Ω). By classical regularity results, χi∈L∞. Using the boundedness of χiandϕ,
there exists M≥0, such that /bardbl∂ϕ
∂xi/bardblL∞/bardblχ
i/bardblL∞<M.L e t
vε=ϕ+/summationdisplay
iε∂ϕ
∂xi(x)χ
i(x
ε).
Then, vε∈Kε, which will allow us to take it as a test function in (1.3). Moreover, vε→ϕ
strongly in L2(Ω). Let us compute Dvε.W eo b t a i n
Dvε=/summationdisplay
i∂ϕ
∂xi(x)(ei+Dχ
i(x
ε)) +ε/summationdisplay
iD∂ϕ
∂xi(x)χ
i(x
ε),
where ei,1≤i≤n, are the elements of the canonical basis in Rn.
Using vεas a test function in (1.3), we can write
Df/integraldisplay
ΩDPεuε/tildewider(Dvε)dx+/integraldisplay
Ωεβ(uε)(vε−uε)dx
≥/integraldisplay
Ωεf(vε−uε)dx+Df/integraldisplay
ΩεDuεDuεdx−a/angbracketleftμε,G(vε)−G(uε)/angbracketright. (4.1)
Denote
ρQej=1
|Y∗|Df/integraldisplay
Y∗(Dχ
j+ej)dy, (4.2)
where ρ=|Y∗|/|Y|. Neglecting the term ε/summationtext
iD∂ϕ
∂xi(x)χ
i(x
ε) which tends strongly to zero, we
can pass easily to limit in the first term of the left-hand side of (4.1). Hence,
Df/integraldisplay
ΩDPεuε/tildewidestDvεdx→/integraldisplay
ΩρQDuDϕ dx. (4.3)
For the second term on the left-hand side of (4.1), let us notice that, exactly like in [4], one can
easily prove that for any zε/arrowrighttophalfzweakly in H1
0(Ω), we get
β(zε)→β(z) strongly in L
q(Ω),

No.1 Claudia: HOMOGENIZATION RESULTS FOR ENZYME CATALYZED REACTIONS 81
where
q=2n
q(n−2)+n. Therefore, we have
/integraldisplay
Ωεβ(uε)(vε−uε)dx→/integraldisplay
Ωβ(u)ρ(ϕ−u)dx. (4.4)
For the first term of the right-hand side of (4.1), we get
/integraldisplay
Ωεf(vε−uε)dx=/integraldisplay
ΩfχΩε(vε−Pεuε)dx→/integraldisplay
Ωfρ(ϕ−u)dx. (4.5)
For the third term of the right-hand side of (4.1), assuming (2.1) for the maximal monotone
graph gand using (3.3) written for Gand for zε=Pεuε,w eg e t
G(Pεuε)/arrowrighttophalfG(u)w e a k l y i n W1,
q
0(Ω).
Combining this with the convergence (3.2) written for h=1 ,w eh a v e
/angbracketleftμε,G(Pεuε)/angbracketright→|∂T|
|Y|/integraldisplay
ΩG(u)dx.
Using a similar technique for the convergence of /angbracketleftμε,G(vε)/angbracketright,w eo b t a i n
a/angbracketleftμε,G(vε)−G(Pεuε)/angbracketright→a|∂T|
|Y|/integraldisplay
Ω(G(ϕ)−G(u))dx. (4.6)
For passing to limit in the second term of the right-hand side of (4.1), let us consider thesubdifferential inequality
D
f/integraldisplay
ΩεDuεDuεdx≥Df/integraldisplay
ΩεDwεDwεdx+2Df/integraldisplay
ΩεDwε(Duε−Dwε)dx, (4.7)
for any wε∈H1
0(Ω). With the same reasoning as before, choosing
wε=
ϕ+/summationdisplay
iε∂
ϕ
∂xi(x)χ
i(x
ε),
where
ϕhas similar properties as the corresponding ϕ, the right-hand side of the inequality
(4.7) passes to limit, and we obtain
lim inf
ε→0Df/integraldisplay
ΩεDuεDuεdx≥/integraldisplay
ΩρQD
ϕD
ϕdx+2/integraldisplay
ΩρQD
ϕ(Du−D
ϕ)dx,
for any
ϕ∈D(Ω) and, by density, for any
ϕ∈H1
0(Ω). Hence, for u∈H1
0(Ω), we have
lim inf
ε→0Df/integraldisplay
ΩεDuεDuεdx≥/integraldisplay
ΩρQDuDu dx. (4.8)
From (4.3)–(4.6) and (4.8), we get
/integraldisplay
ΩρQDuDϕ dx+/integraldisplay
Ωβ(u)ρ(ϕ−u)dx
≥/integraldisplay
Ωfρ(ϕ−u)dx+/integraldisplay
ΩρQDuDu dx−a|∂T|
|Y|/integraldisplay
Ω(G(ϕ)−G(u))dx,
for any ϕ∈D(Ω) and, by density, for any v∈H1
0(Ω). So, finally, we obtain
/integraldisplay
ΩQDuD (v−u)dx+/integraldisplay
Ωβ(u)(v−u)dx≥/integraldisplay
Ωf(v−u)dx−a|∂T|
|Y∗|/integraldisplay
Ω(G(v)−G(u))dx,
which gives exactly the limit problem (2.3). This ends the proof of Theorem 2.1.
Remark The results of this article are obtained for the case n≥3. All of them are still
valid, under our assumptions, in the case n= 2. Of course, for this case, n/(n−2) has to be
replaced by + ∞in (2.1).

82 ACTA MATHEMATICA SCIENTIA Vol.29 Ser.B
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