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A mathematical study of an enzymatic reaction in non homogeneous reactors with substrate and product inhibition: application to cellobiose hydrolysis in a bioreactor with a dead-zone

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A mathematical study of an enzymaticreaction in non homogeneous reactors withsubstrate and product inhibition:application to cellobiose hydrolysis in abioreactor with a dead-zone
A. Saddoud
∗
T. Sari
∗
A. Rapaport
∗
R. Lortie
∗∗
J. Harmand
∗∗∗
E. Dubreucq
∗∗∗∗∗
UMR INRA-SupAgro MISTEA, Montpellier, France E-mails: saddoud@supagro.inra.fr, tewﬁk.sari@uha.fr,rapaport@supagro.inra.fr
∗∗
NRC Biotechnology Research Institute, National Research Council,Montreal, Canada E-mail: Robert.Lortie@cnrc-nrc.gc.ca
∗∗∗
INRA LBE, Narbonne, France E-mail: harmand@supagro.inra.fr
∗∗∗∗
UMR INRA/SupAgro/CIRAD/UMII IATE, Montpellier, France E-mail: dubreucq@supagro.inra.fr
Abstract:
This paper presents a mathematical study for an enzymatic reaction with bothsubstrate and product inhibition in two interconnected Continuous Stirred Tank Reactors(ICSTRs) where one of them is a dead zone.We propose necessary and suﬃcient conditionsthat guarantee the steady states multiplicity. The main purpose of this work is to estimate thekinetic and operating parameters which give, in experimental data, the steady states multiplicityand guarantee the presence of the best steady state that has the optimum biological performancein a non homogeneous reaction system.Keywords: Mathematical study, enzymatic hydrolysis, interconnected reactors, dead zone,multiplicity, biological performance.1. INTRODUCTIONMultiplicity of steady states has been studied in consid-erable detail in diﬀerent types of chemical and biologicalreactors [1, 20, 30, 12]. This subject is of great interestbecause each of those steady states is marked by distinctiveconversion, selectivity and stability and some of them aremore desirable for the manipulator than others. Thus, itis important to predict the conditions for multiplicity tooccur. In biological systems, multiplicity of steady stateshas been reported both in cellular growth and in enzy-matic reactors. Such a multiplicity in enzyme reactor aretheoretically and experimentally discussed [25, 28].Cellulases are a consortium of hydrolytic enzymes whichadsorb on and hydrolyses insoluble cellulose. The classicMichaelis-Menten approach is inadequate to describe thebehavior of cellulases on insoluble cellulose. However, thisapproach is adequate to describe the behavior of cellu-lases on soluble cellulose. For these two cases, modelshave been developed to describe the hydrolysis of cel-lulosic substrate. Indeed, number of kinetic models of enzymatic cellulose hydrolysis have been developed dur-ing the past several decades [32, 31, 7, 6]. These models
⋆
This work is ﬁnancially supported by Agropolis Foundation
⋆⋆
J. Harmand, A. Rapaport and T. Sari are members of the INRA-INRIA project ’MERE’. T . Sari is on leave from University of Mulhouse, France.
cover the entire domain and factors of operations such asphysical properties of the substrate (srcin, composition,cristallinity, degree of polymerisation...etc), enzyme syn-ergy (srcin, composition...etc), mass transfer (substrateadsorption, bulk and pore diﬀusion...etc) and intrinsickinetics [32, 35, 6, 24, 34, 7]. These models belong totwo types of modeling approaches, empirical and mecha-nistic models. Empirical models relate the factors usingmathematical correlations without any insight into theunderlying mechanisms. Mechanistic models are developedfrom the reaction mechanisms, mass transfer considerationand other physical parameters that aﬀect the extent of hydrolysis. Several mechanistic models of cellulose hydrol-ysis are cited in the literature [31, 13, 18, 22]. Most of them can explain very well the initial progression of theenzymatic hydrolysis. However, they fail to predict thelater stages of hydrolysis. Moreover, these models do notincorporate the speciﬁc characteristics of cellulose coupledwith the heterogeneity of the environmental conditionssuch as space heterogeneity.In an eﬀort to contribute to the understanding of enzymereactors involving complex kinetics and space heterogene-ity, we decided to develop a new mechanistic model whichtakes into account the inhomogeneity in medium compo-sition. A series of CSTRs can be used to approximate thebehaviour of a single Packed Bed Reactor (PBR). In mostpractical situations and especially at steady state, PBR
exhibits better performances than CSTRs for catalyticreaction systems. However, in practice, it is preferableto use a number of chemostats in series, especially forthe simplicity of their building and their mathematicalsimulation.Multiplicity of steady states during enzymatic cellulosicsubstrate hydrolysis in heterogeneous conditions has notbeen investigated. We propose here a mathematical anal-ysis in the spirit of the work [12] that was addressingmicrobial ecosystems with “cascade” heterogeneity, andnot enzymatic reactions.The main objectives of this work are, ﬁrstly, to develop amechanistic model for enzymatic hydrolysis of cellobiose,the last intermediate in the hydrolysis of cellulose, in thecomplex system simulating an inhomogeneous environ-ment. Secondly, to ﬁnd the necessary conditions for steadystates multiplicity. Finally, to investigate at steady statethe biological performance of the enzymatic system and toderive the experimental parameters which give the optimalperformance in the domain of multiplicity.The paper is organized as follows. In section 2, the dy-namical model of two interconnected CSTRs where one of them is a dead zone is derived. In section 3, the suﬃcientcondition of steady state multiplicity in the case of onereactor is achieved. Section 4 is devoted to discuss theexistence and the stability of steady states and to deﬁnethe domain of multiplicity in the case of two reactors.In section 5, the evaluation of the biological performanceof the bioprocess under the estimated parameters in thedomain of multiplicity is achieved.2. MATHEMATICAL MODELIt is often observed in CSTR a “dead” zone where themixing is not well achieved. A simple and convenient wayfor taking into account this phenomenon is to consider thata tank of volume
V
is divided into two parts, describedas two abstract chemostats of volumes
αV
and (1
−
α
)
V
,respectively (see Fig. 1). For the process into considerationin this paper, the soluble cellulosic substrate is continu-ously fed in the ﬁrst tank with a constant ﬂow rate
Q
1
,the second one (the dead zone) being connected to theﬁrst one through a diﬀusion with rate
Q
2
. The dynamical
VQ
2
S
out
Q
1
Q
1
S
in0
(1−α)α
V
Fig. 1. Two continuous stirred tank reactorsmodel of the general system represented in Fig. 1 can bewritten as follows:
dS
1
dt
=
Q
1
(
S
0
in
−
S
1
) +
Q
2
(
S
2
−
S
1
)
αV
−
ν
(
S
1
)
dS
2
dt
=
Q
2
(
S
1
−
S
2
)(1
−
α
)
V
−
ν
(
S
2
)(1)where
S
1
and
S
2
are the substrate concentrations in theﬁrst and second reactor, respectively;
Q
1
is the input ﬂowrate in the ﬁrst reactor;
Q
2
is the diﬀusion rate betweenreactors and
αV
and (1
−
α
)
V
are the volumes of reactors.Consider the kinetic function
ν
(
S
) =
V
max
S K
M
[1 +
P/K
ip
] +
S
+
S
2
/K
is
(2)where
P
is the product of the reaction
P
= 2(
S
0
in
−
S
)
,
(3)as two glucose molecules are produced from one moleculeof substrate.
V
max
is the maximum reaction rate,
K
M
is the half-saturation constant,
K
is
and
K
ip
are the substrate andproduct inhibition constants, respectively. Note that, thesystem (1) does not describe the mass balance of theenzymes. Here the enzymes have to be considered as cat-alysts. They are not consumed by the reaction and onlytheir presence in the reactor is important to determine therate of the substrate degradation.We recall the usual positivity properties applied to solu-tions of (1).
Proposition 1.
For every initial condition
S
1
(0)
≥
0,
S
2
(0)
≥
0, and every
t >
0, we have
S
1
(
t
)
>
0,
S
2
(
t
)
>
0.
Proof
. The invariance of the non-negative cone
R
2+
isguaranteed by the fact that:i. If
S
1
= 0 then
dS
1
dt
=
Q
1
S
0
in
αV
+
Q
2
S
2
αV
>
0,ii. If
S
2
= 0 then
dS
2
dt
=
Q
2
S
1
1
−
α
V >
0.
2.1 Dimensionless parameters
Using the expression (3) of the product
P
, we can writethe kinetic function (2) as follows
ν
(
S
) =
V
max
S K
M
[1 + 2
S
0
in
/K
ip
] + [1
−
2
K
M
/K
ip
]
S
+
S
2
/K
is
.
On can recognize the Haldane law, that is often identiﬁedon experimental data, when 1
−
2
K
M
/K
ip
>
0 or equiv-alently 2
K
M
< K
ip
. In the rest of the paper, we shellassume that this last condition is fulﬁlled. Then we havethe following expression
ν
(
S
) =
V
max
1
−
2
K
M
/K
ip
µ
(
S
)where
S
= 1
−
2
K
M
/K
ip
1 + 2
S
0
in
/K
ip
S K
M
,
and
µ
(
S
) =
S
1 +
S
+
S
2
/K
i
,
(4)with
K
i
=
K
is
[1
−
2
K
M
/K
ip
]
2
K
M
[1 + 2
S
0
in
/K
ip
]
.
(5)System (1) depends on nine parameters: the ﬁve operatingparameters
Q
1
,
Q
2
,
α
,
V
and
S
0
in
, and the four kineticparameters
V
max
,
K
M
,
K
is
and
K
ip
. To decrease the
number of parameters in this system we use the followingchange of variables :
S
i
= 1
−
2
K
M
/K
ip
1 + 2
S
0
in
/K
ip
S
i
K
M
, i
= 1
,
2
t
=
V
max
1 + 2
S
0
in
/K
ip
tK
M
Then system (1) becomes
dS
1
dt
=
S
in
−
S
1
αD
1
+
S
2
−
S
1
αD
2
−
µ
(
S
1
)
dS
2
dt
=
S
1
−
S
2
(1
−
α
)
D
2
−
µ
(
S
2
)where
S
in
= 1
−
2
K
M
/K
ip
1 + 2
S
0
in
/K
ip
S
0
in
K
M
(6)and
D
i
=
V
max
K
M
[1 + 2
S
0
in
/K
ip
]
V Q
i
, i
= 1
,
2
.
(7)For convenience we drop the bars. We obtain the followingsystem
dS
1
dt
=
S
in
−
S
1
αD
1
+
S
2
−
S
1
αD
2
−
µ
(
S
1
)
dS
2
dt
=
S
1
−
S
2
(1
−
α
)
D
2
−
µ
(
S
2
)(8)System (8) depends only on ﬁve dimensionless parameters:the four operating parameters
D
1
,
D
2
,
α
and
S
in
, and thekinetic parameter
K
i
. The parameters
K
i
,
S
in
,
D
1
and
D
2
of system (8) are related to the srcinal parameters of system (1) by formulas (5), (6) and (7). The numbers
D
1
and
D
2
are related to the Damk¨ohler numbers
Da
2
=
V
max
K
M
[1+2
S
0
in
/K
ip
]
Q
1
αV
, Da
2
=
V
max
K
M
[1+2
S
0
in
/K
ip
]
Q
2
(1
−
α
)
V
,
by the formulas
Da
1
=
αD
1
, Da
2
= (1
−
α
)
D
2
.
2.2 Steady states and their stability
A steady state of the dimensionless system (8) satisfy thefollowing equations
S
∗
2
=
f
(
S
∗
1
)
, S
∗
1
=
g
(
S
∗
2
) (9)where
f
(
S
1
) :=
αµ
(
S
1
)
D
2
+
D
2
(
S
1
−
S
in
)
D
1
+
S
1
and
g
(
S
2
) := (1
−
α
)
µ
(
S
2
)
D
2
+
S
2
.
Thus, the steady states are obtained as the intersec-tion points of the graphs of functions
S
1
→
f
(
S
1
) and
S
2
→
g
(
S
2
). On Fig. 2, their graphs are plotted forplausible values of the parameters for beta-glucosidase, of NovozymeXX type, hydrolysing cellobiose.
Proposition 2.
Let (
S
∗
1
,S
∗
2
) be an steady state of system(8). Then
S
∗
2
is a ﬁxed point of function
f
◦
g
. System (8)admits at least one steady state.Fig. 2. On the right, the function
f
◦
g
. On the right,the curves of equations
S
2
=
f
(
S
1
) and
S
1
=
g
(
S
2
)for
α
= 0
.
5,
K
i
= 1
.
25,
S
in
= 12
.
5,
D
1
= 12
.
5 and
D
2
= 50. The system has three steady states.
Proof
. From (9) we deduce that
S
∗
2
=
f
(
g
(
S
∗
2
)) =
f
◦
g
(
S
∗
2
)
.
Hence
S
∗
2
is a ﬁxed point of
f
◦
g
. Since
f
◦
g
is a continuousfunction and
f
◦
g
(0) =
−
D
2
S
in
D
1
<
0
, f
◦
g
(
S
in
)
> S
in
,f
◦
g
has at least one ﬁxed point
S
∗
2
∈
]0
,S
in
[.For the study of the stability of steady states we have thefollowing results :
Proposition 3.
Let (
S
∗
1
,S
∗
2
) be the steady state of system(8). If (
f
◦
g
)
′
(
S
∗
2
)
>
1 then (
S
∗
1
,S
∗
2
) is a stable node. If (
f
◦
g
)
′
(
S
∗
2
)
<
1 then (
S
∗
1
,S
∗
2
) is a saddle point.
Proof
. The Jacobian matrix of system (8), evaluated at(
S
∗
1
,S
∗
2
) is given by
J
=
−
1
αD
1
−
1
αD
2
−
µ
′
(
S
∗
1
) 1
αD
2
1(1
−
α
)
D
2
−
1(1
−
α
)
D
2
−
µ
′
(
S
∗
2
)
(where
µ
′
denotes the derivative of
µ
w.r.t.
S
). Thedeterminant and trace of this matrix are:det
J
=
1
αD
1
+ 1
αD
2
+
µ
′
(
S
∗
1
)
1(1
−
α
)
D
2
+
µ
′
(
S
∗
2
)
−
1(1
−
α
)
D
2
αD
2
tr
J
=
−
1
αD
1
−
1
αD
2
−
µ
′
(
S
∗
1
)
−
1(1
−
α
)
D
2
−
µ
′
(
S
∗
2
)Let (
S
∗
1
,S
∗
2
), where
S
∗
1
=
g
(
S
∗
2
) be a steady state. We have(
f
◦
g
)
′
(
S
∗
2
) =
f
′
(
g
(
S
∗
2
))
g
′
(
S
∗
2
) =
f
′
(
S
∗
1
)
g
′
(
S
∗
2
)Since
f
′
(
S
1
) =
αD
2
µ
′
(
S
1
) +
D
2
D
1
+ 1and
g
′
(
S
2
) = (1
−
α
)
D
2
µ
′
(
S
2
) + 1we have(
f
◦
g
)
′
(
S
∗
2
) =
α
(1
−
α
)
D
22
det
J
+ 1
.
If (
f
◦
g
)
′
(
S
∗
2
)
<
1, then det
J <
0. Thus the eigenvalues arereal and of opposite sign. Hence the steady state (
S
∗
1
,S
∗
2
) isa saddle point. If (
f
◦
g
)
′
(
S
∗
2
)
>
1, then we have det
J >
0.Since tr
J <
0, the eigenvalues are of negative real part.We deduce that the steady state (
S
∗
1
,S
∗
2
) is a stable node.When there is only one steady state, then it is a stablenode. Actually the steady state is globally attractive.When there exist three steady states
E
1
,
E
2
,
E
3
we noticethat the stable separators of the saddle point
E
2
separatesthe phase plane into two domains of attraction of the stablenodes
E
1
and
E
3
, see Fig. 3.Fig. 3. The trajectories in red are the stable separatorsof the saddle
E
2
. The green and black trajectoriesconverge toward
E
1
and
E
3
respectively.3. MULTIPLICITY OF STEADY STATESOur main problem is to understand the role of each of theﬁve parameters
D
1
,
D
2
,
α
,
S
in
and
K
i
of system (8). Forthis purpose we consider the case of one reactor which canbe described by the following equation:
dS dt
=
S
in
−
S A
−
µ
(
S
) (10)where
A
is a positive parameter and
µ
(
S
) is the Haldanefunction (4). In this section, we do not provide proofsbecause of lack of space.We ﬁrst propose a necessary condition for having multiplesteady states.
Proposition 4.
System (10) can have multiple steady stateif and only if (
S
in
−
K
i
)
3
> K
i
S
in
.
(11)In the case where this condition is satisﬁed then system(10) has three steady state if and only if
A
−
< A < A
+
where
A
−
=
−
1
µ
′
(
S
−
)
, A
+
=
−
1
µ
′
(
S
+
)and
S
−
< S
+
are the positive roots of equation2
S
3
+ (
K
i
−
S
in
)
S
2
+
K
i
S
in
= 0
.
The domain of the (
K
i
,S
in
) plane of multiple steady stategiven by the condition (11) is shown in Fig. 4. Assume that
0 2 4 6 8 10 12 14 16 18 2005101520253035404550
Fig. 4. If (
K
i
,S
in
) lies upper the curve of equation(
S
in
−
K
i
)
3
= 27
K
i
S
in
then system (10) can havethree steady statescondition (11) is satisﬁed. For
α
ﬁxed in ]0
,
1[ we denote
D
α
=
{
(
D
1
,D
2
) : system (8) has 3 steady states
}
This domain can be determined by using the followingmethod. Let
H
(
S
2
,D
2
)=
α
[
µ
(
g
(
S
2
)) +
µ
′
(
g
(
S
2
))(
S
in
−
g
(
S
2
))]
g
′
(
S
2
)+(1
−
α
)[
µ
(
S
2
) +
µ
′
(
S
2
)(
S
in
−
S
2
)]Notice that
H
(
S
2
,D
2
) depends also on
D
2
since
g
(
S
2
)depends on
D
2
. Let
D
2
be ﬁxed. Assume that equation
H
(
S
2
,D
2
) = 0 has two positive solutions, depending on
D
2
and denoted by
S
−
2
(
D
2
)
< S
+2
(
D
2
) such that0
< b
−
(
D
2
)
< b
+
(
D
2
)
,
where
b
±
(
D
2
) = 1
g
′
(
S
±
2
(
D
2
))
−
1
−
αD
2
µ
′
(
g
(
S
±
2
(
D
2
))
.
From these values of
b
we deduce the bifurcation values of
D
1
D
−
1
(
D
2
) =
D
2
b
−
(
D
2
)
, D
+1
(
D
2
) =
D
2
b
+
(
D
2
)
.
We give now a suﬃcient condition for (
D
1
,D
2
) to belongto
D
α
.
Proposition 5.
For each
α
∈
]0
,
1[ and each
D
2
>
0, if
D
−
1
(
D
2
)
< D
1
< D
+1
(
D
2
) then (
D
1
,D
2
)
∈ D
α
Equation
H
(
S
2
,D
2
) = 0 cannot be solved analytically.It can be solved numerically. Fig. 5 shows the domain of multiplicity
D
0
.
5
.Finally, we obtain a characterization of the domain
D
α
.
Proposition 6.
Assume that
K
i
and
S
in
satisfy (11). Let
α
∈
]0
,
1[. We have the following properties
(a)
D
α
∩{
D
2
≈
0
} ≈
]
A
−
,A
+
[
(b)
D
α
∩{
D
2
≈ ∞} ≈
]
A
−
α
,
A
+
α
[
(c)
D
α
∩{
D
1
≈
0
} ≈
]
A
−
1
−
α
,
A
+
1
−
α
[
(d)
D
α
∩{
D
2
≈ ∞}
=
∅
(e)
If
α
≈
1 then
D
α
≈ {
(
D
1
,D
2
) :
A
−
< D
1
< A
+
}
.
(f)
If
α
≈
0 then
D
α
≈ {
(
D
1
,D
2
) :
A
−
< D
1
+
D
2
< A
+
}
.where
A
−
and
A
+
are deﬁned as in Proposition 4.
Fig. 5. The domain
D
α
for
α
= 0
.
5,
K
i
= 1
.
25 and
S
in
= 12
.
5. Inside the region delimited by the coloredcurves the system has 3 steady states. Outside theseregion the system has only 1 steady state.4. EVALUATING THE BIOLOGICALPERFORMANCE OF THE BIOPROCESSIt is well known that there exists a trade-oﬀ between yield
Y
and productivity
P
in enzymatic production in a singletank of volume
V
, where one has
Y
=
QV
(
S
in
−
S
1
)
P
=
S
in
−
S
1
S
in
Considered as functions of the ﬂow rate
Q
,
Y
is a decreas-ing function while
P
is increasing. Satisfactory trade-oﬀscan be reached maximizing convex combination of thesetwo criteria, as it is done for instance in [4]. Similarly, weconsider here the criterion
J
λ
(
D
1
,D
2
) =
λS
in
−
S
1
(
D
1
,D
2
)
αD
1
+(1
−
λ
)
S
in
−
S
1
(
D
1
,D
2
)
S
in
parametrized by
λ
. But here, the dependency of
S
1
withrespect to the Damk¨ohler numbers is much more complex.For numerical simulations, we have considered the sameplausible values as previously:
K
i
= 1
.
25,
S
in
= 12
.
5,
D
1 = 12
.
5 and
D
2 = 50 and
α
= 0
.
5. In order to studythe inﬂuence of the parameters
D
1
and
D
2
on the criterion
J
λ
(
D
1
,D
2
), we have considered variations of
D
1
and
D
2
about their nominal values.In Figures (6), one can make the following two observa-tions:- In presence of two steady states, one is always betterthat the other one in terms of the criterion
J
λ
,whatever is the choice of
λ
,- The criterion
J
λ
is maximized w.r.t. to
D
2
in theboundary of the domain of existence of two stablesteady states,- Except for large values of
λ
, the criterion
J
λ
ismaximized w.r.t. to
D
1
in the boundary of the domainof existence of two stable steady states.From the application viewpoint, this means that the mosteﬃcient operating point has to be close to instability andshall require a feedback strategy to stabilize the dynamics.
JD
α=0.9α=0.9α=0.1α=0.1
1
λ
5 10 15 200.050.100.150.200.250.300.350.400.45
alpha=0.9alpha=0.1alpha=0.1alpha=0.9D
λ
2
30 35 40 45 50 55 60 65 70 750.100.150.200.250.300.350.400.450.50
Fig. 6. Performance Criterion with with respect to
D
1
and
D
2
5. CONCLUSIONIn conclusion, results showed that two CSTRs, in whicha single bio-reaction with non-Michaelian kinetics occursand one CSTR represents a dead zone, can exhibit multiplesteady states. In literature, the existence of such a multi-plicity was well studied [5, 21, 11, 2, 12, 28]. Recently, ageneric approach for the analysis of interconnected bio-logical systems has been used and results have been validfor any kinetic function which veriﬁes a speciﬁc condition[16]. In these works, diﬀerent methods were used to solvethe problem of multiplicity such as the discriminator roots[20] and the Sturm method [29]. Some other studies solved
the steady states multiplicity by the mean of a methodwhich combines the Sturm method with tangent analysis[9]. In our study, we combine the discriminator roots withtangent analysis to solve multiple steady states of theenzymatic hydrolysis of a cellobiose in an heterogeneousenvironment created by the dead zone.In this paper, we have proposed a mathematical modelfor enzymatic hydrolysis of a cellobiose at steady state,in two interconnected CSTRs with dead zone. We provedthat, under a suﬃcient condition, the system has a mul-tiplicity of steady states where two of them are stableand one is unstable. An enzyme was chosen to derive theexperimental parameters (inlet substrate concentration,operating and kinetic parameters), that give the optimalbiological performance criterion
J
λ
in the domain of sucha multiplicity.Additional work will be done to validate the present math-ematical model by experimental results and to drive theenzymatic system to the best steady state that maximizesthe biological performance
J
λ
.REFERENCES[1] Alvarez-Ramirez J.
Global stabilization of chemical reactors with classical PI control
. Int J Robust Non-linear Contr. 11,735-47, 2001.[2] Vasquez-Bahena J., Montes-Horcasitas M.C., Ortega-Lopez J, Magana-plaza I., Flores-Cotera L.B.
Multiple steady states in a continuous stirred tank reactor: an experimental case study for hydrolysis of sucrose by invertase
. Process Biochem. 39, 2179-2182, 2004.[3] Balakrishnan A. and Yang R. Y. K.
Improvement of chemostat performance via nonlinear oscillations,Part 2. Extension to other systems.
. ACH. ModelsChem. 135, 1-18, 1996.

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