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Artificial Enzyme Membranes as a Tool for Teaching Optimal Control
ARTIFICIAL ENZYME MEMBRANES AS A TOOL FOR TEACHING OPTIMAL CONTROL
D. Thomas E.R.A. nO 338 du C.N.R.S Departement de Genie Biologique Universite de Technologie de Compiegne 60206 - Compiegne - France
J .P. Kernevez
Departement de Mathematiques Appliquees Universite de Technologie de Compiegne 60206 - Compiegne - France
ABSTRACT This paper describes immobilized enzyme systems which are studied by biochemists for theoretical purposes and for applications. In order to achieve these goals, modelling (by P.D.E.s) and numerical analysis of these systems are useful. In order to investigate the possibilities of regulation of enzyme systems, optimal control problems arise. They are chosen as a motivation for the study of optimal control at the M.Sc. level. INTRODUCTION
ARTIFICIAL ENZYME BEARING MEMBRANES MODELLING
Enzymes are catalysts of biological reactions. One can say that death in living organisms is indeed death of enzymes. Typically, and schematically, enzyme E is a catalyst of the reaction transforming substrate S into product P S
When enzyme proteins in reaction is and product easel! are:
E
P •
molecules are bounded to inactive an artificial membrane, the enzyme coupled to diffusion of substrate S P. The equations governing the "model
r ;" s(O,t)
a
s(x,O)
0
2 -2 +
dt
dX
0
s 1+ I si s ( 1 , t)
0
8
where s ~ s(x,t) 1S the substrate concentration as a function of x (0 < x < I) and t (0 < t < T). (Sui tab le uni ties have been chosen for s, x and t). o is a positive coefficient defined by ( 1 .2)
Artificial enzyme bearing membranes are simple systems, easy to modelize by partial differential equations (P.D.E.s). In itself this modelling is an attractive approach to P.D.E.s. (§ I). It gives (non linear) elliptic or parabolic equations whose functional and numerical analysis are a motivation to study the mathematical aspects of boundary value problems arising in physics and engineering (§ 2) and the approximating schemes to solve them (§ 3). An introduction to optimization techniques is offered by problems arising in a natural way either to identify kinetic parameters or to control in an optimal way the enzyme membrane systems (§ 4). Some physical experiments are described in § 5. Enzyme systems are used by us in the Compiegne University of Technology either to introduce P.D.E.s and their numerical analysis to students in bioengineering at the M.Sc. level, either to introduce P.D.E.s theory and optimization techniques to students in applied mathematics at the M.Sc. level. (Table I). I.
( 1. I)
-
o
VM L2
~ DS . . . 1 -3 -1 ( V maX1mum rate of the react1on, 1n mo es cm h , 3 Ki'l~ichaelis constant, in moles cm- , L thickness of the membrane, in cm, DS diffusion coefficient for S in the membrane, in cm 2 h- I .
1 2 / is called the Thiele modulus and is a characterization of the diffusion-reaction coupling properties of the membrane. a and 8 are the substrate boundary values for x ~ 0 and x ~ 1 and are the concentrations of S inside 2 compartments on both sides of the membrane. For more indications about enzyme reactions and modelling of enzyme ~7rbf~)es we ref~r to Banks(I), Ke,n~ve~la~d Thomas ' , Thomas~IO), Thomas et al~1 I),~ 2). 0
2.
MATHEMATICAL ANALYSIS
Because of the monotonicity of the non linear term os / (I + I si) in (I. I), standard methods can be used to prove the existence and uniqueness of a solution for (1.1). Moreover it can be proved that this solution is positive. This can be an occasion for introducing the FaedoGalerkin's method to students in applied mathematics, and to define (or re-define) and use various concepts of functional analysis. Of course the "model case" is only the simplest of the systems which can be studied, and several more complex examples, involving more intricate methods for solving them, are described in Kernevez and Thomas (7) . Some examples can be studi e d in a course on non linear P.D.E.s.
59
being the function
3. NUMERICAL ANALYSIS
F
It is easy to solve (I. I) by using schemes standard for solution of the heat equation. A totally ~m plicit scheme for instance would be
(4.2)
F(s,i)
and i
being defined by
n+1
s.
J
n -s o
n+1 n+1 2 n+1 I+ s . 1- S.
S .
J _
r
J+
~t (3. I)
s
s.
+o ~
n+1 I+s.
(~x)2
o
B
o
at (4.3)
2 a i -2 ax
°
i(O,t)
v( t)
i (x, 0)
0
ai ax
o
(I,t)
0
J
I, ... ,J-I
and
n = 0, ...
s~" s(j ~ x, n~t) J
,N-I
(J~x = I , N ~ t = T).
At every time level. one can solve the non linear equation (3.1) by an iterative procedure n+l,k+1
n
s . _ _ -s --"-_ _... J. _
n+l,k+1 n+l,k+1 2 n+l,k+1 +s r. I - s J +I
S.
(~x) 2
6t
n+l,k+1
S .
o
J l+s~+1 ,k J
+0
(3.2)
sn+l,k+1
n+l,k+1 sJ
B
n
S .
S .
J
J
with a test ~
l.
O
and
O
sand i are respectively a substrate and an inhibitor concentration. From (4.2) it is easy to see that the inhibitor inhibits the reaction. The Neumann boundary condition for x = I means that the membrane is coated along an impermeable electrode. This electrode is a device measuring the substrate concentration s(l,t) . The inhibitor concentration at x=O, v (=v(t)), is a control variable, at our disposal in the set of admissible controls 2 ~d = {v \ v € L (0, T) ,0 ~ v::- M}
We shall call s(x,t;v) the solution of (4.1), (4.2), (4.3) corresponding to the element v in
11ad'
n+l,o
J-I
In (4.1) and (4.3)
(4.4)
= et
o
(3.3)
ai
J
n+1
s~
where
J
n+1
s o I+s+i
\ S.n+l,k+l_ S.n+l,k\ <
j =I
J
J l:
<:
\st;-+l,k+l\
j=o
J
J
Let US call zd(t) a desired value of substrate concentration s(l,t) at time t. This function zd is, for instance, a constant, and corresponds to a desired constant level of substrate concentration along the electrode during the transient state. Or zd can be a linear function of t. The choice of this function zd is made in order to check whetJ:1er simple biochemical systems are able to perform such controls that are observed by bivchemists.
to stop the ite rations. Let us define the cost function In 2 ou 3 dime nsions the elliptic c ase, (3.4)
-~s 1 s
+
=
0
s / (I + \ s \) = 0
g
given on r
in
n
(4.5)
Our objective is to find u such that
can be used to introduce finit e element methods on an open bounded domain n with boundary r . (4.6)
For nume rical results confronted with experimental results one can see Kern e vez and Thomas (7). 4.
CONTROL OF DISTRIBUTED BIOCHEMICAL SYSTEMS
Let us see on a simple example how the biochemical syst e m previously described can be controll e d.
r ,',
The state of the system is given by the equations: - - - 2 + F(s,i) at ax
(4. I)
J(v)
s(O,t) =
et
s(x,O
0
0 as ( I , t) ax
0
[UE,l.l d
1
J(u) ; J(v)
VuE rtLad
Without entering into the details of prQofs, for which we refer to Kernevez and Thomas\7~ we can say that this problem admits at least one solution, such that (4.7)
(J'(u),v-u)
~
0
vv E
'llad
In (4.7) ( , ) denotes the scalar product in L2 (0,T). J'(v) belongs to L2 (0,T) and is defined ~n the following way: given v E ~d ' define i and s by (4.1), (4.2), (4.3). Then, for these functions i and s (th e state of the system corresponding to v), define the adjoint st a te (p,q):
60
of
2
p
o
(5. I)
d s - - - +0 F(s) 2 dx
(5.2)
F(s)
as (4.8)
ap ax-
(I, t)
with 0
2
_ lS. _ U at (4.9)
+ aF p
a/
q (0, t)
o
q (x, T)
o
s(I)
s (0)
a
s / (I +s+ks2)
1200, k
= 0,1
and a
=
75.
For these values of the parameters the system (5.1) admits at least 2 stable solutions (Kernevez and Thomas (7) ) .
o
ai
lS. dX
Then the gradient
J'(v)
~~
(O,t).
(4.10) J'(v) =
=
=
o
(I, t)
o
1S given by
Now if a changes continuously, first increasing, then decreasing, the system is not in the same state for a given boundary value a, according to its past history (Fig. I). A system exhibiting oscillations is the former when, in plus, an activator is modulating o. (Kernevez and Thomas( ), Duban et al(3». 2 d S - - - + 0 a F(s) 2 at dX dS
This very simple expression of the gradient is the starting point for optimization techniques of the gradient type. Surprisingly, even the simple gradient method gives a rapid convergence towards an optimal control.
0
2 da - aa- a +0 a F(s) 2 at ax
0
(5.3) F(s) given by (5.2)
For further optimal control problems stemming from enzyme distributed systems we refer to Brauner a)d Penel(2) , Yvon(13), Kernevez, Quadrat and Viot(9 , where are studied respectively a system whose formulation is more complex that the "model case", an optimal control of a system governed by a variational inequality, and a distributed system in a stochastic environment. In Joly et al(6) is studied the problem of identification of parameters in distributed systems. But, besides these subjects, a very important field is the field of real time control. It is necessary to view enzyme reactors as distributed systems for the purpose of controlling them. This is the work of Henry(5).
s(O,t)
s
a(O,t)
a
s(x,O)
s
dS
0
ax-
( I , t)
0
0
da ( I , t) ax
0
0
a(x,O)
a
0
The calculated substrate profile is oscillating between 2 extreme curves (Fig. 2). In conclusion enzyme distributed systems offer simple examples whose study is a tool to learn optimal control. Moreover this kind of work is very useful for the analysis and control of immobilized enzyme reactors (Gellf et al(4». REFERENCES
The "model case" introduced in § I and which is governed by equations (1.1) is a membrane made of glucose oxidase bounded to albumin. The membrane thickness is L = 5.10- 3 cm. The membrane separates 2 compartments in which there is a solution of glucose. Glucose is diffusing inside the membrane and reacts because of the glucose oxidase, enzyme which is a catalyst of the transformation of glucose into gluconic acid. The glucose coefficient of diffusion is DS = 5. 10- 3 cm 2 h- I . The kinetic parameters are VM = 3,66 10- 3 mole cm- 3h- l , KM = 1,3 IO-5moles cm- 3 , so that 0 = 1,4. A system exhibiting hysteresis is made of urate oxidase cocrosslinked with · an inactive protein. Here the substrate is uric acid and the product is allantoin.
( I)
Banks, H.T., Modeling of Control and Dynamical Systems in the Life Sciences CD S, LECTURE NOTES 73-1, CENTER FOR DYNAMICAL SYSTEMS, Brown University.
(2)
Brauner, C.M., Penel, P., Un probleme de Controle Optimal non Lineaire en Biomathematique, ANNALI DELL'UNIVERSITA DI FERRARA Sezione VII, Scienze Mathematiche. Vol. XVIII.
(3)
Duban, M.C., Kernevez, J.P., Thomas, D., Hysteresis, oscillations and structuration in space in distributed enzyme systems. JOURNAL OF MATHEMATICAL BIOLOGY (in press).
(4)
Gellf, G., Kernevez, J .P., Broun, G., Thomas, D., Water-insoluble enzyme columns: comprehensive analysis of the parameters allowing their
The steady state equations are:
61
control. BIOTECHNOL. BIOENG. (1974), 315-332. (5)
(6)
(7)
(8)
(9)
.!i.,
Henry, J., Modeling, identification and controllability of water-insoluble enzyme columns. SYMPOSIUM ON ANALYSIS AND REGULATION OF IMMOBILIZED ENZYME SYSTEMS, May 1975, to be published by North-Holland - Elsevier . Joly, G., Kernevez, J.P., Thomas, D. Identification of Kinetic Parameters in Biochemical Systems, submitted to JOURNAL OF THEORETICAL BIOLOGY. Kernevez, J.P., Thomas, D., Numerical Analysis and Control of immobilized Enzyme Systems. J OURNAL OF APPLIED MATHEMATICS AND OPTIMIZATION, ("1975),1. , Kernevez, J.P., Thomas, D., Numerical Analysis of artificial enzyme membranes -hysteresis, oscillations and spontaneous structuration, in TECHNICAL CONFERENCE ON OPTIMIZATION TECHNIQUES, LECTURE NOTES IN COMPUTER SCIENCE, Springer Verlag, (in press).
Control of a Non Linear Stochastic Value Problem, pp. 389-3 98 in 5th Conference on Optimization Techniques, Part I, LECTURE NOTES IN COMPUTER SCIENCE, Springer Ver lag, ( 1973). (10)
Thomas, D., Fonctional organization in artificial enzyme membranes: accomplishments and prospects. In "Membranes, dissipative structures and evolution". Ed. by PRIGOGINE I. WILEY, INTER SCIENCE, N.Y. (1974).
(11)
Thomas, D., Caplan, S.R., Artificial Enzyme Membranes, in "Membrane separation processes", ed. by MEARES, P., E.lsevier, Amsterdam (1974).
(12)
Thomas, D., Bourdillon, C., Broun, G., Kernevez, J.P., Kinetic behavior of enzymes immobilized in artificial membranes ; inhibition and reversibility effects. BIOCHEMISTRY, (1974), ~ , 2995.
(13)
Yvon, J.P., Optimal Control of Systems Governed by Variational Inequalities, pp. 265275 in 5th Confer~nce on Optimization Techniques, Part I, LECTURE NOTES IN COMPUTER SCIENCE, Springer Verlag, (1973) .
Kernevez, J.P., Quadrat, J.P., Viot, M.,
200L-__------~--~ >
150
~
> ~ MODELLING OF IMMOBLL IZED ENZYME SYSTEMS:
(J
INTR ODU CTI ON TO P . O.E.s
cc:
100
MATHEMATI CAL STUDY OF THESE EQUATI ONS : INTR OD UCTI ON TO GA LE RKIN' S METHO D AND VA RI AT I ONAL ME THOD S
50
NUMERI CAL STUDY OF THESE EQUATI ONS :
60
70
SUBSTRATE CONCENTRATION
iNT RODUC TI ON TO APP ROXIMATING SCHE MES FOR SO LVIN G P . D. E. s ~
OPTI MAL CONTROL AND IDENTIFICATION OF PARAMETERS:
Hy.t e re. i . phenOMnon : bound.ry flux It • fu.n c tion o f .ub.trate concentr.ti o n . t the bo und. d. • • Cor) 'l v.lue . : 800, IClClO .nd 1200. For e.eh 0 vtlUII! .rro.... indie .te th e dire e tion oC v.dat i on tor sub . trate concentra t i on.
A MOTIVATI ON TO THE USE OF OPTIMIZATI ON TECHNI QUES
REAL TIME CONTRO L OF TUBULAR REACTORS CONTAINING IMMOBILIZED
s
ENZYMES:
100
AN INT RODUC TI ON TO FEEDBACK CONT RO L OF B I OC HEMI CAL PROCE SS ES .
~
Ac tual t o pics which are be ing use d in cours e s at Compiegne. Fr om the experimental work in o ur Laborat o ry s t e m man y examples for the students in mathemati cs .
ncw,z200.lIh.i-.ph_ ' hn.. _ •• pu ..." .. t .. • ~ ...... o of 0 .oo ...... (0 . 0. 2,0 . ' ,0 . 100'.0. 500 ), 0
62
,i_
,rofil.
D. Thomas E.R.A. nO 338 du C.N.R.S Departement de Genie Biologique Universite de Technologie de Compiegne 60206 - Compiegne - France
J .P. Kernevez
Departement de Mathematiques Appliquees Universite de Technologie de Compiegne 60206 - Compiegne - France
ABSTRACT This paper describes immobilized enzyme systems which are studied by biochemists for theoretical purposes and for applications. In order to achieve these goals, modelling (by P.D.E.s) and numerical analysis of these systems are useful. In order to investigate the possibilities of regulation of enzyme systems, optimal control problems arise. They are chosen as a motivation for the study of optimal control at the M.Sc. level. INTRODUCTION
ARTIFICIAL ENZYME BEARING MEMBRANES MODELLING
Enzymes are catalysts of biological reactions. One can say that death in living organisms is indeed death of enzymes. Typically, and schematically, enzyme E is a catalyst of the reaction transforming substrate S into product P S
When enzyme proteins in reaction is and product easel! are:
E
P •
molecules are bounded to inactive an artificial membrane, the enzyme coupled to diffusion of substrate S P. The equations governing the "model
r ;" s(O,t)
a
s(x,O)
0
2 -2 +
dt
dX
0
s 1+ I si s ( 1 , t)
0
8
where s ~ s(x,t) 1S the substrate concentration as a function of x (0 < x < I) and t (0 < t < T). (Sui tab le uni ties have been chosen for s, x and t). o is a positive coefficient defined by ( 1 .2)
Artificial enzyme bearing membranes are simple systems, easy to modelize by partial differential equations (P.D.E.s). In itself this modelling is an attractive approach to P.D.E.s. (§ I). It gives (non linear) elliptic or parabolic equations whose functional and numerical analysis are a motivation to study the mathematical aspects of boundary value problems arising in physics and engineering (§ 2) and the approximating schemes to solve them (§ 3). An introduction to optimization techniques is offered by problems arising in a natural way either to identify kinetic parameters or to control in an optimal way the enzyme membrane systems (§ 4). Some physical experiments are described in § 5. Enzyme systems are used by us in the Compiegne University of Technology either to introduce P.D.E.s and their numerical analysis to students in bioengineering at the M.Sc. level, either to introduce P.D.E.s theory and optimization techniques to students in applied mathematics at the M.Sc. level. (Table I). I.
( 1. I)
-
o
VM L2
~ DS . . . 1 -3 -1 ( V maX1mum rate of the react1on, 1n mo es cm h , 3 Ki'l~ichaelis constant, in moles cm- , L thickness of the membrane, in cm, DS diffusion coefficient for S in the membrane, in cm 2 h- I .
1 2 / is called the Thiele modulus and is a characterization of the diffusion-reaction coupling properties of the membrane. a and 8 are the substrate boundary values for x ~ 0 and x ~ 1 and are the concentrations of S inside 2 compartments on both sides of the membrane. For more indications about enzyme reactions and modelling of enzyme ~7rbf~)es we ref~r to Banks(I), Ke,n~ve~la~d Thomas ' , Thomas~IO), Thomas et al~1 I),~ 2). 0
2.
MATHEMATICAL ANALYSIS
Because of the monotonicity of the non linear term os / (I + I si) in (I. I), standard methods can be used to prove the existence and uniqueness of a solution for (1.1). Moreover it can be proved that this solution is positive. This can be an occasion for introducing the FaedoGalerkin's method to students in applied mathematics, and to define (or re-define) and use various concepts of functional analysis. Of course the "model case" is only the simplest of the systems which can be studied, and several more complex examples, involving more intricate methods for solving them, are described in Kernevez and Thomas (7) . Some examples can be studi e d in a course on non linear P.D.E.s.
59
being the function
3. NUMERICAL ANALYSIS
F
It is easy to solve (I. I) by using schemes standard for solution of the heat equation. A totally ~m plicit scheme for instance would be
(4.2)
F(s,i)
and i
being defined by
n+1
s.
J
n -s o
n+1 n+1 2 n+1 I+ s . 1- S.
S .
J _
r
J+
~t (3. I)
s
s.
+o ~
n+1 I+s.
(~x)2
o
B
o
at (4.3)
2 a i -2 ax
°
i(O,t)
v( t)
i (x, 0)
0
ai ax
o
(I,t)
0
J
I, ... ,J-I
and
n = 0, ...
s~" s(j ~ x, n~t) J
,N-I
(J~x = I , N ~ t = T).
At every time level. one can solve the non linear equation (3.1) by an iterative procedure n+l,k+1
n
s . _ _ -s --"-_ _... J. _
n+l,k+1 n+l,k+1 2 n+l,k+1 +s r. I - s J +I
S.
(~x) 2
6t
n+l,k+1
S .
o
J l+s~+1 ,k J
+0
(3.2)
sn+l,k+1
n+l,k+1 sJ
B
n
S .
S .
J
J
with a test ~
l.
O
and
O
sand i are respectively a substrate and an inhibitor concentration. From (4.2) it is easy to see that the inhibitor inhibits the reaction. The Neumann boundary condition for x = I means that the membrane is coated along an impermeable electrode. This electrode is a device measuring the substrate concentration s(l,t) . The inhibitor concentration at x=O, v (=v(t)), is a control variable, at our disposal in the set of admissible controls 2 ~d = {v \ v € L (0, T) ,0 ~ v::- M}
We shall call s(x,t;v) the solution of (4.1), (4.2), (4.3) corresponding to the element v in
11ad'
n+l,o
J-I
In (4.1) and (4.3)
(4.4)
= et
o
(3.3)
ai
J
n+1
s~
where
J
n+1
s o I+s+i
\ S.n+l,k+l_ S.n+l,k\ <
j =I
J
J l:
<:
\st;-+l,k+l\
j=o
J
J
Let US call zd(t) a desired value of substrate concentration s(l,t) at time t. This function zd is, for instance, a constant, and corresponds to a desired constant level of substrate concentration along the electrode during the transient state. Or zd can be a linear function of t. The choice of this function zd is made in order to check whetJ:1er simple biochemical systems are able to perform such controls that are observed by bivchemists.
to stop the ite rations. Let us define the cost function In 2 ou 3 dime nsions the elliptic c ase, (3.4)
-~s 1 s
+
=
0
s / (I + \ s \) = 0
g
given on r
in
n
(4.5)
Our objective is to find u such that
can be used to introduce finit e element methods on an open bounded domain n with boundary r . (4.6)
For nume rical results confronted with experimental results one can see Kern e vez and Thomas (7). 4.
CONTROL OF DISTRIBUTED BIOCHEMICAL SYSTEMS
Let us see on a simple example how the biochemical syst e m previously described can be controll e d.
r ,',
The state of the system is given by the equations: - - - 2 + F(s,i) at ax
(4. I)
J(v)
s(O,t) =
et
s(x,O
0
0 as ( I , t) ax
0
[UE,l.l d
1
J(u) ; J(v)
VuE rtLad
Without entering into the details of prQofs, for which we refer to Kernevez and Thomas\7~ we can say that this problem admits at least one solution, such that (4.7)
(J'(u),v-u)
~
0
vv E
'llad
In (4.7) ( , ) denotes the scalar product in L2 (0,T). J'(v) belongs to L2 (0,T) and is defined ~n the following way: given v E ~d ' define i and s by (4.1), (4.2), (4.3). Then, for these functions i and s (th e state of the system corresponding to v), define the adjoint st a te (p,q):
60
of
2
p
o
(5. I)
d s - - - +0 F(s) 2 dx
(5.2)
F(s)
as (4.8)
ap ax-
(I, t)
with 0
2
_ lS. _ U at (4.9)
+ aF p
a/
q (0, t)
o
q (x, T)
o
s(I)
s (0)
a
s / (I +s+ks2)
1200, k
= 0,1
and a
=
75.
For these values of the parameters the system (5.1) admits at least 2 stable solutions (Kernevez and Thomas (7) ) .
o
ai
lS. dX
Then the gradient
J'(v)
~~
(O,t).
(4.10) J'(v) =
=
=
o
(I, t)
o
1S given by
Now if a changes continuously, first increasing, then decreasing, the system is not in the same state for a given boundary value a, according to its past history (Fig. I). A system exhibiting oscillations is the former when, in plus, an activator is modulating o. (Kernevez and Thomas( ), Duban et al(3». 2 d S - - - + 0 a F(s) 2 at dX dS
This very simple expression of the gradient is the starting point for optimization techniques of the gradient type. Surprisingly, even the simple gradient method gives a rapid convergence towards an optimal control.
0
2 da - aa- a +0 a F(s) 2 at ax
0
(5.3) F(s) given by (5.2)
For further optimal control problems stemming from enzyme distributed systems we refer to Brauner a)d Penel(2) , Yvon(13), Kernevez, Quadrat and Viot(9 , where are studied respectively a system whose formulation is more complex that the "model case", an optimal control of a system governed by a variational inequality, and a distributed system in a stochastic environment. In Joly et al(6) is studied the problem of identification of parameters in distributed systems. But, besides these subjects, a very important field is the field of real time control. It is necessary to view enzyme reactors as distributed systems for the purpose of controlling them. This is the work of Henry(5).
s(O,t)
s
a(O,t)
a
s(x,O)
s
dS
0
ax-
( I , t)
0
0
da ( I , t) ax
0
0
a(x,O)
a
0
The calculated substrate profile is oscillating between 2 extreme curves (Fig. 2). In conclusion enzyme distributed systems offer simple examples whose study is a tool to learn optimal control. Moreover this kind of work is very useful for the analysis and control of immobilized enzyme reactors (Gellf et al(4». REFERENCES
The "model case" introduced in § I and which is governed by equations (1.1) is a membrane made of glucose oxidase bounded to albumin. The membrane thickness is L = 5.10- 3 cm. The membrane separates 2 compartments in which there is a solution of glucose. Glucose is diffusing inside the membrane and reacts because of the glucose oxidase, enzyme which is a catalyst of the transformation of glucose into gluconic acid. The glucose coefficient of diffusion is DS = 5. 10- 3 cm 2 h- I . The kinetic parameters are VM = 3,66 10- 3 mole cm- 3h- l , KM = 1,3 IO-5moles cm- 3 , so that 0 = 1,4. A system exhibiting hysteresis is made of urate oxidase cocrosslinked with · an inactive protein. Here the substrate is uric acid and the product is allantoin.
( I)
Banks, H.T., Modeling of Control and Dynamical Systems in the Life Sciences CD S, LECTURE NOTES 73-1, CENTER FOR DYNAMICAL SYSTEMS, Brown University.
(2)
Brauner, C.M., Penel, P., Un probleme de Controle Optimal non Lineaire en Biomathematique, ANNALI DELL'UNIVERSITA DI FERRARA Sezione VII, Scienze Mathematiche. Vol. XVIII.
(3)
Duban, M.C., Kernevez, J.P., Thomas, D., Hysteresis, oscillations and structuration in space in distributed enzyme systems. JOURNAL OF MATHEMATICAL BIOLOGY (in press).
(4)
Gellf, G., Kernevez, J .P., Broun, G., Thomas, D., Water-insoluble enzyme columns: comprehensive analysis of the parameters allowing their
The steady state equations are:
61
control. BIOTECHNOL. BIOENG. (1974), 315-332. (5)
(6)
(7)
(8)
(9)
.!i.,
Henry, J., Modeling, identification and controllability of water-insoluble enzyme columns. SYMPOSIUM ON ANALYSIS AND REGULATION OF IMMOBILIZED ENZYME SYSTEMS, May 1975, to be published by North-Holland - Elsevier . Joly, G., Kernevez, J.P., Thomas, D. Identification of Kinetic Parameters in Biochemical Systems, submitted to JOURNAL OF THEORETICAL BIOLOGY. Kernevez, J.P., Thomas, D., Numerical Analysis and Control of immobilized Enzyme Systems. J OURNAL OF APPLIED MATHEMATICS AND OPTIMIZATION, ("1975),1. , Kernevez, J.P., Thomas, D., Numerical Analysis of artificial enzyme membranes -hysteresis, oscillations and spontaneous structuration, in TECHNICAL CONFERENCE ON OPTIMIZATION TECHNIQUES, LECTURE NOTES IN COMPUTER SCIENCE, Springer Verlag, (in press).
Control of a Non Linear Stochastic Value Problem, pp. 389-3 98 in 5th Conference on Optimization Techniques, Part I, LECTURE NOTES IN COMPUTER SCIENCE, Springer Ver lag, ( 1973). (10)
Thomas, D., Fonctional organization in artificial enzyme membranes: accomplishments and prospects. In "Membranes, dissipative structures and evolution". Ed. by PRIGOGINE I. WILEY, INTER SCIENCE, N.Y. (1974).
(11)
Thomas, D., Caplan, S.R., Artificial Enzyme Membranes, in "Membrane separation processes", ed. by MEARES, P., E.lsevier, Amsterdam (1974).
(12)
Thomas, D., Bourdillon, C., Broun, G., Kernevez, J.P., Kinetic behavior of enzymes immobilized in artificial membranes ; inhibition and reversibility effects. BIOCHEMISTRY, (1974), ~ , 2995.
(13)
Yvon, J.P., Optimal Control of Systems Governed by Variational Inequalities, pp. 265275 in 5th Confer~nce on Optimization Techniques, Part I, LECTURE NOTES IN COMPUTER SCIENCE, Springer Verlag, (1973) .
Kernevez, J.P., Quadrat, J.P., Viot, M.,
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MATHEMATI CAL STUDY OF THESE EQUATI ONS : INTR OD UCTI ON TO GA LE RKIN' S METHO D AND VA RI AT I ONAL ME THOD S
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NUMERI CAL STUDY OF THESE EQUATI ONS :
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OPTI MAL CONTROL AND IDENTIFICATION OF PARAMETERS:
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AN INT RODUC TI ON TO FEEDBACK CONT RO L OF B I OC HEMI CAL PROCE SS ES .
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Ac tual t o pics which are be ing use d in cours e s at Compiegne. Fr om the experimental work in o ur Laborat o ry s t e m man y examples for the students in mathemati cs .
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