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Numerical analysis of immobilized enzyme systems

Jean-Pierre Kernevez, Daniel Thomas

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Jean-Pierre Kernevez, Daniel Thomas. Numerical analysis of immobilized enzyme systems. [Research Report] IRIA-RR-028, IRIA. 1973, pp.61. hal-04715951

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IRIA
laboria

laboratoire de recherche
en informatique
et automatique

Numerical analysis
of immobilized
enzyme systems

J.P Kernevez, D. Thomas

Institut de Recherche
d'Informatique
et d'Automatique

Domaine de Voluceau
Rocquencourt
78150 - Le Chesnay
Tél. 954.90.20

Rapport de Recherche n°28

septembre 1973
NUMERICAL ANALYSIS
OF IMMOBILIZED ENZYME SYSTEMS

J.P. Kernevez
D. Thomas

(présenté par J.L. Lions)

RÉSUMÉ

L'objet de ce rapport est la description du comportement cinétique de systèmes porteurs d'enzymes immobilisées.

Les résultats de simulations numériques concernant des membranes et des réacteurs sont donnés dans le contexte expérimental.

Des problèmes d'optimisation (contrôle optimal et identification de paramètres cinétiques) sont également exposés.

ABSTRACT

The aim of this report is to describe the kinetic behaviour of systems bearing immobilized enzymes.

The results of numerical simulations dealing with membranes and biochemical reactors are given within the experimental context.

Optimization problems (optimal control and identification of kinetic parameters) are also introduced.
P R E F A C E

The computer and mathematical work to be presented in this report is
not only USEFUL at the fundamental and theoretical point of vue but also
for a lot of biomedical applications.

The computation of the concentration profiles inside an active layer
has allowed the production of new specific electrodes. They constitute
entirely new tools in the armamentarium of the clinical Chemist, with
unique advantages.

The regulation and control results are the first step to produce
with artificial multienzyme systems the synthesis or modification of
antibiotics, steroids, hormones, vitamins and other molecules in pharma-
cetical industry.

The better numerical knowledge and the optimization of synthetic
enzyme membranes allow also medical applications in the field of prostheses.

The results have particular relevance to the following three prosthetic
devices: artificial kidney, membrane oxygenators and internal prostheses.
The interest of the first two points is obvious, the interest of the third
one is, in particular for the treatment of hereditary enzyme defects
(e.g. phenylketonuria and acatalasemia), certain cancers (e.g. leukemia
with L. asparaginase, and other diseases (gout with uricase).

This report can play a relevant part in the growing importance in
clinical applications of immobilized enzymes.
INTRODUCTION

The artificial binding of proteins onto membrane supports gives a good experimental tool in order to analyse the enzyme kinetics in structured phase. (THOMAS 1971).

The work in this field shows the diffusion and reaction reciprocal effects inside the structure.

This interaction can explain the apparent kinetic constant modifications observed by different authors (THOMAS and CAPLAN, in press). The global enzyme kinetics are deeply modified by the diffusion.

In general, enzymes in vivo act inside structured systems. This is not only true for enzymes bound onto organelles but also for cystoplasme enzymes, because the viscosity is high enough and the convection effects are negligible.

Experimental models can simulate the main properties of these biological systems.

Numerical analysis methods not only allow the kinetic behavior study and an easier generalization of the results but also are a guide for the experimentalist.

With numerical simulation results it is possible to perform the experiments in the best conditions.

NIMS (1968), BLUMENTHAL et al (1967), PASYNSKI et al (1964) have associated free enzyme in solution with an inactive membrane in order to produce transport models. This was a way to study the association between enzyme reaction and metabolite diffusion but this association does not take place inside the structure itself.

GOLDMAN et al (1968a, 1968b, 1971) have published some results obtained with papaïn and phosphatase collodion membranes. The active site repartition inside the membrane produced by this method is not well-defined and it is difficult to numerically simulate this kind of system.

It is convenient to classify the systems which are to be presented into 4 main categories.
a) Heterogeneous kinetics :

Between the classical enzymology in solution and the very compli-
cated biological systems, it is possible to build a step by using artificial
membranes bearing enzymes.

These systems allow the study of the simplest kinetic rules of the
heterogeneous enzymology. Different exemples of enzyme kinetic modulation by
the diffusion are to be presented : reversible reaction, inhibitor effects,
multienzyme systems.

Several papers were published in this field during the recent years.
These papers deal mainly with heterogeneous enzyme kinetics with the station-
nary state assymption and for zero and first order reactions. GOLDMAN et al
(1968<sub>a</sub>,1968<sub>b</sub>, 1971) ; KASHE et al (1971) ; KOBAYASHI and LAIDLER (1973) ;
SUNDARAM and LAIDLER (1971) ; SELEGNY et al (1968) ; THOMAS et al (1972) ;
BLAEDEL and BOGUSLASKI (1972).

b) Transport phenomena

With artificial enzyme membranes it is possible to perform studies
in asymmetrical conditions. This point it very important to study the enzyme
role during transport phenomena, in particular active transport.

The reciprocal effect of potential differences and enzyme reactions
can be studied also in these conditions.

c) Enzyme reactors

Immobilized enzymes can be easily used continuously in reactors.
Numerical analysis dealing with packed bed reactor is to be presented.

d) Identification and control

For both biological models and applications the study of the system
controls is a critical point. It is possible to take into account random or
stochastic processes for a global control of the systems.

The use of the diffusion-reaction properties to identify enzyme
kinetic parameters will be described.

In each section, equations are given and the numerical results are
discussed in the experimental context.

The mathematical aspects of these problems were already published.
(KERNEVEZ 1972).
## II - HETEROGENEOUS ENZYME KINETICS

### 1º) Irreversible monoenzyme system

#### a) - Classical enzymology (recall and notations):

An enzyme is a proteic catalyser. (DIXON and WEBB, 1972).

Scheme of an irreversible monoenzyme reaction

$$ (1) \quad \begin{array}{ccc} & k_1 & k_2 \\ E + S \rightleftharpoons \begin{array}[t]{c} k_1 \\ k_2-1 \end{array} & \rightarrow & E + P \end{array} $$

E free enzyme

S substrate

P product

[i] is the notation for I concentration.

For a general treatment, one can get the system kinetic behavior by using the mass action law.

The quasistationarity hypothesis for [ES] gives an expression so-called Michaelian for the reaction rate (MICHAELIS and MENTEN, 1913).

$$ (2) \quad v = \frac{d[\text{P}]}{dt} = -\frac{d[\text{S}]}{dt} = k_2 E_0 \frac{[\text{S}]}{K_M + [\text{S}]} $$

$$ \text{with } K_M = \frac{k_{-1} + k_2}{k_{+1}} \quad (\text{concentration dimensions}) $$

the MICHAELIS constant which is very often equal to $K_S = \frac{k_{-1}}{k_{+1}}$ (if $k_{-1} \gg k_2$) in the first approximation.
$E_0$ is the active site concentration.

With high substrate concentration, $v$ tends to $k_2 E_0$, maximum rate $v_M$ of the reaction. (Dimensions: number of moles $L^{-3}T^{-1}$).

### b) - Enzyme membrane equations:

Inside artificial membranes the enzymes are immobilized into a macro-molecular carrier. Enzyme molecules are homogeneously distributed inside the support. (THOMAS et al, 1972- a)

#### i) Michaelian hypothesis:

It is impossible to use the global michaelian kinetics now, for it is not possible to give one substrate concentration inside the carrier. This concentration varies from point to point inside the structure.

*But the v expression is always valid for an elementary volume.*

Let us consider a membrane (thickness: e) between two compartments with respectively concentrations of solutions $s_1$ and $s_2$. The reference axis is perpendicular to the membrane surface, $x = 0$ is chosen for the interface between membrane and compartment (1). Obviously $[S]$ is a function of space and time ($x$ and $t$).

For each $x$ inside the membrane, $\frac{\partial [S]}{\partial t}$ is linked to enzyme reaction and metabolite diffusion:

$$ (3) \quad \frac{\partial [S]}{\partial t} = \left(\frac{\partial S}{\partial t}\right)_{\text{diffusion}} + \left(\frac{\partial S}{\partial t}\right)_{\text{reaction}} $$

$$ \left(\frac{\partial S}{\partial t}\right)_{\text{reaction}} = - \frac{k_2 E_0 [S]}{K_M + [S]} \quad ; \text{and according to FICK's second law:} $$

$$ (4) \quad \left(\frac{\partial S}{\partial t}\right)_{\text{diffusion}} = D_S \frac{\partial^2 [S]}{\partial x^2} $$

with $D_S$ substrate diffusion coefficient. ($L^2 T^{-1}$)
$$
(5) \quad \frac{\partial [S]}{\partial t} = D_S \frac{\partial^2 [S]}{\partial x^2} - \frac{k_2 E_O [S]}{K_M + [S]}
$$

and with P

$$
(6) \quad \frac{\partial [P]}{\partial t} = D_P \frac{\partial^2 [P]}{\partial x^2} + \frac{k_2 E_O [S]}{K_M + [S]}
$$

$[P] =$ product concentration

$D_p$ = diffusion coefficient of P.

Let us replace $\frac{x}{e}$ by $x$ and $\frac{t}{e^2/D_S}$ by $t$. $x$ and $t$ are now dimensionless.

$K_M$ is chosen as concentration unit $s = \frac{[S]}{K_M}$; $p = \frac{[P]}{K_M}$

One can write the equation (5) like:

$$
(7) \quad \frac{\partial s}{\partial t} - \frac{\partial^2 s}{\partial x^2} + \sigma \frac{\partial s}{\partial t} = 0
$$

where

$$
(8) \quad \sigma = \frac{V_M}{K_M} \frac{e^2}{D_S}
$$

obviously, $\sigma$ alone is a perfect characterization of a membrane system.

In the same way one can write the p equation

$$
(9) \quad \frac{\partial p}{\partial t} - \frac{D_p}{D_S} \frac{\partial^2 p}{\partial x^2} - \sigma \frac{s}{1+s} = 0
$$

Boundary conditions:

1<sup>st</sup> case:

$\alpha$ and $\beta$ concentrations at the membrane boundaries in the compartments
① and ② are constant.

(DIRICHLET conditions)

$$
(10) \quad s(0,t) = \alpha, \quad s(1,t) = \beta, \quad (\alpha \text{ and } \beta \gg 0).
$$
## 2<sup>nd</sup> case:

Boundary concentrations are mobile.

V is a one compartment volume solution.

Ω is the area of the membrane solution interface.

(the magnitudes are given in the new unity system).

Let us write that the substrate concentrations variation inside the compartment I is due to the flux through the interface:

$$ (11) \qquad V \frac{d}{dt} s(0,t) = \Omega \frac{\partial s}{\partial x}(0,t) $$

$$ (12) \qquad \frac{d}{dt} s(0,t) = \chi \frac{\partial s}{\partial x}(0,t) $$

$$ (13) \qquad \text{where } \dot{x} = \frac{\Omega}{V} = \frac{\text{Membrane volume}}{\text{Compartment 1}} \\ \text{Volume of solution} $$

boundary conditions are in this case

$$ (14) \qquad \frac{\partial s}{\partial t} + \chi \frac{\partial s}{\partial v} = 0 \qquad \text{, initial value of } s = \alpha $$

where:

$$ (15) \qquad \frac{\partial}{\partial v} = \begin{cases} -\frac{\partial}{\partial x} & \text{for } x = 0 \\ +\frac{\partial}{\partial x} & \text{for } x = 1 \end{cases} $$

when $\chi = 0$ , $s$ = constant.

**Note :**

For the second kind of boundary conditions, if the system is studied over a long period of time, it can be demonstrated that the system evolution is ruled by these equations :

$$ (16) \qquad - \frac{\partial^2 s}{\partial x^2} + \sigma \frac{s}{1+s} = 0 $$
$$ (17) \qquad \frac{\partial s}{\partial t} + x \frac{\partial s}{\partial v} = 0 $$

**Note**: The boundary conditions for p are similar, but in this case x is replaced by $\frac{D_p}{D_s} x$.

### 3<sup>rd</sup> case :

When $1/x = 0$ (the interface is impermeable to everything)

$$ (18) \qquad \frac{\partial s}{\partial v} = 0 \quad \text{(NEUMANN condition).} $$

This boundary condition occurs when an active membrane is along an electrode surface. The system is immersed into a substrate solution. Concentrations are measured at the membrane electrode interface.

### 4<sup>th</sup> case :

Boundary conditions of the so-called "immersed membrane system"

When the membrane is immersed into a solution of substrate :

$s(0,t) = s(1,t) = \alpha(t) = \text{substrate concentration in the bulk solution}$
with :

$$ (19) \qquad \frac{\partial \alpha}{\partial t} + x \left[ \frac{\partial s}{\partial x}(1,t) - \frac{\partial s}{\partial x}(0,t) \right] = 0 $$

$\alpha(0) = \alpha_0 = \text{initial concentration.}$
Note:

For the four kinds of boundary conditions previously described we will
demonstrate that the concentration profiles tend to stationary profiles
which are solutions of the equation:

$$
(20) \qquad - \frac{\partial^2 s}{\partial x^2} + \sigma \frac{s}{1+s} = 0
$$

- Initial conditions:

In general, at time t = 0, the membrane is empty of diffusible molecules
and

$$
s(x,0) = 0 ; p(x,0) = 0 , \quad 0 < x < 1
$$

i.i) Equations without michaelian hypothesis (KERNEVEZ 1972)

The system is ruled by these equations:

- for fixed enzyme sites:

$$
(21) \quad \frac{\partial [E]}{\partial t} + k_1 [E] [S] - (k_{-1} + k_2) [E] = 0
$$

- for the substrate

$$
(22) \quad \frac{\partial [E]}{\partial t} - D \frac{\partial^2 [S]}{\partial x^2} + k_1 [E] [S] - k_{-1} [ES] = 0
$$

- for the product

$$
(23) \quad \frac{\partial [P]}{\partial t} - D \frac{\partial^2 [P]}{\partial x^2} - k_2 [ES] = 0
$$

By replacing the space and time units as before and with $\alpha = \frac{[E]}{E_0}$ and
$\theta = \frac{e^2}{D_S}$ the equations are:

$$
(24) \quad \frac{\partial \alpha}{\partial t} + k_1 K_M \theta (\alpha s + \alpha - 1) = 0
$$
$$
(25) \quad \frac{\partial s}{\partial t} - \frac{\partial^2 s}{\partial x^2} + \frac{k_1}{K_2} K_M s \alpha - \frac{k_{-1}}{k_2} (1 - \alpha) = 0
$$

$$
(26) \quad \frac{\partial p}{\partial t} - \frac{\partial^2 p}{\partial x^2} - \sigma(1 - \alpha) = 0
$$

Initial and boundary conditions for s and p are the same, there
is also $\alpha(x,0) = 1$

c) - Numerical results in the experimental context (THOMAS 1971)

Two kinds of studies are to be presented now: the first one is a
quick evolution of a system with DIRICHLET conditions, the second one
is an evolution of a system during a long period with mobile boundary
concentrations.

The first study is a numerical comparison between profile evolutions
inside an active membrane during a transient state with and without the
michaelian stationary state hypothesis on the complex $\tilde{E}_S$ concentration.

i) Transient state comparison with and without michaelian
hypothesis:

Parameter values:

$$
\begin{align*}
k_1 &= 1.5 \ 10^{11} \ \text{mole}^{-1} \ \text{cm}^3 \ \text{h}^{-1} \\
k_2 &= 6.3 \ 10^2 \ \text{h}^{-1} \\
k_{-1} &= 6.3 \ 10^5 \ \text{h}^{-1} \\
V_M &= 3.6 \ 10^{-3} \ \text{mole} \ \text{cm}^{-3} \ \text{h}^{-1} \\
E_O &= 0.57 \ \text{mole} \ \text{cm}^{-3}
\end{align*}
\text{Membrane thickness } e = 5 \ 10^{-3} \ \text{cm}
\text{Diffusion coefficient } D_S = 5 \ 10^{-3} \ \text{cm}^2 \ \text{h}^{-1}
$$
Concentration profile evolutions calculated during a transient state for a monoenzyme membrane with and without michaelian hypothesis are given respectively in figure 1 and figure 2.

Evolutions of calculated concentrations in substrate, in product, and in free sites on abscissa $\frac{e}{2}$ are given in figures 3, 4 and 5 for both cases previously described.

For insoluble enzymes, an increasing of the length of the transient state limits the use of the michaelian hypothesis more than for soluble enzymes.

This result could be used to identify kinetic parameters $k_1$, $k_{-1}$ and $k_2$ of the enzyme.

In solution the useful time for this measurement is shorter than a few ms. In membrane the useful time could be longer than 30 seconds.

### i.i) Evolution of a system with mobile boundary concentrations:

The membrane separates two compartments containing substrate solutions with different concentrations at time zero.

Experimental and numerical results are compared on figure 6.

There is a good agreement between experimental results and numerical simulation.

The membrane was obtained by a cocrosslinking method between albumin and glucose-oxidase.

- Maximum activity by volume unit of membrane

$$k_2^E O = 3.66 \ 10^{-3} \text{moles cm}^{-3} \text{ h}^{-1}$$

- Coefficient of diffusion

$$D_S = 0.5 \ 10^{-2} \text{cm}^2 \text{ h}^{-1}$$

- Michaelis constant

$$(28) \qquad K_M = 1.3 \ 10^{-5} \text{moles cm}^{-3}$$
<figure><img src="image_3.png" /><figcaption>Figure 1</figcaption></figure>

Substrate concentration profile evolution calculated inside an enzyme membrane with the michaelian assumption. (time unit : $\frac{e^2}{D} 10^{-3}$).

<figure><img src="image_4.png" /><figcaption>Figure 2</figcaption></figure>

Substrate concentration profile evolution calculated inside an enzyme membrane without the michaelian assumption (time unit : $\frac{e^2}{D} 10^{-3}$).
<figure><img src="image_5.png" /><figcaption>**Figure 3** : Calculated substrate concentration evolution on the membrane symmetry plane with (-) and without (---) the michaelian assumption.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Figure 4** : Calculated product concentration evolution on the membrane symmetry plane with (-) and without (---) the michaelian assumption.</figcaption></figure>
<figure><img src="image_7.png" /></figure>

**Figure 5** : Calculated free site concentration evolution on the membrane symmetry plane with ( - ) and without ( --- ) the michaelian assumption.

<figure><img src="image_8.png" /></figure>

**Figure 6** : Calculated and measured evolution of substrate concentrations on both sides of an enzyme membrane. Concentrations are given for the donor ( - ) and receptor ( --- ) compartments.
(28) - thickness:

$$ 
\begin{aligned}
e &= 0.5 \ 10^{-2} \text{ cm} \\
\sigma &= \frac{e^2 V_M}{D K_M} = 1.4 \quad ; \quad \chi = 0.05
\end{aligned}
 $$

The profiles obtained were previously discussed (THOMAS et al, 1972)

### i.i.i.) System with a NEUMANN condition: exemple of an enzyme electrode (KERNEVEZ 1971)

Parameter values:

$$ 
\begin{aligned}
V_M &= 1.92 \ 10^{-6} \text{ mole cm}^{-3} \text{ s}^{-1} \\
K_M &= 2 \ 10^{-5} \text{ mole cm}^{-3}
\end{aligned}
 $$

$$ 
\begin{aligned}
(29) \quad e &= 2.5 \ 10^{-2} \text{ cm} \\
D_F \approx D_S &= 1.2 \ 10^{-5} \text{ cm}^2 \text{ s}^{-1} \\
\text{time unit} &= \frac{e^2}{D_S}
\end{aligned}
 $$

The evolution of calculated profiles inside an enzyme coating along an electrode is given in figure 7. Product and substrate diffusion coefficients are the same.

If the substrate concentration is smaller than the Michaelis constant, the reaction kinetics are very close to the first order. The concentration of product along the electrode is directly proportional to the substrate concentration in the bulk solution.

For example an electrode bearing lysine decarboxylase with various lysine concentrations (substrate) from $10^{-5}$ to $10^{-1}$ M, gives a linear relationship between the product concentration ($CO_2$) measured and the concentration of substrate in the bulk solution. (figure 8).
<figure><img src="image_9.png" /><figcaption>Figure 8</figcaption></figure>

<figure><img src="image_10.png" /></figure>

**Figure 7** : Calculated concentration profile evolution for the substrate ( --- ) and the product ( - ) inside the active layer of an enzyme electrode.

**Figure 8** : $P_2^{CO_2}$ Measured with a lysine decarboxylase electrode as a function of the lysine concentration in the external medium.
This is in good agreement with the numerical simulation (figure 9).

Note:

If an homogeneous enzyme membrane is immersed into a solution, the boundaries are symmetrical and there is a symmetry plane. Through this plane, there is no substrate or product diffusion, because the profile slopes are horizontal. This plane is like an impermeable wall. The concentration profiles in a half membrane and in an electrode coating are the same. By using an electrode we can find out the concentrations in the middle of a symmetrical membrane. (Fig. 10).

2°) Reversible monoenzyme system (THOMAS et al, in press)

a) - Equation:

reaction scheme:

$$
(30) \quad E + S \xrightarrow{k_1} K \xrightarrow{k_2} ES \xrightarrow{k_2} E + P \\
K \xrightarrow{k_1} K_1 \xrightarrow{k_1} K_2
$$

with the quasi-stationary assumption on the ES concentration, the
reaction rate is:

$$
(31) \quad \left( \frac{\partial S}{\partial t} \right)_{\text{reaction}} = - V_M \frac{[S] - a [P]}{K_M + [S] + b [P]}
$$

$$
(32) \quad a = \frac{k_{-1}}{k_1} \frac{k_{-2}}{k_2} \qquad b = \frac{k_{-2}}{k_1}
$$

As in (1) with dimensionless variables the equation is:

$$
(33) \quad \frac{\partial s}{\partial t} - \frac{\partial^2 s}{\partial x^2} + \sigma \frac{s - ap}{1 + s + bp} = 0
$$

Boundary and initial conditions are the same as in the irreversible
system in 1°).
<figure><img src="image_11.png" /><figcaption>**Figure 9** : Calculated product concentration along the electrode (Neumann condition) as a function of the substrate concentration in the external medium. Results are given with a logarithmic scale.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Figure 10** : Substrate (---) and product (-) concentration profiles are the same for the active layer of an enzyme electrode and for one half of an enzyme membrane with symmetrical boundaries.</figcaption></figure>
b) - Experimental and numerical results:

The reversibility is the simplest mechanism for a feedback limitation of a product accumulation.

Numerical and experimental studies were performed with the phospho-glucose isomerase which transforms in a reversible process D fructose-6 phosphate into D glucose-6 phosphate.

Experimental parameter values

- Glucose-6 phosphate isomerase activity

$$V_{M1} = (1.13) 10^{-3} \text{ moles cm}^{-3} \text{ h}^{-1}$$

- Michaelis constants

$$K_{M1} = 10^{-7} \text{ moles cm}^{-3}$$

$$K_{M2} = 3.45 10^{-7} \text{ moles cm}^{-3}$$

- Equilibrium concentration ratio

$$ (34) \quad \frac{\text{D-Fructose -6 phosphate}}{\text{D-Glucose -6 phosphate}} = 0.5 $$

- Diffusion coefficient

D-Fructose -6 phosphate

$$D_S = (0.40) 10^{-3} \text{ cm}^2 \text{ h}^{-1}$$

D glucose -6phosphate

$$D_P = (0.47) 10^{-3} \text{ cm}^2 \text{ h}^{-1}$$

- membrane thickness

$$e = (3.5) 10^{-3} \text{ cm}$$

Substrate concentration evolution (F-6 P) for soluble enzyme and for an immersed enzyme membrane are presented in figure 11. The boundary conditions correspond to the 4th case described in (1).
<figure><img src="image_13.png" /><figcaption>**Figure 11** : Reversible mono enzyme system : calculated and experimental substrate concentration evolution for enzyme in solution (---) and in a membrane (-) . The amounts of enzyme activity and the bulk solution volumes are the same in both cases.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Figure 12** : Instantaneous concentration profiles of substrate (-) and product (---) for a reversible monoenzyme membrane.</figcaption></figure>
Inside the membrane the substrate concentration is lower and the product concentration higher than in the bulk solution. In this way the reversibility effect is bigger with the enzyme membrane.

A couple of instantaneous profiles is presented on figure 12.

### 3°) Monoenzyme systems with inhibitors or activators

(THOMAS et al, in press)

It was necessary to study the effector effects in order to perform regulation and control of the described systems.

#### a) - Equations

##### - Competitive inhibition

By a way similar to the previously described systems, it is possible to write equations:

$$ (35) \quad \frac{\partial s}{\partial t} - \frac{\partial^2 s}{\partial x^2} + \sigma \frac{s}{1 + \frac{K_M}{K_i} i + s} = 0 \quad \text{(for the substrate)} $$

with $i = \frac{[I]}{K_M}$ and $K_i$ is the inhibition constant

$$ (36) \quad \frac{\partial i}{\partial t} - \frac{D_I}{D_S} \frac{\partial^2 i}{\partial x^2} = 0 \quad \text{(for the inhibitor)} $$
<figure><img src="image_15.png" /><figcaption>Figure 13</figcaption></figure>

Enzyme solution : α coefficient value as a function of the competitive

inhibitor concentration for different substrate concentrations.

Enzyme membranes : α coefficient value as a function of the concentration

for different substrate concentrations.

<figure><img src="image_16.png" /><figcaption>Figure 14</figcaption></figure>
- Non-competitive inhibition:

$$ (37) \quad \frac{\partial s}{\partial t} - \frac{\partial^2 s}{\partial x^2} + \sigma \frac{1}{1 + \frac{K_M}{K_i} i} \frac{s}{1 + s} = 0 \quad \text{(for the substrate)} $$

$$ (38) \quad \frac{\partial i}{\partial t} - \frac{D_I}{D_S} \frac{\partial^2 i}{\partial x^2} = 0 \quad \text{(for the inhibitor)} $$

- Activator:

$$ (39) \quad \frac{\partial s}{\partial t} - \frac{\partial^2 s}{\partial x^2} + \frac{a}{1+a} \frac{s}{1+s} = 0 \quad \text{(for the substrate)} $$
with $a = \frac{A}{K_M}$

$$ (40) \quad \frac{\partial a}{\partial t} - \frac{D_A}{D_S} \frac{\partial^2 a}{\partial x^2} = 0 \quad \text{(for the activator)} $$

Initial and boundary conditions for the different substances are the conditions described in 1º) b - i.

b) - Experimental and numerical results

We shall now present the numerical studies of inhibitor effects. These results are given in the experimental context.

In order to compare the inhibitor kinetic effects it is necessary to define a coefficient $\alpha$ between 0 and 1. V is the enzyme reaction rate without inhibitor, $V_\alpha$ is the reaction rate in presence of an inhibitor.

i) Competitive inhibition:

The calculated variation of $\alpha$, as previously defined, as a function of inhibitor concentration for several substrate concentration values is given for a michaelian enzyme in solution (figure 13) and an immersed membrane (figure 14).
The enzyme in a membrane is less sensitive to the inhibitor effect, especially for very small substrate concentration values. Diffusion limitation effects are nil for a zero order kinetics, that is to say when the substrate concentration is big enough.

Numerical results are compared with experimental results obtained with uricase immobilized into a protein membrane. The enzyme substrate is uric acid, and xanthine is a competitive inhibitor.

## Parameter values

- Maximum activity

$$V_M = 1.5 \ 10^{-5} \text{ mole cm}^{-3} \text{ h}^{-1}$$

- Uric acid diffusion coefficient

$$D_S = 2.8 \ 10^{-4} \text{ cm}^2 \text{ h}^{-1}$$

(41) - Michaelis constant

$$K_M = 2.7 \ 10^{-8} \text{ moles cm}^{-3}$$

- Inhibition constant

$$K_i = 1.5 \ 10^{-8} \text{ moles cm}^{-3}$$

- Substrate concentration

$$S_1 = 2 \ 10^{-8} \text{ moles cm}^{-3}$$

Coefficient α value as a function of inhibitor concentration (xanthine) is given on figure 15. Curves are from the numerical simulation and points are given for experimental results. Uricase is less sensitive to the inhibition after immobilization.

Experimental variation of α as a function of the substrate concentration for a constant inhibitor concentration ($2 K_i$) is given figure 16.

These results are similar to the numerical simulation (figure 20).
<figure><img src="image_17.png" /><figcaption>**Figure 15** : Experimental enzyme activity as a function of the competitive inhibitor concentration for a constant substrate concentration (2K<sub>M</sub>) with enzyme in solution (---) and inside a membrane (-).</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>**Figure 16** : Experimental enzyme activity as a function of substrate concentration for a constant competitive inhibitor concentration (2K<sub>I</sub>) with enzyme in solution (---) and in a membrane (-).</figcaption></figure>
### i.i.) Non-competitive inhibition:

The calculated variation of $\alpha$ previously defined, as a function of inhibitor concentration for several substrate concentration values is given for a michaelian enzyme in solution and in an immersed membrane (figure 17). With any substrate concentration there is only one curve for the enzyme in solution. It is a typical non competitive behavior.

After immobilization $\alpha$ value as a function of inhibitor concentration is dependent of the substrate concentration. Every enzyme membrane curves are above the enzyme solution curve.

### Comparison between numerical simulation and experimental results obtained with L.D.H. protein membrane:

This enzyme was studied with pyruvate like substrate and tartrate like inhibitor.

### Parameters values:

- Maximum activity

$$V_M = 5 \ 10^{-4} \text{moles cm}^{-3} \text{h}^{-1}$$

- Pyruvate diffusion coefficient

$$D = 10^{-3} \text{ cm}^2 \text{ h}^{-1}$$

(42) - Michaelis constant

$$K_M = 6.4 \ 10^{-6} \text{moles cm}^{-3}$$

- Tartrate inhibition constant

$$K_i = 4.2 \ 10^{-5} \text{moles cm}^{-3}$$

The experimental value of $\alpha$ as a function of the tartrate concentration is presented figure 18.
<figure><img src="image_19.png" /><figcaption>Figure 17</figcaption></figure>

Calculated α coefficient value as a function of a non competitive inhibitor concentration for different substrate concentrations. For enzyme in solution (-) and in a membrane (---).

<figure><img src="image_20.png" /><figcaption>Figure 18</figcaption></figure>

Experimental enzyme activity as a function of the non competitive inhibitor concentration for a constant substrate concentration (2K_m) with enzyme in solution (---) and in a membrane (-).
The experimental value of α as a function of substrate concentration for a constant inhibitor concentration ( 2 Kᵢ ) is presented figure 19.

The experimental results are similar to the numerical simulation
(figure 20).

4°) *Multienzyme systems* (LECOQ et al, 1973)

a) - Simple bienzyme system

i.) Equation:

Two sequential enzymes are studied:

<figure><img src="image_21.png" /></figure>

The reaction kinetics are michaelian. The system is studied with enzymes
in solution and immobilized into a membrane.

- Kinetic parameters:

V<sub>M1</sub> and V<sub>M2</sub> maximum activities for both enzymes

K<sub>M1</sub> and K<sub>M2</sub> Michaelis constants

- Physical parameters:

e thickness of the membrane

D diffusion coefficient (D<sub>S</sub> = D<sub>P</sub> = D<sub>Q</sub>)

System in solution

S equation

$$
(44) \qquad \frac{dS}{dt} = - V_{M1} \frac{S}{K_{M1} + S}
$$
<figure><img src="image_22.png" /><figcaption>**Figure 19** : Experimental enzyme activity as a function of substrate concentration for constant competitive inhibitor concentration with enzyme in solution (---) and in a membrane (---).</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>**Figure 20** : Calculated α coefficient value as a function of substrate concentration for the different previously described systems (in inhibitor concentration = 2K<sub>T</sub>).</figcaption></figure>
Equation for P

$$ (45) \qquad \frac{dP}{dt} = V_{M1} \frac{S}{K_{M1} + S} - V_{M2} \frac{P}{K_{M2} + P} $$

Equation for Q

$$ (46) \qquad \frac{dQ}{dt} = V_{M2} \frac{P}{K_{M2} + P} $$

## Membrane system

Equations are written in order to get dimensionless parameters and variables.

$$ s = \frac{S}{K_{M1}} \qquad p = \frac{P}{K_{M2}} \qquad q = \frac{Q}{K_{M2}} $$

$$ (47) \qquad \frac{x}{e} \rightarrow x \qquad \frac{t}{e^2/D} \rightarrow t $$

$$ \sigma_1 = \frac{V_{M1}}{K_{M1}} \cdot \frac{e^2}{D} \qquad \sigma_2 = \frac{V_{M2}}{K_{M2}} \cdot \frac{e^2}{D} $$

Equations for S, P and Q in a membrane system:

$$ (48) \qquad \frac{\partial s}{\partial t} = \frac{\partial^2 s}{\partial x^2} - \sigma_1 \frac{s}{1 + s} $$

$$ (49) \qquad \frac{\partial p}{\partial t} = \frac{\partial^2 p}{\partial x^2} - \sigma_2 \frac{p}{1+p} + \frac{K_{M1}}{K_{M2}} \sigma_1 \frac{s}{1+s} $$

$$ (50) \qquad \frac{\partial q}{\partial t} = \frac{\partial^2 q}{\partial x^2} + \sigma_2 \frac{q}{1+q} $$

Boundary conditions are conditions so-called of the immersed membrane (4th case 1º) - b -i.)
At time zero, in the bulk solution there is only substrate (concentration $K_{M1}$), and the membrane is empty.

### i.i.) Numerical results:

Numerical results are given on the figures 21 and 22, respectively for enzymes in solution and immobilized enzymes. The production of Q is quicker in the membrane system than in solution.

This acceleration is due to a higher product concentration in the membrane than in the bulk solution.

The bienzyme membrane system is more efficient than enzymes dispersed in solution.

### b) - Bienzyme system with a feed-back

MATTIASSON and MOSBACH's work (1970) deals with a simple bienzyme system, without activation or inhibition effects.

A system with a feed-back effect is presented now. Its study deals with the two enzymes in solution, the two enzymes in the same membrane and the two enzymes into two different membranes.

A theoretical analysis of a system with two enzymes immobilized onto a membrane was published by GOLDMAN and KATCHALSKI (1971). This study was performed with the stationary state assumption, with first order reactions and without diffusion inside the active layer.

### i.) Equations:

The enzymes are β-glucosidase and glucose-oxidase reaction scheme:

<figure><img src="image_24.png" /></figure>
<figure><img src="image_25.png" /><figcaption>**Figure 21** : S,P and Q concentration evolution for a bienzyme system in solution.</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>**Figure 22** : S,P and Q concentration evolution in the bulk solution for an immersed bienzyme membrane.</figcaption></figure>
Note:

In this case, the system is too complex to allow the use of new dimensionless variables.

Symbols used:

S salicine concentration

P glucose concentration

I gluconolactone concentration

Q gluconic acid concentration

K ratio $\frac{V_M}{K_M}$ for the β-glucosidase activity

$K_i$ constant of inhibition of the gluconolactone

$V_M$ and $K_M$ are written for the glucose oxidase activity

K constant of gluconolactone decomposition

Equations in solution for S, P, I and Q

$$ (51) \qquad \frac{dS}{dt} = -k \frac{S}{1 + \frac{I}{K_i}} $$

$$ (52) \qquad \frac{dP}{dt} = - \frac{V_M P}{K_M + P} + \frac{kS}{1 + \frac{I}{K_i}} $$

$$ (53) \qquad \frac{dI}{dt} = + \frac{V_M P}{K_M + P} - KI^2 $$

$$ (54) \qquad \frac{dQ}{dt} = + KI^2 $$
Study of the one membrane system

It is necessary to take account of production, consumption and
diffusion.

$$
(55) \quad \frac{\partial S}{\partial t} = D_S \frac{\partial^2 S}{\partial x^2} - \frac{kS}{1 + \frac{I}{K_i}}
$$

$$
(56) \quad \frac{\partial P}{\partial t} = D_P \frac{\partial^2 P}{\partial x^2} + \frac{kS}{1 + \frac{I}{K_i}} - \frac{V_M P}{K_M + P}
$$

$$
(57) \quad \frac{\partial I}{\partial t} = D_I \frac{\partial^2 I}{\partial x^2} + \frac{V_M P}{K_M + P} - KI^2
$$

$$
(58) \quad \frac{\partial Q}{\partial t} = D_Q \frac{\partial^2 Q}{\partial x^2} + KI^2
$$

where Ds, DP, DI and DQ are respectively diffusion coefficient for
salicine, glucose, gluconolactone and gluconic acid.

Initial conditions:

$$
S(o,o) = 3,5 \cdot 10^{-6} \text{ moles cm}^{-3}
$$

$S(x,0) = 0$

concentrations of P, I et Q are nil at time zero.

Boundary conditions:

$$
(59) \quad \left[ \frac{\partial S}{\partial t} \right]_{\text{solution}} = - \frac{J_S \Omega}{V} \quad \text{with} \quad J_i = D_i \frac{\partial i}{\partial v}
$$

$$
(60) \quad \left[ \frac{\partial P}{\partial t} \right]_{\text{solution}} = - \frac{J_P \Omega}{V}
$$
$$
(61) \quad \left[ \frac{\partial I}{\partial t} \right]_{\text{solution}} = - \frac{J_I \Omega}{V} - KI^2
$$

$$
(62) \quad \left[ \frac{\partial Q}{\partial t} \right]_{\text{solution}} = - \frac{J_Q \Omega}{V} + KI^2
$$

$\Omega$ is the solution membrane interface area and $V$ solution volume.

Study of the two membrane system

For the first membrane:

$$
(63) \quad \frac{\partial S}{\partial t} = D_S \frac{\partial^2 S}{\partial x^2} - k \frac{S}{1 + \frac{I}{K_i}}
$$

$$
(64) \quad \frac{\partial P}{\partial t} = D_P \frac{\partial^2 P}{\partial x^2} + \frac{kS}{1 + \frac{I}{K_i}}
$$

$$
(65) \quad \frac{\partial I}{\partial t} = D_I \frac{\partial^2 I}{\partial x^2} - KI^2
$$

$$
(66) \quad \frac{\partial Q}{\partial t} = D_Q \frac{\partial^2 Q}{\partial x^2} + KI^2
$$

For the second membrane:

$$
(67) \quad \frac{\partial S}{\partial t} = D_S \frac{\partial^2 S}{\partial x^2}
$$

$$
(68) \quad \frac{\partial P}{\partial t} = D_P \frac{\partial^2 P}{\partial x^2} - \frac{V_M P}{K_M + P}
$$

$$
(69) \quad \frac{\partial I}{\partial t} = D_I \frac{\partial^2 I}{\partial x^2} + \frac{V_M P}{K_M + P} - KI^2
$$

$$
(70) \quad \frac{\partial Q}{\partial t} = D_Q \frac{\partial^2 Q}{\partial x^2} + KI^2
$$
### Initial conditions:

$$ 
\begin{array}{l}
\text{(71)} \quad \begin{array}{l} S(0,0) = 3,5 \cdot 10^{-6} \text{ moles cm}^{-3} \\ S(x,0) = 0 \text{ for both membranes.} \\ P, I \text{ and Q are nil at time zero.} \end{array}
\end{array}
$$

### Boundary conditions:

$$ (72) \quad \left[ \frac{\partial S}{\partial t} \right]_{\text{solution}} = - \frac{J_{S1}\Omega_1}{V} - \frac{J_{S2}\Omega_2}{V} $$

$$ (73) \quad \left[ \frac{\partial P}{\partial t} \right]_{\text{solution}} = - \frac{J_{P1}\Omega_1}{V} - \frac{J_{P2}\Omega_2}{V} $$

$$ (74) \quad \left[ \frac{\partial I}{\partial t} \right]_{\text{solution}} = - \frac{J_{I1}\Omega_1}{V} - \frac{J_{I2}\Omega_2}{V} - KI^2 $$

$$ (75) \quad \left[ \frac{\partial Q}{\partial t} \right]_{\text{solution}} = - \frac{J_{Q1}\Omega_1}{V} - \frac{J_{Q2}\Omega_2}{V} + KI^2 $$

## i.i.) Experimental and numerical results:

### Parameter values

$$ \text{initial salicine concentration : } 3,5 \cdot 10^{-6} \text{ moles cm}^{-3} $$

### β glucosidase

$$ \text{in solution} \quad \frac{V_M}{K_M} = k = 1,4 \text{ h}^{-1} $$

$$ \text{in a membrane} \quad k = 270 \text{ h}^{-1} $$

### glucose oxidase

$$ K_M = 4 \cdot 10^{-6} \text{ moles cm}^{-3} $$

$$ \text{in solution} \quad V_M = 5,42 \cdot 10^{-6} \text{ moles cm}^{-3} \text{ h}^{-1} $$

$$ \text{in a membrane} \quad V_M = 1,04 \cdot 10^{-3} \text{ moles cm}^{-3} \text{ h}^{-1} $$
lactone

$$ \text{inhibition } K_i \text{ constant} = 10^{-8} \text{ moles cm}^{-3} $$

$$ \text{dissociation k constant} = 2,745 \cdot 10^4 \text{ mole}^{-1} \text{ cm}^3 \text{ h}^{-1} $$

$$ \text{Volume of the bulk solution : } 50 \text{ cm}^3 $$

$$ \text{Membrane thickness } e = 6,2 \cdot 10^{-3} \text{ cm} $$

$$ \text{area } s = 42 \text{ cm}^2 $$

diffusion coefficient

$$ \text{salicine } D_S = 1,2 \cdot 10^{-3} \text{ cm}^2 \text{ h}^{-1} $$

$$ \text{glucose } D_P = 10^{-3} \text{ cm}^2 \text{ h}^{-1} $$

$$ \text{lactone } D_I = 10^{-3} \text{ cm}^2 \text{ h}^{-1} $$

$$ \text{gluconic acid } D_Q = 10^{-3} \text{ cm}^2 \text{ h}^{-1} $$

Maximum enzyme activities are the same in the three different systems.

Raw experimental results are given on figure 23. However the plotting of $\frac{v}{v_o}$ ratio as a function of time gives more quantitative information. (figure 24).

($v$ is the instantaneous value of $\frac{ds}{dt}$ in the bulk solution. $v_o$ is the value at time zero, that is to say without inhibitor.

By using numerical simulation it is possible to get several curves. Firstly saligenine, glucose and gluconolactone concentrations in the bulk solution as a function of time for free enzyme (Figure 25), one membrane system (figure 26) and two membranes system (figure 27).

Calculated $\frac{v}{v_o}$ ratio as a function of time is plotted on figure 28.

It is not possible to superimpose curves from figure 24 and 26 but the calculated and experimental results are very closed.
<figure><img src="image_27.png" /><figcaption>**Figure 23** : Experimental saligenine concentration variations as a function of time for the bienzyme systems in solution -●-, in a membrane -O- and in two different membranes.</figcaption></figure>

<figure><img src="image_28.png" /><figcaption>**Figure 24** : Experimental value of the $\frac{v}{v_0}$ ratio as a function of time for the systems in solution -●-, in a membrane -O- and in solution -Δ-.</figcaption></figure>
<figure><img src="image_29.png" /><figcaption>Figure 25 : Saligenine (1), Glucose (2) and lactone (3) concentration evolution in a solution obtained on computer.</figcaption></figure>

<figure><img src="image_30.png" /><figcaption>Figure 26 : Saligenine (1), glucose (2) and lactone (3) concentration evolution for the one membrane system obtained on computer.</figcaption></figure>
<figure><img src="image_31.png" /><figcaption>**Figure 27** : Saligenine (1), glucose (2) and lactone (3) concentration evolution for the two membrane system obtained on computer.</figcaption></figure>

<figure><img src="image_32.png" /><figcaption>**Figure 28** : $\frac{v}{v_0}$ ratio value as a function of time for the solution (1), one membrane (2), and two membranes (3) systems.</figcaption></figure>
<figure><img src="image_33.png" /><figcaption>**Figure 29** : Gluconolactone concentration profile evolution inside a β-glucosidase glucose oxidase membrane.</figcaption></figure>
with

$x_i$ = characteristic function of $]i-1,i[$

$$ (77) \quad F(y_i) = \sigma \frac{y_i}{1 + |y_i|}, \quad \sigma = 10 $$

**Boundary conditions**

$$ y_1(o) = 1, \quad y_1(10) = 0 $$

$$ y_i(o) = y_i(10) = 0 \quad 2 \le i \le 11 $$

A slightly different model is obtained when the last enzyme product ($y_{11}$) is an inhibitor of the first one.

First reaction term is replaced by

$$ (78) \quad \chi_1 F(y_1, y_{11}) = \chi_1 \sigma \frac{|y_1|}{1+|y_1|+|y_{11}|} $$

**i.i.) Numerical simulation:**

Figure 30 gives the relationship between $J_p$, flux leaving for $x = 10$ of the last product ($y_{11}$) and $J_s$, substrate flux entering for $x = 0$

**d) - Multienzyme systems, bidimensional case.**

In this case each enzyme $E_j$ is located in an elementary square of a mosaic.

<table><tr><td rowspan="4">Γ</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>E<sub>1</sub></td><td></td><td></td><td></td><td></td></tr><tr><td>E<sub>2</sub></td><td>E<sub>3</sub></td><td>E<sub>7</sub></td><td>E<sub>8</sub></td><td>E<sub>9</sub></td></tr><tr><td></td><td>E<sub>4</sub></td><td>E<sub>5</sub></td><td>E<sub>6</sub></td><td>E<sub>10</sub></td></tr></table>
<figure><img src="image_34.png" /><figcaption>**Figure 30** : Multienzyme monodimensional system : relationship between the substrate concentration for x = 0 and the $\frac{J_p}{J_s}$ ratio ($J_s$ is the substrate entering flux for x = 0 and $J_p$ is the leaving product flux for x = 10).</figcaption></figure>
Equations are like in (c) only $-\frac{d^2}{dx^2}$ is replaced by $-\Delta$,
and $\chi_j$ is the characteristic function of $E_j$ square.

Boundary conditions are similar

$y_1 = 1$ on the boundary $\Gamma$ of $\Omega$, domain of the mosaic

$y_i = 0 \quad 2 \le i \le 11$

Another kind of this system is a multienzyme with feedback. The $y_8$
product is an inhibitor of $E_2$. This system is a loop at both metabolic
and spatial points of view.

<figure><img src="image_35.png" /></figure>

The $E_2$ reaction term is:

$$ (79) \qquad \chi_2 \sigma \frac{y_2}{1+|y_2|+|y_8|} $$

i.i.) Numerical results:

The results are to be published.
III - TRANSPORT PHENOMENA

THOMAS et al (1972)

THOMAS (1971)

1º) - *Two active layers system, active transport:*

This system is an example of a transport against the concentration gradient. There is in the middle two protein active layers and on both sides two selective layers. These films are impermeable to glucose-6-phosphate, but permeable to glucose.

<table><tbody><tr><td rowspan="10">SOLUTION 1</td><td>C</td><td></td><td></td><td></td></tr><tr><td>O</td><td></td><td></td><td>O</td></tr><tr><td>U</td><td></td><td></td><td>U</td></tr><tr><td>C</td><td>H</td><td>P</td><td>C</td></tr><tr><td>H</td><td>E</td><td>H</td><td>H</td></tr><tr><td>E</td><td>X</td><td>O</td><td>E</td></tr><tr><td>O</td><td></td><td>S</td><td></td></tr><tr><td>S</td><td>K</td><td>P</td><td>S</td></tr><tr><td>E</td><td>I</td><td>H</td><td>E</td></tr><tr><td>L</td><td>N</td><td>A</td><td>L</td></tr><tr><td>E</td><td>A</td><td>T</td><td>E</td></tr><tr><td>C</td><td>S</td><td>A</td><td>C</td></tr><tr><td>T</td><td>E</td><td>S</td><td>T</td></tr><tr><td>I</td><td></td><td>E</td><td>I</td></tr><tr><td>V</td><td></td><td></td><td>V</td></tr><tr><td>E</td><td></td><td></td><td>E</td></tr></tbody></table>

SOLUTION 2

In the enzyme layers produced by the crosslinking method there are hexokinase and phosphatase activity.
In this asymmetrical membrane, the glucose is temporarily phosphorylated
according with the scheme.

$$
(30) \quad \text{glucose} + \text{ATP} \xrightarrow{\text{hexokinase } (E_1)} \text{glucose-6 phosphate} + \text{ADP}
$$

$$
(81) \quad \text{glucose-6 phosphate} \xrightarrow{\text{phosphatase } (E_2)} \text{glucose} + \text{inorganic phosphate}
$$

Chemically, this system behaves like a simple ATPase.

$$
(82) \quad \text{ATP} \longrightarrow \text{ADP} + \text{inorganic mineral phosphate}
$$

i.) Equation:

- In the first layer ($0 < x < \frac{1}{2}$), the enzyme $E_1$, ($S \xrightarrow{E_1} P$) is immobilized with the reaction rate:

$$
(33) \quad F(x,s,p) = \sigma \frac{s}{1 + \frac{K_M}{K_p} p + s} \quad \text{for } 0 < x < \frac{1}{2}
$$

The variables have been defined in (II - 3º - a).

- In the second one ($\frac{1}{2} < x < 1$), $E_2$ is immobilized, ($P \xrightarrow{E_2} S$)

The reaction rate is:

$$
(s4) \quad v = - \sigma' \frac{P}{\frac{\dot{K}'_M}{K_M} + P}
$$

with

$$
(85) \quad \sigma' = \frac{v_{M'}}{k_{M'}} \frac{e^2}{D_S}
$$

$v_{M'}$ and $k_{M'}$ Michaelis constants of $E_2$

Boundary conditions for P : NEUMANN

Boundary conditions for S are of DIRICHLET type for x = 0, and of mobile
boundary concentration type for x = 1.
Equations:

$$ (86) \quad \frac{\partial s}{\partial t} - \frac{\partial^2 s}{\partial x^2} + F(x,s,p) = 0, \quad \frac{\partial p}{\partial t} - \frac{D_P}{D_S} \frac{\partial^2 p}{\partial x^2} - F(x,s,p) = 0 $$

$$ \frac{\partial p}{\partial v} = 0 \quad \text{for } x = 0 \text{ and } x = 1 $$

$$ s(0,t) = \alpha \quad \text{and} \quad s(1,t) = \beta \quad (\alpha, \beta > 0) $$

$$ s(0,t) = \alpha \quad \text{and} \quad \begin{cases} s(1,0) = \beta \\ \frac{\partial s}{\partial t} + \chi \frac{\partial s}{\partial v} = 0 \quad \text{for } x = 1 \end{cases} $$

$$ s(x,0) = p(x,0) = 0 $$

Note:

Films are selective only for concentration smaller than h

$$ (87) \quad \frac{\partial p}{\partial v} = \begin{cases} 0 & \text{for } p \le h \\ -ap & \text{for } p \ge h \end{cases} \quad (a>0) $$

i.i.) Numerical results and experimental context:

Concentration profiles in a membrane between two compartments containing substrate solutions $S_1$ and $S_2$. In the initial conditions $S_1 = S_2$ and during the evolution $S_2$ increases, $S_1$ is constant (the first compartment is big enough in comparison with the second one).

With a well defined spatial distribution of two different enzymes it is possible to transform the diffusion-reaction phenomenon in a pumping effect against the concentration gradient.

$$ V_M = V_{M'} = 3,5 \text{ moles cm}^{-3} \text{ h}^{-1} $$

$$ K_M = K_{M'} = K_p = 10^{-6} \text{ moles cm}^{-3} $$

$$ e = 5 \times 10^{-3} \text{ cm} $$

$$ (88) \quad D_S = D_P = 5 \times 10^{-3} \text{ cm}^2 \text{ h}^{-1} $$
<figure><img src="image_36.png" /><figcaption>**Figure 31** : Concentration profiles for a double layer bienzyme membrane. Glucose is given in -and G 6 P in ----- .</figcaption></figure>

<figure><img src="image_37.png" /><figcaption>**Figure 32** : Receptor compartment glucose concentration as a function of time for different initial concentrations.</figcaption></figure>
$$
\begin{align*}
(88) \quad & \alpha = 1 \quad \text{and} \quad \beta = 1 \quad \text{Dirichlet conditions for } S \\
& \alpha = 1, \quad \beta = 0 \quad \text{and } \chi = 0.1 \quad \text{with mobile concentration for } x = 1. \\
& a = 0.1 \quad \text{for a P boundary condition which depends from P value.}
\end{align*}
$$

Experimentally at time zero, glucose solutions are equal in the compartments. ($S_1 = S_2$). The first one is 150 times bigger than the second one, and first concentration is constant during all the experiment.

Receptor compartment concentration is measured, as a function of time
(figure 32) for three different initial conditions.

At the beginning there is a linear increasing of S₂ until a plateau.
This plateau is explained by the profile B on figure 3J. After the S₂
concentration decreases when all the ATP is consumed.

The transport system is saturated for S₁ values bigger than 10⁻²M,
initial accumulation rates and the level of the resulting gradient remains
constant.

Profile evolution from an empty ideal membrane shows that concentration
130 times bigger than in the external medium can be obtained in the receptor
compartment. (Fig. 33 and 34).

2º) *Diffusion-reaction linked to an acido basic metabolism*
*A dissipative structure example.* (THOMAS et al (1973))

i.) Equation:

If there is a pH effect, V_M is replaced by αV_M. α is a function of pH with a value between 0 and 1.
<figure><img src="image_38.png" /><figcaption>Figure 33 and 34: Calculated concentration profile evolution inside the double layer bienzyme system: Instantaneous profiles (fig. 33) and final profile at the stationary state (fig. 34).</figcaption></figure>

<figure><img src="image_39.png" /><figcaption>Figure 34</figcaption></figure>
Two enzymes $E_1$ and $E_2$ are homogeneously immobilized.

The reaction $E_i$ is:

$$S_i \xrightarrow{E_i} P_i \quad i = 1,2$$

<figure><img src="image_40.png" /></figure>

with an excess of substrates reaction rates are:

$$v_i = \alpha_i \sigma_i \quad (i = 1,2)$$

$P_1$ is an acid, $E_1$ decreases the pH value

$P_2$ is a base, so that $E_2$ increases the pH value.

<table><thead><tr><th>enzyme</th><th>range of activity</th><th>substrate</th><th>product</th><th>Kind of product</th><th>pH variation</th></tr></thead><tbody><tr><td>E<sub>1</sub></td><td>low pH</td><td>S<sub>1</sub></td><td>P<sub>1</sub></td><td>acid</td><td>decreasing</td></tr><tr><td>E<sub>2</sub></td><td>high pH</td><td>S<sub>2</sub></td><td>P<sub>2</sub></td><td>base</td><td>increasing</td></tr></tbody></table>
E_i activity is maximum at pH = a_i (i = 1,2).

When the two enzymes are homogeneously immobilized inside a membrane, there is no variation of pH for a pH value a_o.

With a pH value of a_o + ε in the first compartment and of a_o - ε in the second one there will be amplification of the concentration difference On both sides there will be an autocatalytic reaction.

Equations:

Product concentrations are ruled by

$$ (89) \quad \frac{\partial P_i}{\partial t} - \frac{\partial^2 P_i}{\partial x^2} + \alpha_i = 0 \quad i = 1,2 $$

with σ₁ = σ₂ = σ

pH is linked to P₁ and P₂ by

$$ (90) \quad \text{pH} = (\text{pH})_0 + \beta(p_2 - p_1) $$

It is possible to write

$$ (91) \quad \frac{\partial \text{pH}}{\partial t} - \frac{\partial^2 \text{pH}}{\partial x^2} + \beta \sigma (\alpha_2 - \alpha_1) = 0 $$

The function (α₂ - α₁) is described by the below curve.
<figure><img src="image_41.png" /></figure>

**Initial conditions:**

For $t = 0$ the pH through all the system is equal to $a_0$.

**Boundary conditions:**

$$ (92) \quad \frac{\partial \text{pH}}{\partial t} + \chi \frac{\partial \text{pH}}{\partial v} = 0 $$

**i.i.) Numerical results:**

The concentration profile inside the membrane is presented figure 35.

This result is in good agreement with the experimental observations.
<figure><img src="image_42.png" /><figcaption>**Figure 35** : pH profile evolution inside the homogeneous glucose oxidase-urease membrane after asymmetrical fluctuations.</figcaption></figure>
### 3°) - Ion diffusion and enzyme ion production inside an artificial membrane

DAVID et al (1973)

An artificial enzyme membrane separates two compartments containing electrolyte solution ($I^+$, $I^-$) with different concentrations. Inside the membrane the enzyme reaction transforms a neutral substrate S into a product with two ions $P^+$ and $P^-$ according with this above scheme:

$$ (93) \qquad E + S \underset{k_{-1}}{\overset{k_1}{\rightleftharpoons}} ES \quad \underset{k_2}{\overset{k_2}{\rightleftharpoons}} E + P^+ + P^- $$

A potential difference can be maintained between the membrane boundaries, the systems can also spontaneously evolve.

In presence of symmetrical concentrations of substrate the reaction term is:

$$ (94) \qquad F(s) = \sigma \frac{s}{1+s} $$

Units are like in III 1° - b-i for time, space, and concentration. Concentrations of $S$, $I^+$, $I^-$, $P^+$, $P^-$ are written respectively $s$, $y_1$, $y_2$, $y_3$ and $y_4$

and it is possible to define

$$ (95) \qquad \phi(x,t) = \frac{F}{RT} \cdot \text{potential difference (in mV).} $$

Fluxes are defined by:

$$ (96) \qquad J_1 = - \left[ \frac{\partial y_1}{\partial x} + y_1 \frac{\partial \psi}{\partial x} \right] $$
$$ (97) \qquad J_2 = - \frac{D_-}{D_+} \left[ \frac{\partial y_2}{\partial x} - y_2 \frac{\partial \psi}{\partial x} \right] $$

$$ (98) \qquad J_3 = - \left[ \frac{\partial y_3}{\partial x} + y_3 \frac{\partial \psi}{\partial x} \right] $$

$$ (99) \qquad J_4 = - \frac{D_-}{D_+} \left[ \frac{\partial y_4}{\partial x} - y_4 \frac{\partial \psi}{\partial x} \right] $$

and equations for $s, y_i$ ($i = 1,2,3,4$) are

$$ (100) \qquad \frac{\partial s}{\partial t} - \frac{\partial^2 s}{\partial x^2} + F(s) = 0 $$

$$ (101) \qquad \frac{\partial y_1}{\partial t} + \frac{\partial J_1}{\partial x} = 0 \quad ; \quad \frac{\partial y_2}{\partial t} + \frac{\partial J_2}{\partial x} = 0 $$

$$ (102) \qquad \frac{\partial y_3}{\partial t} + \frac{\partial J_3}{\partial x} = F(s) \quad ; \quad \frac{\partial y_4}{\partial t} + \frac{\partial J_4}{\partial x} = F(s) $$

Boundary conditions for $s$ and $y_i$ ($i = 1,2,3,4$) are of Dirichlet type:

$$ (103) \qquad s(0,t) = s(1,t) = 10 $$

$$ (104) \qquad y_1(0,t) = y_2(0,t) = 10 \quad ; \quad y_1(1,t) = y_2(1,t) = 1 $$

$$ (105) \qquad y_3(0,t) = y_3(1,t) = y_4(0,t) = y_4(1,t) = 0 $$

and initial conditions

$$ s(x,0) = 0 $$

$$ (106) \qquad y_1(x,0) = y_2(x,0) = -9x + 10 $$

$$ y_3(x,0) = y_4(x,0) = 0 $$

It is possible to write the **electroneutrality assumption**

$$ (107) \qquad y_1 + y_3 = y_2 + y_4 $$
When the potential difference is maintained at the membrane boundaries

$$ (108) \qquad \psi(0,t) = 0 \qquad \psi(1,t) = V $$

When the systems evolve "spontaneously", it is possible to write the current zero assumption.

$$ (109) \qquad J_1 + J_2 = J_3 + J_4 $$

### i.i.) Numerical results:

Diffusion coefficient values:

$$ D_+ = D_S = D_I^+ = D_P^+ = 10^{-3} \text{ cm}^2 \text{ h}^{-1} $$

$$ D_- = D_I^- = D_P^- = 10^{-4} \text{ cm}^2 \text{ h}^{-1} $$

Membrane thickness:

$$ e = 5.10^{-3} \text{ cm} $$

$$ K_M = 10^{-6} \text{ mole cm}^{-3} $$

$$ V_M = 10^{-3} \text{ mole cm}^{-3} \text{ h}^{-1} $$

$$ \sigma = \frac{V_M}{K_L} \frac{e^2}{D_S} = 25 $$

Concentration profile evolutions are presented figures 36, 37, 38, 39 and 40 for respectively S, I⁺, I⁻, P⁺, P⁻, and the potential profile evolution figure 41.
<figure><img src="image_43.png" /><figcaption>Figure 36 : Substrate concentration profile evolution</figcaption></figure>

<figure><img src="image_44.png" /><figcaption>Figure 37 : I⁺ Concentration profile evolution</figcaption></figure>
<figure><img src="image_45.png" /><figcaption>Figure 38 : I⁻ Concentration profile evolution</figcaption></figure>

<figure><img src="image_46.png" /><figcaption>Figure 39 : P⁺ Concentration profile evolution</figcaption></figure>
<figure><img src="image_47.png" /><figcaption>Figure 40 : P⁻ Concentration profile evolution</figcaption></figure>

<figure><img src="image_48.png" /><figcaption>Figure 41 : Potential profile evolution.
Biochemical reactors</figcaption></figure>
IV - ENZYME REACTORS

GELLF (1972)

GELLF et al (in press)

HENRY and GELLF (1973)

Studies about the kinetic behavior of enzyme porous particles used in a packed bed are to be presented now.

It is possible to classify immobilized enzyme reactors into three categories:

- Immobilized enzyme is suspended in a batch reactor. The leaving flow goes through a filtration system. Maltose hydrolysis to glucose by glucoamylase activity with this method was studied by O'NEILL et al (1971) and SMILEY (1971).

- The enzyme is covalently bound onto an ion exchange sheet of paper. A continuous flow of substrate solution goes through this sheet. L.D.M. and pyruvate kinase were studied in this kind of reactors by WILSON et al (1968 and 1968).

But with this system there is a lot of experimental troubles.

- The enzyme is adsorbed or bound onto porous particles. These particles are used in packed bed columns. A continuous flow of substrate solution goes through the columns.

This system is the most efficient. BAR. ELI and KATCHALSKI (1963), NIKOLAIEV and MAR DASHEV (1961 and 1962), ARNOLD (1966), LILLY et al (1966) have given a relation in which the percentage of substrate transformed is linked to the system parameters.

No study for an evolutive process was previously described.
a) First model:

Let us consider an enzyme column: $l$ is the distance between the entrance and the exit, $s$ is the section area, and a flow of substrate solution goes through the column with a linear velocity $C$. The axis is the revolution axis of the enzyme column. The abscissa $0$ is chosen at the column entrance.

In each point of abscissa $x$ inside the reactor $\frac{\partial S}{\partial t}$ is ruled by three simultaneous phenomena: enzyme activity, solution convection, and substrate diffusion. In this first model the diffusion inside the particles and the unstirred layers is not taken into account.

Equations:

$$ (110) \qquad \frac{\partial S}{\partial t} = \left(\frac{\partial S}{\partial t}\right)_{\text{diffusion}} + \left(\frac{\partial S}{\partial t}\right)_{\text{convection}} + \left(\frac{\partial S}{\partial t}\right)_{\text{reaction}} $$

$$ (111) \qquad \left(\frac{\partial S}{\partial t}\right)_{\text{reaction}} = - \frac{V_M' S}{K_M' + S} $$

$V_M'$ and $K_M'$ are experimental apparent constants of immobilized enzyme.

$$ (112) \qquad \left(\frac{\partial S}{\partial t}\right)_{\text{convection}} = - \frac{\partial S}{\partial x} \cdot \frac{\partial x}{\partial t} = - C \frac{\partial S}{\partial x} $$

The partial differential equation is

$$ (113) \qquad \frac{\partial S}{\partial t} = D_S \frac{\partial^2 S}{\partial x^2} - C \frac{\partial S}{\partial x} - \frac{V_M' S}{K_M' + S} $$
It is possible to simplify the equations by writing dimensionless variables:

$\frac{x}{l}$ is $x$

$$s = \frac{s}{K_M}$$

$\frac{t}{l^2/D}$ is $t$

$x$, $s$ and $t$ are dimensionless variables.

$$ (114) \qquad \lambda = \frac{c l}{D_s} \quad \text{and} \quad \sigma = \frac{v_M'}{K_M'} \cdot \frac{l^2}{D_s} $$

dimensionless parameters

The equation (113) is now:

$$ (115) \qquad \frac{\partial s}{\partial t} = \frac{\partial^2 s}{\partial x^2} - \lambda \frac{\partial s}{\partial x} - \sigma \frac{s}{1+s} $$

for $0 < x < 1$ and $t > 0$

**Initial condition:**

$$ s(X,0) = 0 $$

**Boundary conditions:**

$$ s(0,t) = s_1 \quad \text{and} \quad s_1 > 0 $$

$$ \frac{\partial s}{\partial x}(1,t) = 0 $$
<figure><img src="image_49.png" /><figcaption>**Figure 42** : S₁α as a function of log (1-α) for an experimental system.</figcaption></figure>

<figure><img src="image_50.png" /><figcaption>**Figure 43** : Scheme of the model.</figcaption></figure>
In fact, in this model the diffusion is negligible and an analytical
solution is obviously obtained with the stationary state assumption.

$$
(116) \quad s_1^\alpha - K_M' \log(1-\alpha) = \frac{V_M' \ell}{C} \quad \text{with } \alpha = \frac{s_1 - s_2}{s_1} \quad \text{for} \quad \begin{cases} x = 1 \\ s = s_1 \end{cases}
$$

If this model is valid the plottings of s₁α as a function of
Log (1-α) with different linear flow velocities would be some straight lines
with the same slope K'M' and V'M'C' like value at the intersection point.

The different authors got a linear relationship between s₁α and
Log (1-α) for each flow velocity (C). But the apparent constant values
vary with this velocity.

For example, several curves from an α chymotrypsin packed bed reactor
are given figure 42. V'M' and K'M' apparent values vary with the linear
flow velocity. LILLY, MORNBY and CROOK with ficin bound on C.M.C., O'NEILL,
DUNNIL and LILLY with a D.E.A.E. amyloglucosidase and MONSAND with bentonite
invertase described the same behavior.

Michaelis constant and maximum activity are some enzyme molecular pro-
perties and it is absolutely impossible to get a variation of these values
with the flow velocity. This apparent variation occurs only to compensate
the inadequacy of this model. It is necessary to take into account in plus
the diffusion inside the particles and the unstirred layers and the flow
velocity profile between the particles.

b) - Second model:

There is the same behavior inside any elementary column inside the
packed bed reactor and this behavior is identical to the behavior of the
all reactor.

Let us consider a liquid vessel surrounded by a solid phase.
Equations for this system: (figure 43)

- Solid phase:

$$ (117) \quad \frac{\partial s}{\partial t} - D_M \frac{\partial^2 s}{\partial y^2} + V_M \frac{s}{K_M + s} = 0 $$

$$ (118) \quad \frac{\partial p}{\partial t} - D_M \frac{\partial^2 p}{\partial y^2} - V_M \frac{s}{K_M + s} = 0 $$

- Liquid phase:

$$ (119) \quad \frac{\partial s}{\partial t} - D_L \frac{\partial^2 s}{\partial y^2} + C(y) \frac{\partial s}{\partial x} = 0 $$

$$ (120) \quad \frac{\partial p}{\partial t} - D_L \frac{\partial^2 p}{\partial y^2} + C(y) \frac{\partial p}{\partial x} = 0 $$

$$ (121) \text{ with } C(y) = y(y-3e) \frac{4}{9e^2} C_{\text{max}} $$

(C is equal to zero at the solid-liquid interface).

- Interface conditions:

For $y = 0$ and $y = 3e$

$$ (122) \quad D_M \frac{\partial s}{\partial y_M} = D_L \frac{\partial s}{\partial y_L} $$

- Others conditions:

$$ (123) \quad \frac{\partial s}{\partial y} \text{ and } \frac{\partial p}{\partial y} \text{ are equal to zero for } y = -\frac{e}{2} \text{ and } y = 3e + \frac{e}{2} \\ 0 < x < \ell, \text{ and } s = s_1 \text{ and } p = 0 \text{ for } x = 0 \text{ and } 0 < y < 3e $$
### - Initial conditions:

For $t = 0$ in every point $s = 0$ and $p = 0$

## Numerical solutions

### - Parameter values:

$e = 4 \ 10^{-3} \text{ cm}$ ; $e = 11,5 \text{ cm}$

$C_{\text{max}} = 600 \text{ cm h}^{-1}$

$D_M = 10^{-4} \text{ cm}^2 \text{ h}^{-1}$ ; $D_L = 5 \ 10^{-4} \text{ cm}^2 \text{ h}^{-1}$

$V_M = 2 \cdot 10^{-4} \text{ moles cm}^{-3} \text{ h}^{-1}$

$K_M = 10^{-6} \text{ moles cm}^{-3}$

The problem is solved by numerical methods (HENRY 1973).

Concentration profiles are calculated inside the liquid and solid phases for stationary states or during evolutive processes. It is possible, in particular, to get the leaving concentration $S_2$ as a function of the entering one $S_1$ if the other parameters are well-known.

The relationship between $S_1\alpha$ and $\text{Log}(1-\alpha)$, from the results of the numerical simulation, is plotted on figure 44.

There is a good agreement between the experimental results and the numerical simulation for the middle values of the linear flow velocity (C). For very slow or very quick processes the relative shift between simulation and experimental results can attain 20 %.

It is very important to note that the simulation results give a linear relationship between $S_1\alpha$ and $\text{Log}(1-\alpha)$ (Figure 44). This fact is in agreement with the experimental results previously described by the different authors.

The last model not only gives a more precise analysis of the packed bed reactors but also explains the apparent variations of the enzyme kinetic constants.
<figure><img src="image_51.png" /><figcaption>**Figure 44** : S₁α as a function of Log (1-α) from the simulation on computer.</figcaption></figure>
V - CONTROL AND IDENTIFICATION PROBLEMS

When a model or a biochemical reactor is realized or designed, it is
important to know how it is possible to control its behavior.

a) - Substrate flux control by a boundary
inhibitor concentration

This system was described in II - 3º -a.

The boundary concentration of inhibitor is written v(t).

$$
\begin{align*}
& \text{(124)} \quad \left\{
\begin{array}{@{}l@{\quad}l@{}}
\displaystyle \frac{\partial y}{\partial t} - \frac{\partial^2 y}{\partial x^2} + F(y,i) = 0 & \text{where } F(y,i) = \frac{y}{1+y+\lambda i} \\[1em]
\text{y}(0,t) = \alpha, \quad \text{y}(1,t) = \beta, \quad \text{y}(x,0) = 0 \\[1em]
\displaystyle \frac{\partial i}{\partial t} - \frac{\partial^2 i}{\partial x^2} = 0, & \text{i}(0,t) = i(1,t) = v(t), \quad i(x,0) = 0
\end{array}
\right.
\end{align*}
$$

The observation is $t \rightarrow \frac{\partial y}{\partial v}(0,t)$, entering flux.

$t^\gamma \frac{\partial y}{\partial v}$ is defined and belongs to $L^2(0,T)$ for $\gamma > \frac{1}{2}$

It is possible to take for cost function:

$$
\begin{align*}
(125) \quad J(v) &= \int_{t_1}^{t_2} t^{2\gamma} (Cy(t) - z_d(t))^2 dt \\
&\quad (0 \le t_1 \le t_2 \le T).
\end{align*}
$$

The convex of admissible controls is

$$
(126) \quad \mathcal{U}_{\text{ad}} = \{ v \mid v \in L^{\infty}(0,T) \quad , \quad 0 \le v \le M \}
$$
and we are looking for some $u \in \mathcal{U}_{ad}$ such that

$$ (127) \qquad J(u) = \inf_{v \in \mathcal{U}_{ad}} J(v) $$

**Note:**

Experimental aspect of this control:

- to increase $v$, some inhibitor is added in the compartments

- to decrease $v$, some inhibitor is consumed, for example by a chemical reaction.

**Note:**

A variant of this problem is:

The equations are the same, but with

$$ i(0,t) = v_o(t) \quad , \quad i(1,t) = v_1(t) \quad , \quad (v_o, v_1 \in \mathcal{U}_{ad}) $$

and we wish to minimize

$$ (128) \qquad J(v) = J(v_o, v_1) = \int_{t_1}^{t_2} t^{2\gamma} \left( \frac{\partial y}{\partial v} (0,t) - (z_d)_o (t) \right)^2 dt \\ \qquad \qquad + \int_{t_1}^{t_2} t^{2\gamma} \left( \frac{\partial y}{\partial v} (1,t) - (z_d)_1 (t) \right)^2 dt. $$

**Note:**

A similar problem is the substrate concentration control in the middle of a membrane by a boundary inhibitor concentration.
The observation is now

$$t \rightarrow y(\frac{1}{2}, t) \quad (\text{electrode})$$

and we wish to minimize

$$(129) \quad J(v) = \int_{t_1}^{t_2} (Cy(t) - z_d(t))^2 dt \quad \text{where} \quad Cy(t) = y(\frac{1}{2}, t).$$

## b) - Active transport control

This bienzyme system has been described in III 1º.

It would be interesting to control the active transport velocity against the concentration gradient.

It is possible to perform this control because the presence of an activator ($M_g^{++}$) is a necessity for the enzyme reaction in the first layer.

Equations of this system:

$(y_1, y_2, y_3$ are substrat, product and activator concentrations).

$$(130) \quad \frac{\partial y_1}{\partial t} - \frac{\partial^2 y_1}{\partial x^2} + F(y_1, y_2, y_3) = 0 \quad x \in ]0,1[$$

$$(131) \quad \frac{\partial y_2}{\partial t} - \frac{\partial^2 y_2}{\partial x^2} - F(y_1, y_2, y_3) = 0 \quad t \in ]0,T[$$

$$(132) \quad \frac{\partial y_3}{\partial t} - \frac{\partial^2 y_3}{\partial x^2} = 0 \quad \text{where}$$

$$(133) \quad F(y_1, y_2, y_3) = \sigma \frac{y_3}{1+y_3} \frac{y_1}{1+y_1+y_2} \\ \text{if } 0 < x < \frac{1}{2} \quad (\sigma > 0)$$

$$(134) \quad F(y_1, y_2, y_3) = -\sigma \frac{y_2}{1+y_2} \quad \text{if } \frac{1}{2} < x < 1$$
<figure><img src="image_52.png" /><figcaption>**Figure 45** : Active transport control, to achieve a flux of product p varying linearly with time. At the beginning, condition 137 constrains the flux to be nil. Boundary flux as a function of time.</figcaption></figure>
Boundary conditions:

$$
(135) \quad y_1(0,t) = \alpha \qquad \alpha > 0
$$

$$
(136) \quad \begin{cases} \frac{\partial y_1}{\partial t} + x \frac{\partial y_1}{\partial v} = 0 & \text{for } x=1 \\ y_1(1,0) = \beta & \beta \ge 0 \end{cases}
$$

$$
(137) \quad \frac{\partial y_2}{\partial v} = \begin{cases} 0 & \text{if } y_2 \le h \\ -ay_2 & \text{if } y_2 \ge h \end{cases}
$$

$$
(138) \quad y_3(0,t) = 0, \quad y_3(1,t) = v(t) = \underline{\text{control}}.
$$

Initial conditions:

$$
y_1(x,0) = 0, \quad y_2(x,0) = 0, \quad y_3(x,0) = 0.
$$

The difference between the boundary conditions for $y_1$ in $x=0$ and $x=1$ is linked to the compartment relative sizes. The first compartment is large enough to maintain a constant concentration, the second one is smaller by far.

The control is the activator concentration in the second compartment.

This control is in the set of admissible controls.

$$
(139) \quad \mathcal{U}_{\text{ad}} = \{ v \mid v \in L^2(0,T) , \quad 0 < v(t) < M \}
$$

The cost function $J$ is defined by

$$
(140) \quad J(v) = \int_{0}^{T} (y_{1}(1,t) - z_{d}(t))^{2} \, dt.
$$

i.i.) Numerical results:

See figure (45).
### c) - Stochastic control (KERNEVEZ, QUADRAT and VIOT)

#### i.) Problem presentation:

We present now an optimal feedback control of a system described by the equations of the irreversible monoenzyme system with competitive inhibition (II 3º- a) and with a random environment.

We assume that this system advances by successive quasi-steady states.

In this case the equations are

$$ (141) \qquad \begin{cases} y''(x) = \frac{\sigma y(x)}{1+i(x)+|y(x)|} ; & y(0) = y(1) = 1 \\ i(x) = wx + (1-x)\theta \end{cases} $$

In these equations 2 parameters are random, $\sigma$ and $w$.

<figure><img src="image_53.png" /></figure>
σ depends on random fluctuations of temperature, pH, and cofactors concentration.

w is the inhibitor concentration in the 2nd compartment and evolves randomly.

To control the system we have at our disposal θ, inhibitor concentration in the 1st compartment.

Moreover, the control, to be efficient, should occur in a closed feedback loop from an observation of the system.

In the present case, the observation is the flux of substrate entering into the membrane at $x = 0$, i.e. $-y'(0)$.

$$ \theta = u(y'(0)) $$

where u is some function from $\mathbb{R}$ into $\mathbb{R}_+$.

The cost function to minimize is the average value of deviation between $y'(0)$ and a fixed value $z_d$.

$$ \min_{u \in \mathcal{U}_{ad}} E|y'(0) - z_d|^2 $$

where $\mathcal{U}_{ad}$ is some fixed subspace of functions with values in $\mathbb{R}_+$.

### i.i.) Numerical results:

In pictures (46), (47) and (48) we shall see respectively:

- In the 1st, σ is random on the interval (30,40) with a uniform distribution. w is fixed at 6 and we looked for an optimal open loop to approach $z_d = -2$. All these curves are between the 2 extreme ones corresponding to $σ = 30$ and $σ = 40$.
<figure><img src="image_54.png" /><figcaption>Figure 46 :</figcaption></figure>
<figure><img src="image_55.png" /><figcaption>Figure 47</figcaption></figure>
<figure><img src="image_56.png" /><figcaption>Figure 48</figcaption></figure>
- In the 2^nd^ one, σ is again random and w is fixed, like in the 1^st^ picture, but now we are looking for a linear feedback control i.e. of the form

$$u(y'(0)) = \alpha(y'(0) - z_d) + \beta$$

which is optimal and the values of y'(0) for all the curves between σ = 30 and σ = 40 are very near from z_d = -2.

- In the 3^rd^ picture, σ is fixed at 36 and w is a variable which is equally distributed between 3.75 and 8.25.

This time again an optimal linear feedback control gives a good minimization of the deviation between y'(0) for every curve and z_d = -2.

### d) - Identification problem:

In (II 1^o^ b - i.i.) we gave the "model case" equations in the non-michaelian hypothesis.

In addition to K_M and σ, which can be easily measured by classical methods, appear parameters whose measure is difficult or impossible: k_1, k_2, k_{-1}, E_0.

Between these parameters we have the relations:

$$ 
\begin{cases} 
V_M = k_2 E_0 \\ 
K_M = \frac{k_2+k_{-1}}{k_1} 
\end{cases} 
$$

V_M and K_M been known.
The individual effect of each kinetic parameter $k_1, k_2, k_{-1}$ appears
in the following experimental conditions:

- When the membrane begins to fill up

- When the filling up is performed, the boundary substrate concentration is fiercely put to 0 and the membrane becomes empty.

The "electrode system" is used and we measure, in these 2 phases,
the evolution $t \rightarrow s(1,t) = z_d(t)$ of the substrate concentration
against the electrode.

If we call $s(1,t;k_1,k_2,k_{-1},E_0)$ this concentration when the parameters have the values $k_1,k_2,k_{-1},E_0$, we are faced to the following optimization problem:

minimize $J(k_1,k_2,k_{-1},E_0)$ under the constraints ( ) where

$$J(k_1,k_2,k_{-1},E_0) = \int_0^T |s(1,t;k_1,k_{-1},k_2,E_0) - z_d(t)|^2 dt.$$
CONCLUSION

This report is the result of a collaboration between Biochemists and Mathematicians. The experimental systems above are performed in the "E.R.A. N° 338 du C.N.R.S. - Université de Technologie de Compiègne". The discussion about the interest of these systems at both fundamental and applied points of view can be found in THOMAS and CAPLAN (1973).

In this report, numerical results are given within the experimental context and obviously they are very useful for a better knowledge of the system behaviours.

At the mathematical and numerical point of view it is possible to get more details from : KERNEVEZ (1972, 1973) for a general survey of the mathematical problems, KERNEVEZ et al (1973) for a stochastic feed back control, BRAUNER and PENEL (1972) for systems without michaelian assumption, DUBUS (1972) for a multienszyme system and YVON (1973) for optimal control.
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Thesis-Paris.

- BROUN, G., THOMAS, D. and SELEGNY, F. (1972)
J. of Membrane Biol, **8**, 313.

- DAVID, A., THOMAS, D., BROUN, G. and METAYER, M. (1972)
Symposium on chemically grafted and cross-linked protein. Rouen.
Communication n° 21.

- DUBUS, D. (1972)
Thesis. Paris.

- GELLF, G. (1971)
Thesis. Rouen.
- GELLF, G., HENRY, J., THOMAS, D. and BROUN, G. (1973)
  F.E.B.S. Meeting, Dublin, 31, 86.

- GELLF, G., HENRY, J., KERNEVEZ, J.P. and THOMAS, D.
  C.R. Acad. Sc. D. (in press).

- GELLF, G., KERNEVEZ, J.P., THOMAS, D. and BROUN, G.
  Biotechnol. Bioeng. (to be published).

- GOLDMAN, R., KEDEM, O., SILMAN, M.I., CAPLAN, S.R. and KATCHALSKI, E. (1968)<sup>a</sup>
  Biochemistry, 7, 486.

-GOLDMAN, R., KEDEM, O. and KATCHALSKI, E. (1968)<sup>b</sup>
  Biochemistry, 7, 4518.

- GOLDMAN, R., KEDEM, O. and KATCHALSKI, E. (1971)
  Biochemistry, 10, 165.

- GOLDMAN, R., and KATCHALSKI, E. (1971).
  J. Theor. Biol. 32, 243.

- GRUBMOFFER, N. and SCHLEITH, L. (1954)
  Z. Physiol. Chem., 297, 108.

- GUILBAUD, G.G. (1972)
  Biotechnol. Bioeng. Symp., 3, 361.

- HICKS, G.P. and UPDIKE, S.J. (1966)
  Anal. Chem., 38, 726.

- KASCHE, M., LUNDOUIST, M., BERGMANN, R. and AXEN, R. (1971)
  Biochem. Biophys. Res. Commun., 41, 615.

- KERNEVEZ, J.P., (1972)
  Dr. Sc. Thesis. Paris n° A.O. 7246.

- KERNEVEZ, J.P., QUADRAT, J.P. and VIOT, M. (1973)
  Control of a non linear stochastic boundary value problem,
  IFIP 5<sup>th</sup> conference on optimization Techniques, Rome 1973,
  (Springer Verlag).
- KERNEVEZ, J.P. (1973)
  Analysis of some biochemical systems, "applied Mathematics and Optimization" to be published.

- KOBAYASHI, T. and LAIDLER, K.J. (1973)
  Biochem. Biophys. Acta, **302**, 1.

- LECOCQ, D., THOMAS, D., BROUN, G., and KERNEVEZ, J.P.
  Eur. J. Biochem. (to be published).

- LILLY, M.D., HORNBY, W.E. and CROOK, E.M. (1966).
  Biochem. J., 100, 718.

- MATTIASSON, B. and MOSBACH, K. (1971).
  Biochem. Biophys. Acta, **235**, 253.

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  B.Z., **49**, 333.

- MITZ, M.A. (1956)
  Science, **123**, 1076.

- MONSAND, P. (1971).
  Thesis. Toulouse.

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  Biokhimia, **26**, 565.

- NILOLAIEV, A. (1962)
  Biokhimia, **27**, 843.

- NIMS, L.F. (1968)
  Current in modern Biology, **1**, 353.

- OGATA, K., OTTESEN, M. and SVENDSEN I. (1968)
  Biochim. Biophys. Acta, **159**, 403.
- O'NEILL, S., DUNNILL, P. and LILLY, M.D. (1971)
  Biotechnol. Bioeng., 13, 337.

- O'NEILL, S., (1971)
  Biotechnol. Bioeng., 13, 319.

- PASYNSKI, A.G., MOISSEYEVA, L.N. and ZVYAGIL'SKAYA, R.A. (1964)
  Biokhimiya, 29, 2.

- REESE, E.T. and MANDELS, M. (1958)
  J. Am. Chem. Soc., 80, 4625.

- SELEGNY, E., AVRAMEAS, S., BROUN, G., and THOMAS, D. (1968)
  C.R. Acad. Sc. C, 266, 1431.

- SMILEY, K.L. (1971)
  Biotechnol. Bioeng. 13, 309.

- SUNDARAM, P.V and LAIDLER, K.I. (1971) in
  "Chemistry of the cell Interface" (Brown, M.D., ed.) pp 255-
  296. Academic Press, New York.

- THOMAS, D. (1971) Dr. Sc. Thesis. N° A.O. 5407.

- THOMAS, D., BROUN, G. and SELEGNY, E. (1972)
  Biochimie, 54, 229.

- THOMAS, D., TRAN, M.C., GELLF, G., DOMURADO, D., PAILLOT, B., JACOBSEN, R. and BROUN, G. (1972)
  Biotechnol. Bioeng. Symp. 3, 299.

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  by I. PRIGOGINE.
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In "Membrane separation processes" edited by P. MEARES.
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- THOMAS, D., BOURDILLON, C., BROUN, G. and KERNEVEZ, J.P.

Biochemistry, (to be published).

- WILSON, R.J., KAY, G. and LILLY, M.D. (1968<sub>a</sub>)

Biochem. J., 108, 845.

- WILSON, R.J., KAY, G. and LILLY, M.D. (1968<sub>b</sub>)

Biochem. J., 109, 137.

- YVON, J.P. (1973)

Optimal Control of systems governed by variational inequalities
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(Springer Verlag).