Chapter 4 New kinetic model structure

125

Chapter 4

New kinetic model structure

As mentioned in the previous chapters, first principles modelling implies the a priori selection of a kinetic model structure. This choice is made particularly difficult by the lack of detailed knowledge about the bioprocess under consideration and the profusion of kinetic laws, which can be of various types, including physical (Bastin and Dochain, 1990) and black-box models (Montague and Morris, 1994; Chen et al., 2000; Oliveira 2004).

The most widespread black-box models are the artificial neural networks (Psichogios and Ungar, 1992; Montague and Morris, 1994; Feyo de Azevedo et al., 1997; Chen et al., 2000; Oliveira; 2004; Vande Wouwer et al., 2004).

However, we have shown in Chapter 3 that neural networks should be avoided to model unknown kinetics. Indeed, they present a lack of physical meaning and a lot of identification inconveniences (see section 3.3.2). Moreover, the results obtained with such model structures are often of lower quality.

Regarding physical laws, Monod-type models (Bastin and Dochain, 1990) are the most widespread. However, it is often difficult to choose among the numerous laws which are able to characterize the same behaviour. Moreover, the use of such kinetic models can lead to time-consuming optimization. These structures are nonlinear and, in most cases, non linearizable w.r.t. their parameters. Hence, there is no systematic method to determine a unique first

estimation of the different parameters, which would make the identification easier.

In order to keep physical interpretation of the parameters while avoiding the classical problems associated with Monod-type laws, Bogaerts et al. (1999) have proposed a new general and systematic kinetic structure. This one allows to represent the activation and/or the inhibition effects of each macroscopic species in the culture. Moreover, this structure is linearizable with respect to its parameters and it is therefore possible to determine unique initial estimates of the different physical parameters.

However, it still suffers from a lack of generality. This structure does not allow to represent saturation by a macroscopic species. The structure is forced to model this behaviour by an inhibition which compensates a stronger activation.

This, of course, alters the physical meaning.

Hence, in this chapter, we propose to generalize the original kinetic model structure (section 4.1) in order to improve the flexibility and the physical interpretation while keeping a systematic identification method (section 4.2).

Section 4.3 reminds how to compute the covariance matrix and to reduce the number of parameters. The performances of the improved model are tested on 4 case studies (section 4.4): two simulated case studies and two real bioprocesses, the industrial bioprocess described in section 3.2.2 and a yeast culture performed in the Department of General Chemistry and Biosystems of the Université Libre de Bruxelles.

Note that concerning the optimization algorithms to be used for parameter identification with this new kinetic model structure, the reader may still refer to section 3.1.4.

4.1 Generalized kinetic model structure

Let us remind the formalism of the kinetic model structure (3.46) of Bogaerts (1999)

[[[[

^M

]]]]

j e

t t

I l

t R

h h j j

l lj

j

hj( ) 1,

ξ )

,

( ^ξ()

*

∈

∈

∈

∈

=

= =

= ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏

∈

∈

∈

∈

−−−

−

∈

∈

∈

∈

β

α

γ

ϕ

ξ ^(4.1)

The objective of this chapter is to allow a possible saturation by the kinetic model structure. On this purpose, we suggest to replace the activation function

γhj

ξh in (4.1) by

((((

¹

^{− − − −}

^{e−−−−κhjξh(t)}

))))

^{γhj, where} κhj

^{≥ ≥ ≥ ≥}

⁰ is the saturation coefficient of the component h in reaction j.

The generalized kinetic model structure becomes therefore:

((((

^e

))))

^{e j}

^[[[[

^M

^]]]]

α t

I l

β t R

h

t γ κ j

j

l lj

j

h hj

hj 1,

1 )

,

( ^{* ξ () ξ()}

*

∈

∈ ∈

∈

−

−

−

−

=

= =

= ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏

∈∈

∈∈

−−

−−

∈∈

∈∈

−−

−−

ϕ

ξ ^(4.2)

Note that the original activation function ξ^γ_h^hj is actually a particular case of the generalized one. Indeed, when the saturation is weak (κ_hj

→ → → →

0 and thus

0 ξ_h

→ → → →

κhj ), the Taylor series expansion of

((((

¹

^{− − − −}

^{e−−−−κhjξh(t)}

))))

^{around 0} and limited to the first order, is given by

((((

^{1 e ξ ()}

))))

^κhj^ξh^(t)

κhj h t

≈ ≈

≈ ≈

− −

− −

⁻⁻⁻⁻ (4.3)

This observation is illustrated in figure 4.1.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0

0.5 1 1.5

X

k x (1-exp(-k x))

Figure 4.1: activation functions

----: generalized activation function ^___: linear approximation of this latter

Hence, when κ_hjξ_h

<< << << <<

, the generalized kinetic model can be expressed as follows:

((((

κ ^t

))))

^{e j}

[[[[

^M

]]]]

α t

I l

β t R

h

γ h hj j

j

l lj

j

hj 1,

) ( ξ )

,

( ^{* ξ()}

*

∈

∈

∈

∈

≈

≈

≈

≈ ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏

∈

∈

∈

∈

−

−

−

−

∈∈

∈∈

ϕ

ξ ^(4.4)

or

(((( ))))

κ ^{t e j}

[[[[

^M

]]]]

α t

I l

β t R

h γ h R

h

γ hj j

j

l lj

j hj

j

hj ξ () 1,

) ,

( ^{* ξ()}

*

*

∈

∈

∈



∈













≈

≈

≈

≈ ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏

∈

∈

∈

∈

−

−−

−

∈∈

∈∈

∈∈∈

∈

ϕ

ξ ^(4.5)

which corresponds to the expression (4.1) where

(((( ))))

∏

∏ ∏

∏

∈∈∈∈

≈

≈ ≈

≈

*

*

j hj

R h

γ hj j

j α κ

α ^(4.6)

<<

<< <<

<<

h

κhjξ

This property is used in the following section in order to initialize the identification procedure, i.e. to systematically determine a first estimation of the parameters.

4.2 Systematic identification procedure

Besides the ability of model structure (4.2) to describe various effects (activation, inhibition, saturation), it is desirable to establish a systematic parameter identification procedure. The idea is to build it upon the procedure described in 3.1.2.1 and originally developed for model (4.1).

If we first assume weak saturation effects (κ_hj

<< << << <<

), the generalized kinetic model structure (4.2) can be approached by the original one (4.1). Hence, the procedure of section 3.1.2.1 can be used to determine a first estimation of the original parameters α_j^,γ_hj^,β_lj. Based on these parameter values and a selection of small values for κ_hj, the parameters α^*_j can be computed according to (4.6).

Starting from these initial parameter estimates, nonlinear identification of the parameters α^*_j^,γ_hj^,β_lj ^and κ_hjcan then be achieved thanks to the Markov estimator (3.60) where

[[[[

^αj ^γh j ^βl j ^κh j

]]]]

^j

[[[[

^M

]]]]

T

= = = =

^* K K K

∈ ∈ ∈ ∈

1,

θ ^(4.7)

and under the constraints θˆ

≥

0.

Finally, in a last identification step, the whole set of parameters is re- estimated with the initial component concentrations. Once more, the Markov estimator (3.60) is used with

[[[[

^αj ^γh j ^βl j ^κh j ^T

]]]]

^j

[[[[

^M

]]]]

T

= = = =

^* ξ (0)

∈ ∈ ∈ ∈

1,

θ K K K (4.8)

under the constraints θˆ

≥

0.

But how to select initial small values of κ_hj? As mentioned previously, κ_hj values have to be sufficiently small in order to allow the approximation of the activation function

((((

¹

^{− − − −}

^{e−−−−κhjξh(t)}

))))

by its Taylor series expansion limited to the first order around 0 (4.3). Hence, we make an initial guess of these parameters by considering that we accept

(

100

−

^x

)

% of disparity between the nonlinear activation function and its linear version at the highest concentration level of the considered component. So, we compute κ_hj values on the basis of the following equation

((((

¹

^{− − − −}

^{e−−−−κhjξmaxh}

)))) ^{= = = =}

₁₀₀^{x κhjξmaxh (4.9)}

where ξ^max_h is the highest concentration level for the component ξ_h and x represents the confidence in the absence of saturation effect.

By approaching

((((

¹

^{− − − −}

^{e−−−−κhjξmaxh}

))))

by its Taylor series expansion limited to the second order around 0, we obtain

((((

^max

))))

² _max

max ξ

100 2

ξ_h ^hjξ^h _{hj h}

hj κ x κ

κ

− − − − ≈ ≈ ≈ ≈

^(4.10)

which leads to the following expression of the initial values of κ_hj

ξmax

1 100 2

h hj

x

κ



 





− − − −

=

= =

=

^(4.11)

At the end of the above mentioned nonlinear identification steps, all the parameters have been estimated as accurately as possible but the model has to be

validated through direct and cross validation tests and the study of the covariance matrix of the parametric errors. This latter study usually leads to parameter reduction. This is the subject of the following subsection.

4.3 Covariance matrix and parameter reduction

The Markov estimator, used to identify nonlinearly the different kinetic parameters, is identical to the estimator (3.60) used for the original kinetic model structure. Hence, the covariance matrix can be computed thanks to the equations (3.61-3.64).

The study of the covariance matrix can help reducing the number of parameters. Indeed, hardly assessable coefficients can be suppressed after checking that their information is covered by other parameters. In concrete terms, coefficients with high variance and a sufficient correlation with other parameters can be cancelled out. This cancellation reduces the number of parameters, and in turn the effect of a component (e.g. activation, saturation or inhibition).

We suggest using the parameter reduction procedure explained in section 3.1.2.3. However, it has to be refined in order to take some model particularities into account. Indeed, when we consider the structure of the activation function

((((

¹

^{− − − −}

^{e−−−−κhjξh(t)}

))))

^γhj, we observe that the cancellation of a saturation coefficient implies the absence of activation. Actually, if a component activates a reaction with no saturation effect, the saturation coefficient should be cancelled out while keeping an activation effect. As the easiest way to achieve this is to come back to the original model structure, we propose to replace the activation/saturation function

((((

¹

^{− − − −}

^{e−−−−κhjξh(t)}

))))

^γhj by the original activation function ξ^γ_h^hj in this case.

On the other hand, inhibition and saturation coefficients of a given component cannot be eliminated at the same reduction step. Actually, as they model similar macroscopic behaviours, they are closely correlated and the elimination of both parameters could lead to a loss of important information.

Hence, only the most inaccurate parameter can be eliminated at each step of the proposed reduction procedure. If the second parameter is still inaccurate at the next step, it will be cancelled out at this moment.

4.4 Case studies

In order to test the capabilities of the new kinetic model structure to reproduce simulated behaviours as well as real bioprocesses, four different case studies are considered: two simulated case studies and two real bioprocesses.

The first simulation example consists of a simple microbial growth whose kinetics is strongly limited by the biomass and a unique substrate. The second simulation example is inspired from an animal cell culture described in Perrier et al. (2000). These case studies are aimed at demonstrating the flexibility and capability of the kinetic structure to reproduce various effects, and particularly saturation effects.

The experimental case studies consider modelling of the industrial bioprocess described in section 3.2.2 and a yeast culture performed in the Department of General Chemistry and Biosystems of the Université Libre de Bruxelles.

4.4.1 Simulated case studies

Let us describe our two simulated case studies and the corresponding databases.

4.4.1.1 Simple microbial growth

Our first database consists of simple microbial cultures in which only one substrate and the biomass are considered. At the macroscopic level, the biomass growth on this unique substrate may be described by the following reaction scheme:

X S k

ϕ

→

^(4.12)

where S denotes the substrate concentration, X the biomass concentration, k the pseudo-stoichiometric coefficient and

ϕ

the reaction rate.

The mass balances corresponding to this reaction scheme are given by

Sin

D t S D t X t S dt k

t

dS( )

= = = = − − − −

( ( ), ( ))

− − − −

( )

+ + + +

ϕ

^(4.13)

) ( ) ( )) ( ), ( ) (

( S t X t D t X t

dt t

dX

= = = = ϕ − − − −

^(4.14)

where D is the dilution rate, Sⁱⁿ the substrate concentration in the feeding medium and

ϕ

the reaction rate which follows a theoretical Monod-type kinetic law that implies saturation by the components:

) ( ) ( ) ( ) )) (

( ), (

( _max

t X K

t X t S K

t t S

X t S

X m

+ + + + + + + +

= =

= = µ

ϕ

^(4.15)

Finally, the model {(4.12), (4.13)} can be rewritten under the following matricial form

) ( ) ( ) ( ) ( ) , ) (

( t D t t t t

dt t

dξ K ξ ξ F Q

− +

−

= ϕ

^(4.16)

where K is the yield coefficient matrix









−

=

0

K k ^(4.17)

and 



 



=



=

−

( ) ( ) ( ) 0

) (

in

in S

t D t

D t

t Q ξ

F ^(4.18)

The numerical values of the model parameters are the following: k

= = = =

0.5gl⁻⁻⁻⁻¹, 12 ⁻1

=

gl

K_m ,

µ

_max

=

1.4h^{−1 and} K_X

= = = =

10¹¹celll⁻⁻⁻⁻¹.

Simulation of this model in various conditions allows the creation of a database. It consists of 4 batches (_Sⁱⁿ _=0gl⁻¹) of 50 hours which present the same initial concentration in biomass ( ^{11 1}

0====1.410 celll⁻⁻⁻⁻

X ) and different initial concentrations in substrate ( _{[18 24 12 30](} ¹₎

0

= gl−

S ). These latter

concentrations are chosen in order to ensure significant substrate saturation at the beginning of the experiments and strong biomass saturation at the end. The sampling time for the simulated substrate and biomass concentration measurements is 2 hours. Among the different experiments, the first two are used for the identification while the others are kept for cross-validation, to test the generalization of the models.

4.4.1.2 Simulated animal cell culture

The second simulated database involves a more complex model which consists of the animal cell culture exposed in Perrier et al. (2000), a human embryo kidney cell culture. At the macroscopic level, the cells consume glucose in two macroscopic reactions: a growth reaction and a second reaction with lactate production. The corresponding macroscopic reaction scheme is given by

L X G

X G

32 22

21

2 1

ν ν

ν

ϕ ϕ

+ + + +

→ →

→ →

→ →

→ →

(4.19)

where G denotes the glucose concentration, X the biomass concentration and L the lactate concentration.

ν

_ij are the pseudo-stoichiometric coefficients and

ϕ

i the reaction rates.

The mass balances corresponding to this reaction scheme are given by

X dt D

dX

= = = = ϕ

₁

+ + + + ϕ

₂

− − − −

^(4.20)

DGin

dt DG

dG

= = = = − − − − ν

₂₁

ϕ

₁

− − − − ν

₂₂

ϕ

₂

− − − − + + + +

^(4.21)

dt DL

dL

= ν

₃₂

ϕ

₂

−

^(4.22)

where D is the dilution rate, Gⁱⁿ the substrate concentration in the feed medium and

ϕ

_i the reaction rates which follow Monod-type laws:

L K

K G K X G

L L R

m

+ + + + + + + +

=

= =

=

₁

1

µ

ϕ

^(4.23)

G K X G

F

m

+ + + +

= =

= =

₂

2

µ

ϕ

^(4.24)

Finally, the model {(4.20)-(4.22)} can be rewritten under the following matricial form

) ( ) ( ) ( ) ( ) , ) (

( t D t t t t

dt t

dξ K ξ ξ F Q

− +

−

= ϕ

^(4.25)

where K is the yield coefficient matrix

















−

−

=

32 22 21

0

1 1

ν ν ν

K ^(4.26)

and 



 



=



= =

=

= =

= =

− −

− −

) 0 ( )

( ) ( ) (

in

in G

t D t

D t

t Q ξ

F ^(4.27)

The numerical values of the model parameters are the following:

X

G mmol

mmol / 7

.

21

= = = =

1

ν

^,

ν

₂₂

= = = =

8.5mmol_G/mmol_X^,

ν

₃₂

= = = =

17mmol_L /mmol_X^,

10 ⁻⁻⁻⁻1

=

=

=

=

mmoll

K_R ^, K_L

= = = =

50mmoll^{−−−−1,} K_F

= = = =

10mmoll^{−−−−1,}

µ

_m1

= = = =

0.055h^{−−−−1and}

1 2

= = = =

0.045h⁻⁻⁻⁻

µ

m ^.

The simulated database consists of 9 fed-batch cultures (Gⁱⁿ ====50mmoll⁻⁻⁻⁻¹) of 110 hours which present the same initial concentrations in glucose and lactate (_{G 21mmol/l,L 0.13mmol/l}

0

0 ==== ==== ) but three different initial concentrations in biomass (_X0====[0.18 0.4 0.8]_mmol/_l) and different profiles of the external feed rate Fⁱⁿ(t) (Fⁱⁿ(t)====

[[[[

0.5t 0.1t 0.01t

]]]]

(l/h)). The sampling time for the simulated measurements is 10 h. Among the different experiments, only three (_(X0(1),Fⁱⁿ₍₂₎₎, _(X0(2),Fⁱⁿ₍₂₎₎, _(X0(3),Fⁱⁿ₍₂₎₎) are kept for the identification while the other ones will be used for the cross-validation, to test the generalization of the models.

4.4.2 Pilot yeast cultures

Our second real bioprocess is a yeast culture operated in the department of General Chemistry and Biosystems of the Université Libre de Bruxelles. The used microorganism is a Saccharomyces cerevisiae (ATCC 20820). Seven

experiments have been carried out in our bioreactor (Biostat C_DCU3 –B-Braun Biotech International) in order to build our database. They consist in 3 batches and 4 fed-batches. The experimental protocol is proposed in section 4.4.2.2 while the experiments are described in 4.4.2.3. Section 4.4.2.1 reminds the main phenomena occurring in a yeast culture.

4.4.2.1 The main phenomena in a yeast culture

The yeast growth has already been the subject of numerous studies and papers (Sonnleitner and Käppeli, 1986; Dantigny, 1995; Rose and Harrison, 1995; Di Serio et al., 2001; Jones and Kompala, 1999; Karakuzu et al., 2006).

All of them agree concerning the macroscopic behaviour of baker’s yeast, there are three different metabolic pathways

1. the glucose oxidation: G K₄₁O₂ K₁₁ X

ϕ1

→

+

^(4.28)

2. the fermentation: G K₁₂ X K₃₂E

2

+

→

ϕ

(4.29) 3. the ethanol oxidation: E K₄₃O₂ K₁₃ X

ϕ3

→

+

^(4.30)

where X, G, E and O₂ are respectively the biomass, glucose, ethanol and dissolved oxygen. The parameters Kij are the pseudo-stoichiometric coefficients while the vector φ contains the reaction rates.

If the macroscopic reaction scheme (4.28-4.30) is well known, the kinetics are more complex. Indeed, the different metabolic routes are not used by the yeast simultaneously. According to the culture conditions (i.e. the absence of oxygen and the concentrations in glucose or ethanol in the broth), the different pathways are employed or not, repressed or inhibited. Actually, the respiratory capacity of the cells is considered to govern glucose and ethanol metabolism. If no oxygen is available, the glucose is necessarily consumed by the fermentation

pathway, no oxidation route can be used. In addition, the glucose oxidation is repressed at high glucose concentrations, the complete glucose flux cannot be oxidized, the respiratory capacity is saturated and the glucose rest is consumed by the fermentation pathway. The respiratory capacity represents a bottleneck (Sonnleitner and Käppeli, 1986) for oxidative substrate utilisation. Whether the respiration is not saturated by the glucose flux, ethanol and glucose can be oxidized. However, when the glucose quantity increases, the cells prefer to consume a maximum of glucose. These concepts are illustrated in figure 4.2.

Further this complex behaviour related to the respiratory capacity, the yeast growth can be inhibited by ethanol. This component is toxic for the yeast above 10 g/l.

In order to highlight the different phenomena explained in this section, we have performed 7 experiments, 3 batches and 4 fed-batches. Before describing these cultures, let us present our experimental protocol.

Figure 4.2: Limited respiratory capacity, bottleneck (1) respiration not saturated, Glucose (G) and Ethanol (E) oxidized

(2) critical respiration: G oxidized, E not consumed

(3) saturated respiration: maximum of G oxidized and the rest fermented.

Maximum respiratory capacity

G E G

(1) (2)

G

E (3)

4.4.2.2 Experimental protocol and measurement analysis

The experimental protocol used in our laboratory presents 3 steps: 2 pre- cultures (solid and liquid) for the scaling up, and finally the culture in the bioreactor.

The first pre-culture is solid. 100 µl of cells are taken from a lab seed frozen at -80°C and set on a petri plate. The culture medium of this box consists of Agar 20 g/l, glucose 10 g/l, Leucine 3 g/l, Histidine 4 g/l and finally Yeast Nitrogen Base (YNB Difco laboratories), a definite minimum medium especially appropriate for yeast cultures. The Petri’s box is set in an incubator at 30°C during 20h ± 2h.

The second pre-culture is liquid. The cells in the Petri’s box are divided among 4 Erlenmeyer flasks which contain 200 ml medium composed of Yeast Nitrogen Base, glucose 20 g/l, Leucine 3 g/l and Histidine 4 g/l. The flasks are set in an incubator-shaker at 30°C and 200 rpm during 20h ± 2h.

After these pre-cultures, the cell quantity is sufficient to inoculate the pilot bioreactor, a Biostat C_DCU3 (B-Braun Biotech International) with a working volume of 15 l. The culture conditions are the following:

• Temperature: 30°C;

• Airflow: 20 Nl/min;

• Dissolved oxygen: 20% regulated by agitation;

• pH regulated by NH4OH at 5.0;

• Head pressure: 0.5 bar;

• Antifoam (SAG471): 4 ml added before inoculation.

The 100% of dissolved oxygen probe is fixed before inoculation with airflow equal to 30 Nl/min and the agitation equal to 1000 rpm. Let us note that the dissolved oxygen is fixed at 20% in order to avoid limitation by the oxygen.

Regarding the initial content of the bioreactor, it is, for fed-batch fermentations, 5 l with YNB, Leucine 3 g/l, Histidine 4 g/l, glucose 0.15 g/l and the inoculum, i.e. the 800 ml of the liquid pre-culture. As for a batch experience, the bioreactor initially contains more glucose and a different culture medium.

Indeed, YNB is a minimum medium; only elements absolutely necessary for the growth are in minimum and definite quantities. It is a selection and not a production medium. It does not allow to obtain high biomass densities. So, we use a rich medium containing yeast extract, peptone and dextrose (Bacto YPD broth Difco Laboratories) for the initial medium of batch cultures. Moreover, for the above mentioned reasons, this YPD is also used for the feeding solution in fed-batch experiments.

As for the different measurements, we distinguish two types. First, on-line data are acquired by a personal computer equipped with MFCS/win 2.0. It stores the following signals: agitation, temperature, pH, dissolved oxygen in the broth, gaseous outputs (oxygen and carbon dioxide), added NH4OH and feeding profile. As for the concentrations in ethanol, glucose and biomass, we take offline samples which are analysed thanks to a spectrophotometer, in order to determine the optical density (biomass measurement), and an YSI 7100 MBS in order to determine enzymatic measurements of glucose and ethanol concentrations.

4.4.2.3 Experiments and measurement errors

Seven experiments have been carried out according to the protocol briefly presented in the previous section. We have chosen these cultures in order to have a complete database which contains all the possible behaviours described in the figure 4.2.

Hence, we have performed three batches which present initial glucose concentrations respectively equal to 20.5, 15.3 g/l and 19.05 g/l (Exp3, Exp4 and Exp7). These cultures provide information about fermentation and ethanol oxidation pathways. The first one is used for the identification while the other ones are used for the cross validation.

Regarding the glucose oxidation, we have operated fed-batch experiments based on an exponential feeding profile which is optimized to maintain a sufficiently small glucose supply in order to avoid fermentation. As above mentioned, the feeding medium is based on the rich medium YPD. It is concentrated 5 times and mixed with Leucine (3 g/l), Histidine (4 g/l) and glucose in order to obtain a glucose concentration of 250 g/l. The following fed- batch experiments have been performed:

• Exp1: we have applied the exponential profile;

• Exp2: idem;

• Exp3 and Exp4: after the end of the batch, i.e. the complete consumption of ethanol, a fed-batch begins. The profile corresponds to the exponential one;

• Exp5 and Exp6: as the exponential profile gives cultures with weak fermentation, we test two cultures with larger profiles respectively 1.5 and 2 times the initial one.

The profile values are proposed in Appendix 4.1.

Among the seven experiments, we have chosen the experiments 1, 3 and 5 to identify our parameters. Thanks to these cultures, we dispose of sufficient data to identify the different parameters ((12+15+12)*3 components (biomass, glucose and ethanol concentrations) = 117 measurements for 21 kinetic parameters).

Regarding the measurement errors, they are assumed to be corrupted by a white noise with zero mean and a constant variation coefficient for 5% for the biomass, 5% for the glucose and 2% for the ethanol concentrations (advices of YSI 7100 MBS).

4.5 Model identification – Results and discussion

4.5.1 Reaction scheme and pseudo-stoichiometry

As the determination of the pseudo-stoichiometry is not the subject of this chapter, we choose, for the simulated databases, their real pseudo-stoichiometry (4.11 and 4.18). Regarding the industrial bioprocess, we consider the best C- identifiable reaction scheme identified in section 3.3.1 (3.131).

For the yeast culture performed at ULB, we desire a reaction scheme which involves three macroscopic species: biomass, glucose and ethanol. The dissolved oxygen is neglected because it is not limiting thanks to the regulation by the agitation. This choice implies the inability to determine a best C- identifiable reaction scheme. If the number of reactions is equal to the number of components, K_a must be equal to K (section 3.1.1.2). But the ethanol is produced in the fermentation reaction and consumed in the ethanol oxidation pathway. So, this component can not be considered in K_a, this matrix can not be equal to K and, so, a C-identifiable reaction scheme does not exist.

Hence, we adopt the well known reaction scheme (4.28-30) without dissolved oxygen:

X K E

E K X K G

X K G

13

32 12

11

3 2 1

ϕ ϕ ϕ

→

+

→

→

(4.31)

But at this stage, we do not know the values of the pseudo-stoichiometric parameters. However, they are essential to obtain a good initial estimation of the kinetics (sections 3.1.2.1 and 4.2). Hence, we choose to use the well known yeast model of Sonnleitner and Käppeli (1986) to obtain good initial values of the pseudo-stoichiometric parameters. Moreover, we re-identify all its parameters to improve the correspondence with our data.

Appendix 4.2 explains the structure of this model and how to re-identify its parameters, while Table 4.1 presents the re-estimated yield coefficients related to biomass, glucose and ethanol (

[

^TE

]

T G T X

T K K K

K = ).

Sonnleitner

& Käppeli

[[[[ ]]]] [[[[ ]]]] [[[[ ]]]]

[[[[ ]]]]

^{}















−

−−

−

∈

∈

∈

∈

−

−−

−

−

−

−

−

∈

∈∈

∈

∈

∈

∈

∈

∈

∈

∈

∈

=

==

=

1 0.38

0

0 1

1

1.09 0.049

0.70 K

39 . 0 , 37 . 0

13 . 1 , 04 . 1 049

. 0 , 0487 . 0 72

. 0 , 69 . 0

Table 4.1: Yield matrix for yeast cultures with their 99% confidence intervals

As these pseudo-stoichiometric values are specifically adapted to the kinetics of Sonnleitner and Käppeli (1986), we propose to re-identify these parameters with the generalized kinetic ones as well as the initial concentrations during the last nonlinear identification step. The vector of parameters therefore becomes

[[[[

αj γh j βl j κh j ^Kl ^{K T}

]]]]

^j

[[[[

^M

]]]]

T 1 34 (0) 1,

*

∈ ∈ ∈ ∈

=

= =

=

ξ

θ K K K K (4.32)

under the constraints θˆ

≥

0.

4.5.2 Kinetic identification - Results and discussion

In order to reveal the quality of our new kinetic structure, we now systematically compare the new model with the one originally proposed by Bogaerts et al. (1999). For the third database, the results are also compared with a classical extended Monod model of the form

[[[[

M

]]]]

K j K µ K

t

I

l jl l

jl R

h actjh h

h j

j

j

, ξ 1

ξ ) ξ

, (

*

∈

∈ ∈ + ∈

+ + + +

+ +

= +

=

=

= ∏ ∏ ∏ ∏ ∏ ∏ ∏ ∏

∈∈∈

∈ ∈

∈

∈∈

ϕ

ξ ^(4.33)

where K_acthj

≥ ≥ ≥ ≥

0 and K_jl

> > > >

0 are respectively activation/saturation and inhibition parameters. Such a model is not identified for the two simulated databases as their simulation models are already particular cases of the general form (4.33). Regarding the yeast cultures, we prefer considering the model of Sonnleitner and Käppeli (1986) which should provide better results than a simple extended Monod model.

Note that the Monod parameters are identified thanks to a Markov criterion and that the enzyme is supposed to be a product which does not activate any reaction.

In order to study the performances of the considered models, we propose 3 tables for each database:

1. The first one presents results obtained with the two kinetic model structures (the three for the experimental case studies) in direct and cross validation, w.r.t. values of the minimized Markov criterion (3.60). Three values of the parameter x (which represents the confidence in the absence of saturation) are tested in order to observe the influence of this parameter on the nonlinear identification procedure (detection of local minima depending on the a priori

assumption of the presence or absence of saturation effects). Our arbitrarily chosen values are the following: 95%, 70% and 50%.

2. The second table presents the corresponding parameter values with their 95% confidence intervals.

3. The third table details the parameter reduction procedure steps.

Direct validation Cross validation Original kinetic model

structure 0.029 0.573

95% 0.029 0.573

Generalized kinetic

model structure

70%

50% 0.0095 0.1418

Table 4.2: Simulated microbial growth

Markov criterion values (3.60) for the different tested models in direct and cross validation

Parameter values with 95% confidence intervals Original

kinetic model structure

[[[[

^0.03,0.22

]]]]

*====0.08∈∈∈∈ α

[[[[ ]]]]

====

[[[[

∈∈∈∈

[[[[

0.05,0.72

]]]]

∈∈∈∈

[[[[

0.35,0.72

]]]] ]]]]

=

==

= 0.20 0.76

γ γX γS

[[[[ ]]]]

====

[[[[

∈∈∈∈

[[[[

0.001,0.12

]]]]

∈∈∈∈

[[[[

0.003,0.157

]]]] ]]]]

==

== 0.007 0.020

β β_X β_S

95%

[[[[

0.03,0.22

]]]]

*====0.08∈∈∈∈ α

[[[[ ]]]] [[[[ ]]]]

[[[[

∈∈∈∈0.05,0.72 ∈∈∈∈0.35,0.72

]]]]

==

== 0.20 0.76 γ

[

0 0

]

κ=

[[[[ ]]]] [[[[ ]]]]

[[[[

∈∈∈∈^{0.001,0.12 ∈}∈∈∈^0.003,0.157

]]]]

==

== 0.007 0.020

Generalized β kinetic model structure

70%- 50%

[[[[

^0.66,1.69

]]]]

*====1.06∈∈∈∈ α

[[[[ ]]]] [[[[ ]]]]

[[[[

∈∈∈∈0.023,2.85 ∈∈∈∈0.59,2.24

]]]]

==

== 0.26 0.81 γ

[[[[ ]]]] [[[[ ]]]]

[[[[

∈∈∈∈0.007,3.101 ∈∈∈∈0.011,0.461

]]]]

=

=

=

= 0.148 0.071

κ

[

^{0 0}

]

β=

Table 4.3: Simulated microbial growth Parameter values with 95% confidence intervals

Eliminated coefficients first step second step third step fourth step Original kinetic

model structure - - - -

95% κX,κS - - -

70% βS β_X - -

Generalized kinetic model

structure

50% - βX,βS - -

Table 4.4: Simulated microbial growth: Parameter reduction procedure

Direct validation Cross validation Original kinetic model

structure 1.00 22.97

95% 0.17 2.6

70% 0.23 2.57

Generalized kinetic

model structure

50% 0.099 0.83

Table 4.5: Simulated animal cell culture

Markov criterion values (3.60) for the different tested models in direct and cross validation

Eliminated

coefficients first step second step third step fourth step Original kinetic

model structure β₂^X,β₂^L β_1L,γ_2L - γ_1L 95%

G X

L L

κ₁ , , β

κ γ

L G X

β β₂ , ₂

,

κ - -

70%

L L X L

κ β γ

1 2 1

, ,

β κ_1X

L X G

κ κ β

2 2 1

, γ₂^L

Generalized kinetic model

structure

50%

L L X

κ β

1

, 2

β

X G

κ β

1

1 -

L X G L

κ κ β

2 2 2

, γ

Table 4.6: Simulated animal cell culture Parameter reduction procedure