Engineering and Manufacturing for Biotechnology - Marcel Hofman & Philippe Thonart

R. Takors, D. Weuster-Botz, W. Wiechert, C.

considered (Holmberg, 1981; Nihilitä and Virkkunen, 1977) if these data were used for model fit. Thus, batch experimental design strategies have been investigated to overcome this problem (Yoo et al., 1986).

Beyond it, several authors tested new experimental design approaches for the identification of kinetic models based on fed-batch fermentations. Roels (1983) developed a concept of time varying feeding rates in fedbatch processes to estimate maintenance constants for growth. Focussed on baker’s yeast, a special feeding strategy for model parameter estimation was presented by Ejiofor et al. (1994). Using the Fisher information matrix Munack (1989) developed a general methodology for the identification of Monod-type models by fed-batch experiments that was derived from foregoing investigations of Goodwin and Pain (1973). The use of the Fisher information matrix has been shown to be a successful tool for experimental design. Therefore it was tested by Baltes et al. (1994) for fed-batch experimental design, by Munack (1991) to develop optimum sampling strategies and by Schneider and Munack (1995) to estimate bio-process parameters on-line. Furthermore Van Impe et al. (1997) presented an E- optimal experimental design for fed-batch processes and Takors et al. (1997) developed a D-optimal experimental design considering closed-loop substrate controlled steady- state fermentations.

All design strategies have in common, that they are model specific. Only if the appropriate modelling approach is known, an optimal experimental design strategy could be identified. Unfortunately experimentalists are sometimes faced with the problem that principles of microbial kinetics are unknown before kinetic experiments have been carried out. For instance it could be unknown whether product formation is growth coupled or not or whether substrate and/or product inhibition occurs. Thence an alternative experimental design strategy is needed, that takes into account uncertainties of model structure. It should be the task of this approach to design experiments with the aim to clearly discriminate the “right” kinetic model among a group of competing models. This approach is called model discriminating design.

In 1995, Cooney and McDonald presented a test of two different model discrimination approaches, both principally based on a comparison of model responses of a set of four competing kinetic models. As a result, they identified the choice of the discriminating function to be extremely important and favoured the minimum difference between any two model responses for model discrimination.

This paper aims to present a more general approach for model discrimination and model discriminating experimental design. The methodology is based on the calculation of model probabilities derived from the formulation of model system entropy taking into account model parameter inaccuracies as well as model prediction uncertainties.

The model discrimination approach is not limited by the number of competing models that could be considered.

The quality of the approach for “simple” model discrimination is shown using experimental data of steady-state Candida boidinii fermentations. Experiments with the methylotrophic yeast have been carried out using a closed-loop substrate control for methanol (nutristatic fermentation control) that enables steady-state growth even under substrate inhibiting conditions. It could be shown that the kinetic model that is derived from these steady-state experiments could also be used for the modelling of batch and

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fed-batch fermentations. Furthermore simulation results using the kinetics of the anaerobic bacterium Zymomonas mobilis are presented to show the quality of the approach for model discriminating experimental design.

2. Theoretical concept

For modelling of fermentations a system state vector X could be defined consisting of variables like cell-dry-weight, substrate and product concentration or liquid and gaseous streams. Using X , a rate vector for the description of e.g. (spec, growth rate), (spec. substrate consumption rate) and (spec. product formation rate) could be formulated including model parameters like etc. in

It is a basic characteristic of the model discriminating design approach that model discrimination is achieved sequentially. Based on n (“old”) observations additional experiments n+1, n+2, ... are suggested using the system state vector x as a design vector. Information of the last n fermentations is used for non-linear parameter

regression to estimate and COV

A set of competing models is defined including all modelling approaches that might be appropriate to describe microbial kinetics of the biological system. For instance, if substrate inhibition may occur, two different growth models (one with and one without substrate inhibition) form a set of competing models for model discriminating design. Model parameters of each model are calculated using the “old” observations.

A detailed description of the model discrimination approach is given by Takors (1997).

2.1. MODEL DISCRIMINATION

Assumed that observations	of n “old” fermentations are available, it is the task
of the model discrimination	approach to identify the most appropriate model

considering a set of competing models. Thence model probabilities for the i th of m competing models are defined that could be calculated as follows:

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R. Takors, D. Weuster-Botz, W. Wiechert, C.

This Bayes approach was first published by Box and Hill (Box and Hill, 1967) who

introduced a model probability density function The function considers normally distributed measurement errors with constant measurement variance and model predictions

Additionally a model variance was estimated to include effects of measurement

errors for model predictions into model probability calculation.

Often it is useful to consider variable measurement errors instead of constant values. Thus equation (3) could be extended to

including the variable measurement variance

Equation (2) represents a univariate model discrimination approach. However macrokinetic models usually consist of several equations e.g. for growth rate, substrate consumption and product formation. Hence a single model probability could be derived from a multivariate model consisting of k equations as following

2.2. MODEL DISCRIMINATING DESIGN

2.2.1. Extended entropy approach

Based on the a priori model probability estimation (2), Box and Hill derived a model discriminating design criterion that had to be extended to fulfil biological constraints. In principal, experiments are proposed such that (model) system entropy is reduced which

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analogously causes an increase of information content. This leads to the maximisation functional

including in pair's considerations of m model probabilities based on n+1 probability density functions.

Due to variable measurement errors of equation (4) the integration of equation (6) now leads to equation 7

This equation can be transformed to the Box and Hill result assuming

(see Box and Hill, 1967).

To extend the originally univariate approach, the following summation functional is used to estimate the experimental design vector

Hence a multivariate model consisting of k model equations could be used for experimental design.

2.2.2. Model predictive design

It should be pointed out that the extended entropy approach demands the calculation of model variances. Thus model parameter regression must be carried out before the approach could be used. Therefore an alternative design strategy should be developed to overcome this start problem.

Assuming a set of start parameters		model	predictions	could be
calculated using a system state vector		It is the aim of model predictive design to
estimate	such that maximum model	prediction	discrepancies	of competing

models i and j become obvious

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Thus, if multivariate models with k model equations are considered an optimisation functional R could be defined as

This approach only uses measurementvariances considering n “old” fermentations.

3. Material and methods

3.1. FERMENTATION

Fermentations were carried out at pH 5, temperature 28°C and pressure 1 bar using the methylotrophic yeast Candida boidinii (ATCC 32195). The yeast was cultivated under aerobic conditions with methanol as only carbon source and formate dehydrogenase (FDH) as internal product. Aerobic stirred reaction vessels with 7.51 (fermenter Fl) and 201 (fermenter F2) total volume (Chemap, Switzerland) equipped with standard measuring and control units and an optimised mineral medium for maximum growth

(Weuster-Botz and Wandrey, 1994) were used for fermentation. Medium was sterilised by microfiltration (pH-capsule 0.2M, Sartorius, Germany) without antifoam agent, which was autoclaved separately. For titration 4n NaOH was used. To receive on-line information about methanol concentrations a NDIR analyser (Rosemount, USA) was installed in the exhaust gas stream together with a paramagnetic oxygen detector (Rosemount, USA).

For steady-state fermentations Candida boidinii was cultivated in 41 reaction volume using a 7.51 fermenter F1. Batch and fed-batch cultivations were carried out in a 201 fermenter F2 using 121 start reaction volume. Fl and F2 were used to start batch and fed-batch fermentations. In F1. Candida boidinii was cultivated in steady-state culture with a dilution rate of 0.033 1/h. To start batch or fed-batch fermentations a defined sample was taken out of Fl via a harvesting tube and directly pumped into F2. Thus it was ensured that only “active” cells are used as inoculum for batch or fed-batch experiments. F2 was already filled with 121 cultivation medium. For fed-batch fermentations F2 was equipped with an additional methanol feed only including 16g/l methanol and desalted water.

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3.2. ANALYTICAL METHODS

Cell homogenisation for off-line analysis of intracellular FDH is performed with a laboratory vibrator mill (Retsch, Germany) within 15 min using 1.2 g glass beads (diameter 0.5 mm) and 600 1 cell sample. After centrifugation the enzyme activity was essayed spectrophotometrically at 340 nm. The 10 mm cuvettes were thermostated at 30°C. The assay mixture was composed of 2.0 ml sodium phosphate buffer (0.1M, pH 7.5), 0.5 ml 0.01M NAD, 0.1 ml sample and 0.5 ml 1.0 M sodium formate. Off-line analysis of methanol was carried out by gas chromatograph (Chrompack, Germany) with a fused silica capillary column. Dry cell mass was determined gravimetrically by use of 0.45 filters.

3.3. NUMERICAL AND PROGRAMMING TOOLS

All numerical calculations for experimental design, fermentation analysis and simulation are implemented into the C++ coded program PARAGLIDE. Matrix and vector calculations are facilitated by use of ROGUE WAVE libraries (Rogue Wave Software, Inc. Oregon, USA). For graphical applications STARVIEW libraries (STAR DIVISION, Hamburg, Germany) are taken. Numerical optimisations were carried out using the derivation-free Simplex/Nelder-Mead approach.

4. Results and discussion

4.1 MODEL DISCRIMINATION OF STEADY-STATE FERMENTATIONS

The model discrimination approach is tested using experimental results of steady-state fermentations with Candida boidinii. The methylotrophic yeast was cultivated with methanol as carbon source under aerobic conditions. Altogether 19 steady-state fermentations were carried out. These experiments were suggested by help of D-optimal design as well as by intuition (Takors et al, 1997). Kinetic results of growth rate substrate consumption rates of methanol and oxygen and product formation rate of FDH are presented in figures (1) and (2).

To model microbial kinetics 10 different unstructured approaches are used. They form a set of competing models as introduced in section 2. Derived from former research (Wedy, 1992) a start model (0) was already known:

Based on this model, different modelling approaches are created and put together in a set of competing models. For instance methanol and oxygen effects on growth rate are described by help of Monod-type or Andrews-type kinetic equations. Cell mass/substrate or FDH/cell mass yields are expected to be constant or growth rate dependent. Product formation is assumed to be completely or partially growth coupled. Altogether the number of model parameters varies from 7 to 11. An overview of the competing models is given in table (1).

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As indicated in figure (3) model (5) is clearly identified as the most suitable approach obtaining a model probability of 53.7%. This probability is more than double the probability of the second ranked model (0) and more than three times the probability of

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the third ranked modelling approach (3). Table (1) indicates that model (5) differs from model (0) by the formulation of oxygen/cell mass yield. While model (0) assumes a growth rate dependent oxygen/cell mass yield, model (5) simplifies this relation to a constant yield, which is appropriate to describe the experimental results. As a consequence the number of necessary model parameters is reduced to 9. The third ranked model (3) differs from model (5) by the assumption of a partially growth coupled FDH production instead of a completely growth coupled product formation. Figure (2) indicates that no growth uncoupled product formation could be observed experimentally. This corresponds with metabolic pathway constraints as the enzyme formate dehydrogenase is used in dissimilation for the final oxidation of formate to carbon dioxide.

Summing up, it may be said that the model discrimination approach clearly identified a suitable macrokinetic model in agreement with former research results. A model selection only based on sum-of-squares analysis would not have been successful (see figures above columns in figure (3)). This approach would have lead to the identification of model (2) that assumes a growth inhibition caused by high dissolved oxygen concentrations. The inhibition was not experimentally observed.

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4.2 BATCH AND FED-BATCH FERMENTATION MODELLING

To test the applicability of the identified model for the description of instationary fermentations, batch and fed-batch experiments were carried out. As pointed out in section 3.1 cell samples were taken out of steady-state conditions (fermenter F1, residence time 30h) to inoculate fermenter F2. As a consequence the inoculum undergoes strong environmental changes from methanol limitation in F1 to optimal growing conditions in F2. These changes result in an undefined lag-time that is needed by the cells to adapt to the new environmental situation. Thus the identified model must be extended to consider the lag-time for cell adaptation.

To prevent large model structure changes the identified model was simply extended with the following approach:

is defined as the time needed to achieve a 5% decrease of dissolve oxygen

concentration in F2 after inoculation. Thus an easy determinable phenomenological parameter is used.

Using this extended model, batch and fed-batch fermentations were simulated based on start conditions that have been determined experimentally immediately after inoculation. Experimental results are presented in figures (4) and (5).

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As shown simulation results of methanol and cell-dry-weight are in good agreement with experimental results, if a 10% measurement error is assumed. As indicated in figure (5) two feeding phases consisting of 0.05 and 0.1 l/h with 16 g/l methanol were tested. In figures (4) and (5) experimentally determined lag-times (6 and 7h) are presented that were used for simulations. It is shown that especially at the beginning of product formation model predictions do not fit measured values as well as the simulation courses of cell mass and methanol concentration do. This could be a result of missing kinetic information of the preliminary experiments. Figure (2) indicates that no steady-state measurements are available for growth rates lower than 0.033 l/h.

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