1 Introduction

Pyruvic acid and its salts are important chemicals used in the pharmaceutical, food, agrochemical, and cosmetic industry [1, 2, 5]. Pyruvate represents one of the most important metabolites in the central metabolism of living cells because of its role in the glucose uptake (via carbohydrate phosphoenolpyruvate:phosphotransferase system (pts)), its impact as a precursor for amino acid synthesis, its relevance as an intermediate of glycolysis, etc., thus making the metabolite one of the most widely used reactants in the E. coli metabolic network [3].

Basically, there are two different approaches known for the production of pyruvate: (i) the classical chemical route considering the energy-intensive pyrolysis of tartaric acid [4] and (ii) the biotechnological access using purified enzymes, non-growing, immobilized or living cells [1, 2, 5, 7]. Currently, the fermentation is regarded as one of the most promising routes for the production of pyruvate. As an example, the conversion of glucose to pyruvate with non-growing, acetate auxotrophic cells of E. coli YYC202 ldhA::Kan was published, which achieved a maximum volumetric productivity (Q P) of 145 gPyruvate/L/d, a maximum pyruvate/glucose yield (Y P/G) of 1.78 molPyruvate/molGlucose, and maximum product titers (c P) of about 720 mM (approx. 65 g L−1), thus offering good starting conditions for subsequent downstream processing [1, 6, 7].

For further process development (optimization) and scale-up, the quantitative understanding of the basic cellular characteristics, such as the cellular demand for the sole carbon source glucose and for the auxotrophic substance acetate, are of outstanding importance. Therefore, modeling studies were performed to identify simple, easy-to-use, and robust models that are suitable to support the engineering tasks of process optimization and design.

For this purpose, “black-box” unstructured cell models were chosen because they offer a significant simplicity at the same time providing a general understanding of the dominant metabolic processes in the production strain. Although these type of models require very rigorous constraints such as “balanced growth” (i.e. “pseudo” steady-state metabolic conditions) or “one-component cell systems”, they nevertheless showed their applicability and suitability for a variety of different modeling tasks and they additionally represent a valuable basis for subsequently establishing structured models that aim at describing the intracellular metabolism in detail [8, 9]. Unstructured models usually use simple Michaelis–Menten type kinetic equations with few variables, each (more or less) contributing a physical meaning while structured models consist of a wide number of variables thus trying to reflect the microbial complexity on a biochemical level. Because of the simplified model structure “black-box” models usually have problems in describing lengthy cell-lag phases [10], which necessitates their careful application for highly unstable process conditions. However, they can legitimately be applied to systems in which “balanced growth” occurs [11, 12].

The estimation of kinetic parameters in unstructured growth models with high parameter accuracy is essential for successful model validation. Parameter estimation in unstructured growth models is often performed with the aid of continuous fermentation [13]. These experiments are usually time-consuming and their practical realization is not always feasible because of the (relatively) complex experimental set-up (e.g. complex cell recycle system, complex pump–balance system). In contrast, simpler experiments can be achieved using batch fermentations, but the kinetic parameters of the Michaelis–Menten type models cannot be uniquely identified from noisy batch measurements [11, 14]. The extension of the batch experiment by a fed-batch phase with time-variant feed rate leads to a higher accuracy of the parameter estimates [15]. However, it is noteworthy that feed rate profiles obtained for optimal process performance are not necessarily optimal for parameter estimation [16]. For instance, Baltes et al. [11] emphasized that changes in feeding rates should be as small as possible to avoid fast dynamic system responses, which are not described by unstructured models. Nevertheless, our study was based on fed-batch experiments for the estimation of kinetic parameters. This decision was motivated by the aim to identify kinetic parameters of a process that resembles the optimal production conditions as much as possible. Therefore, the glucose-feeding profile kept constant and the acetate feed followed previous findings [2].

Thus the aim of this work was to develop an unstructured, mathematical model of the pyruvate production process considering the acetate auxotrophic strain E. coli YYC202 ldhA::Kan. Because of primary uncertainties regarding the microbial kinetics (and the underlying mechanism) several models for microbial growth and product formation were formulated as a “competing” set of model candidates. To our knowledge, this represents the first approach to model E. coli based pyruvate formation with a production strain.

2 Modeling

In our previous experimental studies [1, 7] we observed that pyruvate formation followed a typical pattern, which can be described as a mixture of growth and non-growth associated product formation. Glucose was assumed to be the only carbon source for cell growth and product formation. However, the acetate auxotrophy had to be considered as well. Furthermore, high extracellular pyruvate levels were identified as an inhibiting factor for cell growth and/or pyruvate formation. Taking these “vague” basic characteristics into account, the following constraints were formulated and used to define primary, verbal models:

  • Glucose and acetate are limiting substrates. All other nutrients such as nitrogen, phosphate, and growth factors are ample;

  • Cell growth occurs simultaneously using glucose and acetate, however acetate is essential for growth;

  • There was no oxygen effect on either biomass growth or bioconversion of glucose to pyruvate—at least at the dissolved oxygen range investigated (from 40 to 100%);

  • Product formation kinetics combined growth associated and non-growth associated characteristics;

  • For simplification, the bioconversion of glucose to pyruvate is regarded as a one-step enzymatic reaction:

$${\text{C}}_{{\text{6}}} {\text{H}}_{{{\text{12}}}} {\text{O}}_{{\text{6}}} {\text{ }}\xrightarrow{{{\text{O}}_{{\text{2}}} }}{\text{ 2C}}_{{\text{3}}} {\text{H}}{}_{{\text{4}}}{\text{O}}_{{\text{3}}} {\text{ }} + {\text{ 2H}}_{{\text{2}}} {\text{O;}}$$

  • Pyruvate production and/or biomass growth are inhibited by high pyruvate concentrations;

  • The suspension viscosity in the reactor remains constant during the experiment;

  • Potential mixing effects of the highly concentrated feeds with the cultivation medium are neglected for the sake of the model simplicity.

2.1 Microbial growth

To model the microbial specific growth rate various, well-known approaches were tested. The “multiple substrates Monod kinetics” was used in models 1 and 2 [17, 18, 19]. However, it is known that this kinetic term has an inherent disadvantage if many essential substrates are considered. Even if the concentrations of, for example, five essential substrates are so high that their contribution of the growth rate reaches 90% of the saturation value, the total growth rate will be limited to 60% of the maximum possible value.

In models 3 and 4, growth inhibition by-product formation was considered according to the Levenspiel approach [20]. This model presumes a critical inhibitor (product) concentration, c P,max, as an upper limit for cell growth. As a consequence, the parameters of the Monod equation are dependent on the product concentration.

In models 5, 6, and 9 inhibition of growth by-product formation was described by the model of Jerusalimsky [21]. This model represents an approximate analogy to the non-competitive substrate inhibition, which is often used in pure enzyme kinetic models.

In models 7 and 8, acetate was assumed to be only limiting substrate for growth, which, strictly speaking, was not experimentally observed. However, this approach was motivated by the strain genotype, namely its acetate auxotrophy.

Finally, in model 10, the well-known Andrews kinetics [22] were considered with respect to a potential acetate (substrate) inhibition of growth at high acetate levels. In models 7, 8, and 10, glucose was only used for pyruvate production and not for growth.

2.2 Product formation

In models 1, 3, 5, 7, and 10 the kinetics of product formation was based on the Luedeking–Piret equation [23] to cope with the experimental observations mentioned in the introductory part of this section. As an alternative, pyruvate formation was modeled using a modified Michaelis–Menten equation for non-competitive product inhibition [24] (see models 2, 4, 6, and 8). Finally, model 9 includes a modified Luedeking–Piret equation together with the Levenspiel term to describe pyruvate formation and product inhibition. The proposed models are summarized in Table 1.

Table 1 Kinetic equations used for description of biomass growth and product formation in the developed models
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2.3 Substrate uptake

The biomass dependent substrate consumption rates are expressed in Eqs. 1 and 2. Yields such as biomass/glucose Y X/G and biomass/acetate Y X/A were assumed to be functions of the biomass growth and maintenance energy demand (Eqs. 3 and 4).

$$r_{{\text{G}}} = \frac{\mu } {{Y_{{{\text{X/G}}}} }}$$
(1)
$$r_{{\text{A}}} = \frac{\mu } {{Y_{{{\text{X/A}}}} }}$$
(2)
$$ Y_{{{\text{X/G}}}} = \frac{{Y_{{{\text{X/G,max}}}} \cdot \mu }} {{Y_{{{\text{X/G,max}}}} \cdot m_{{\text{G}}} + \mu }} $$
(3)
$$ Y_{{{\text{X/A}}}} = \frac{{Y_{{{\text{X/A,max}}}} \cdot \mu }} {{Y_{{{\text{X/A,max}}}} \cdot m_{{\text{A}}} + \mu }} $$
(4)

A simplified overview of the basic model characteristics is given in Table 2.

Table 2 Kinetics used for description of biomass growth and product formation in the developed models. Microbial specific growth rate approaches: multiple substrate Monod kinetics [17, 18, 19], growth inhibition by product formation according to the Levenspiel [20], growth inhibition by product formation described by the Jerusalimsky [21], Monod kinetics and acetate growth inhibition described by Andrews kinetics [22]. Product formation kinetics: product formation based on the Luedeking–Piret [23], product formation according to the Michaelis–Menten equation for non-competitive inhibition with product [24] and product formation and product inhibition described by the modified Luedeking–Piret equation together with the Levenspiel term (L–P Levenspiel)
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2.4 The mass balance model for the fed-batch fermentation

The mathematical model (Eqs. 5, 6, 7, 8, and 9) for the fed-batch fermentation of recombinant E. coli based on the mass balance of the components in a fed-batch reactor is represented by the following differential equations:

$$ \frac{{dc_{{\text{X}}} }} {{dt}} = - \frac{{q_{{\text{V}}} }} {V} \cdot c_{{\text{X}}} + \mu \cdot c_{{\text{X}}} $$
(5)
$$ \frac{{dc_{{\text{G}}} }} {{dt}} = - \frac{{q_{{\text{V}}} }} {V} \cdot c_{{\text{G}}} + \frac{{q_{{{\text{VG}}}} }} {V}c_{{{\text{G,0}}}} - r_{{\text{G}}} \cdot c_{{\text{X}}} - r_{{\text{P}}} \cdot c_{{\text{X}}} $$
(6)
$$ \frac{{dc_{{\text{A}}} }} {{dt}} = - \frac{{q_{{\text{V}}} }} {V} \cdot c_{{\text{A}}} + \frac{{q_{{{\text{VA}}}} }} {V}c_{{{\text{A,0}}}} - r_{{\text{A}}} \cdot c_{{\text{X}}} $$
(7)
$$ \frac{{dc_{{\text{P}}} }} {{dt}} = - \frac{{q_{{\text{V}}} }} {V} \cdot c_{{\text{P}}} + r_{{\text{P}}} \cdot c_{{\text{X}}} \cdot Y_{{{\text{P/G}}}} $$
(8)
$$ \frac{{dV}} {{dt}} = q_{{{\text{VG}}}} + q_{{{\text{VA}}}} = q_{{\text{V}}} $$
(9)

where V is the biosuspension volume, q V is the time-dependent overall volumetric flow rate, q VG and q VA are the analogues for glucose (G) and acetate (A), and c G,0 and c A,0 are glucose and acetate concentration in the feed.

3 Experimental

3.1 Strain and medium

E. coli YYC202 ldhA::Kan [1, 6, 7] was stored at −80°C in LB medium containing glycerol (50%). The culture was reactivated by inoculating a frozen aliquot (500 μL) into an Erlenmeyer flask with 200 mL mineral medium, which was incubated at 37°C on a rotary shaker (3033, GFL GmbH, Burgwedel, Germany) for 15 h.

The fermentation medium contained per liter: 1.50 g NaH2PO4·H2O, 3.25 g KH2PO4, 2.50 g K2HPO4, 0.20 g NH4Cl, 2.00 g (NH4)2SO4, 0.50 g MgSO4, 1 mL trace element solution, 11 g glucose monohydrate, and 0.79 g potassium acetate. Trace element solution contained per liter: 10.00 g CaCl2·2H2O, 0.50 g ZnSO4·7H2O, 0.25 g CuCl2·2H2O, 2.50 g MnSO4·H2O, 1.75 g CoCl2·6H2O, 0.12 g H3BO3, 2.50 g AlCl3·6H2O, 0.50 g Na2MoO4·2H2O, 10.00 g FeSO4. Shake flask growth medium contained per liter: 3.00 g Na2HPO4, 1.50 g KH2PO4, 0.25 g NaCl, 0.50 g NH4Cl, 0.05 g MgSO4, 0.01 g CaCl2, 2.00 g glucose monohydrate, 0.10 g potassium acetate. The fed-batch fermentation feed medium consisted of 700 g L−1 glucose monohydrate and 109 g L−1 potassium acetate.

3.2 Fed-batch fermentation

All experiments were carried out in a 7.5 L bioreactor (INFORS AG, Bottmingen, Switzerland) containing 2.25 L fermentation medium, which was equipped with standard control units for pH, pressure, temperature, aeration, stirrer speed, etc.

After sterilization of the bioreactor and peripheral equipment, fermentation medium was filled into the bioreactor through a sterile microfiltration unit (0.2 μm cut-off, Sartobran, Sartorius AG, Göttingen, Germany) and pH was adjusted at 7.0 by 25% ammonia titration, also during the later fermentation process. All experiments were carried out at the temperature of 37°C. Sufficient aeration (DO (dissolved oxygen) ≥ 40%) was obtained by vigorous stirring (200–1,800 rpm), airflow rate (1–10 L min−1), and reactor overpressure (0.2–0.8 bar). A reflux cooler condensed the outlet gas stream. The condensate was returned to the medium. Exhaust gases were analyzed by the gas analyzer (Binos 100 2 M, Rosemont Analytical/Process Analytical Division, Orville, OH, USA). The bioreactor was inoculated with 10% of the working volume of a preculture.

The fed-batch experiments were performed under different glucose-feeding conditions. All considered a constant feed at 10, 20, and 30 mL h−1, respectively. The acetate was fed by applying the previously developed indirect acetate control ensuring acetate saturating conditions. According to the heuristic approach, acetate consumption was calculated based on the on-line estimated CO2 production rate. As a result of our experimental studies an optimum equimolar ratio between acetate consumption rate and CO2 production rate was identified [1, 2].

3.3 Analytical methods

The optical density (OD) was measured in a double beam spectrophotometer (UV-160, Shimadzu, Kyoto, Japan) at 600 nm. Cell dry weight was measured by filtration of 2.5 mL fermentation broth using pre-weighed microfilters (0.2 μm cut-off, Schleicher & Schuell, Dassel, Germany). After drying for 48 h at 80°C, the filters were cooled in an exsiccator for another 48 h. After filter weighing, the cell dry mass was calculated. Glucose concentration was measured off-line by the enzyme-based biosensor appliance Accutrend (F. Hoffmann–La Roche Ltd, Basel, Switzerland).

The concentrations of pyruvate and acetate in the fermentation supernatant were measured using the HPLC. Determination of the organic acid concentration involves the use of two Aminex HPX-87H (Bio-Rad Laboratories GmbH, Munich, Germany) columns in series. The separation was performed with 0.2 M H2SO4 solution at a flow rate of 0.5 mL min−1 (PUMP S1000, Sykam Chromatografie Vertriebs GmbH, Eresing, Germany) and the detection was at a wavelength of λ=254 nm (diode array detector, DAD). The signal was analyzed with an integrator (C-R3A, Sykam Chromatografie Vertriebs GmbH) and the sample volume used was 100 μL.

3.4 Data handling

The model parameters were estimated by non-linear regression analysis and they were optimized using the Nelder–Mead algorithm [25]. The numerical values of the parameters were evaluated by fitting the model to the experimental data with the “Scientist” [26] software (MicroMath Inc., St. Louis, MO, USA). The model equations were solved numerically by the fourth order Runge–Kutta algorithm, which is also offered in the same software. The set of estimated parameters has been used for the simulation (Figs. 1, 2, and 3).

Fig. 1
figure 1

Data obtained by model simulation (model 5, solid line ; model 7, dashed line ; model 9, gray line), and experimental data for biomass (square) and pyruvate (circle) concentration in fed-batch process at constant glucose volumetric flow rate of q VG=10 mL h−1

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Fig. 2
figure 2

Data obtained by model simulation (model 5, solid line ; model 7, dashed line ; model 9, gray line), and experimental data for biomass (square) and pyruvate (circle) concentration in fed-batch process at constant glucose volumetric flow rate of q VG=20 mL h−1

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Fig. 3
figure 3

Data obtained by model simulation (model 5, solid line ; model 7, dashed line ; model 9, gray line), and experimental data for biomass (square) and pyruvate (circle) concentration in fed-batch process at constant glucose volumetric flow rate of q VG=30 mL h−1

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The calculated data were compared with the experimental data, recalculated in the optimization routine and fed again to the integration step until minimal errors between experimental and integrated values was achieved (built-in Scientist). The residual sum of squares was defined as the sum of the squares of the differences between experimental and calculated data. For discrimination of various models, the minimal value of the residual sum of squares and the model selection criterion (MSC) [27] were used as trial functions. The MSC is defined as (Eq. 10):

$$ {\text{MSC}} = \ln {\left( {\frac{{{\sum\limits_{i = 1}^n {w_{i} \cdot {\left( {Y_{{{\text{obs}}_{i} }} - \overline{Y} _{{{\text{obs}}}} } \right)}^{2} } }}} {{{\sum\limits_{i = 1}^n {w_{i} \cdot {\left( {Y_{{{\text{obs}}_{i} }} - Y_{{{\text{cal}}_{i} }} } \right)}^{2} } }}}} \right)} - \frac{{2 \cdot p}} {n} $$
(10)

where n is the number of points, w i is the weight applied to each points, \( \overline{Y} _{{{\text{obs}}}} \) is the weighted mean of the observed data, \( Y_{{{\text{obs}}_{i} }} \) is the weighted value of observed data, \( Y_{{{\text{cal}}_{i} }} \) represents the weighted value of calculated data and p is the level of significance of the simulation. The MSC attempts to represent the “information content” of a given set of parameter estimates by relating the coefficient of determination to the number of parameters (or equivalently, the number of degrees of freedom) that were required to obtain the fit. When comparing two models with different numbers of parameters, this criterion imposes a burden on the model with more parameters. The most appropriate model will be the one with the largest MSC.

The “Episode” algorithm for a stiff system of differential equations, implemented in the “Scientist” software package, was used for the simulations. It uses variable coefficient Adams–Moulton and backward differentiation formula methods in the Nordsieck form, treating the Jacobian matrix as full or banded.

4 Results and discussion

4.1 Estimation of parameters

As shown in the preceding section, all models consisted of a set of five differential equations (Eqs. 5, 6, 7, 8, and 9) thus representing five dependent state variables and up to 13 parameters. State variables such as biomass concentration c X, glucose concentration c G, acetate concentration c A, pyruvate concentration c P, and reaction volume were taken from the experimental studies presented elsewhere [1, 2]. At the beginning, a primary model identification was performed using the data obtained from the fed-batch fermentation with q VG=10 mL h−1. As already mentioned, the model identification based on the least-square method to minimize difference between experimental and calculated values of state variables. Model validation was performed using two alternative data sets, namely those of the fed-batch fermentations when glucose feed rates of q VG=20 and 30 mL h−1 were installed. In each case, the MSC criterion was used to qualify the model suitability for reflecting the experimental findings.

Initial values of parameters Y X/G,max, Y X/A,max, K S G, and K S A were those previously identified for growth of E. coli K12 on a single substrate [28]. The initial value of the parameter Y P/G was calculated from the experimental data (fed-batch fermentation performed at q VG=10 mL h−1). In all models c P,max was set to be 63.6 g L−1, which represents the experimentally observed value [1, 2]. All other model parameters were initially estimated so that negative estimates of simulated concentration curves were avoided.

Table 3 provides an overview of the residual sum of squares and MSC values obtained after the modeling of the first data set. As indicated, the models, 5, 7, and 9 have the similar and the lowest sum of square errors. Additionally, the MSC values are the highest for models 5, 7, and 9 thus stressing their convenience. As a consequence, these models were favored for further analysis.

Table 3 Residual sum of squares and model selection criterion (MSC) for different models
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Interestingly, models 2, 4, 6, and 8, in which pyruvate formation was described according to the modified Michaelis–Menten equation with non-competitive product inhibition, achieved the lowest accuracy level. However, the consideration of the Luedeking–Piret approach (models 1, 3, 5, 7, and 9) enabled better model predictions. Furthermore, some parameter estimates of the models 2 and 4 were negative, which is meaningless in a biological sense (data not shown). It is also noteworthy that the multiple substrate Monod kinetics (models 1 and 2) and multiple substrate Monod kinetics with the Levenspiel term for product inhibition were not able to describe the behavior of the process adequately. Furthermore, it is known that high acetate concentrations in the fermentation medium lead to inhibition of biomass growth. This was the reason for considering Andrews kinetics [22] in model 10. Unfortunately, the kinetics were not able to describe experimental results acceptably (Table 3).

For the sake of brevity, only the parameter estimates (together with the confidence intervals) of the most favored models, namely models 5, 7, and 9, are listed in Table 4. The calculated values obtained by simulation compared to the experimental data points, for biomass and pyruvate concentrations, are shown in Fig. 1.

Table 4 Estimated values and confidence intervals of parameters for the models 5, 7, and 9
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As it can be seen, all models are able to mirror the dynamic behavior of the process. The simulated biomass concentration curves obtained from model 5 were significantly lower than the measured values. Additionally, the simulated pyruvate curves of model 7 were higher than the experimental data (Fig. 1). In general, relatively large error bars were estimated for almost all parameters of model 5 (except for the parameters α and β of the Luedeking–Piret approach), which are probably caused by inadequate kinetic assumptions. On the contrary, relatively high parameter accuracies were achieved using the models 7 and 9.

Despite the high parameter accuracy and the relatively high model predictive quality of model 7 this model should be not favored because acetate was assumed to be the only limiting substrate for growth. As already stated, this simplifying assumption motivated the acetate auxotrophy of the E. coli strain. However, the strain was not capable of growing without glucose, which, basically, would necessitate considering additional intracellular biochemical reactions and thus obviously contradict the initial motivation to solely use unstructured models.

The predicted biomass and pyruvate curves of model 9 reflect well the experimental data. On the basis of a single criterion, the residual sum of squares, model 9 should thus be favored because it achieved the lowest residual sum of squares. Additionally, the estimated parameters in this model have acceptable confidence interval and the model is mechanistically correct. Furthermore, model 9 is mechanistically the most accurate one because it contains all experimentally observed effects: growth inhibition by pyruvate and pyruvate inhibited product (pyruvate) formation.

Some parameter values given in Table 4, can be qualified as biologically meaningless and are the result of the numerical Nelder–Mead parameter identification considering partially non-appropriate macrokinetic models. For example, the yield coefficients YX/G,max and also μ max appear to be too high. However, if large confidence intervals of these parameters (μ max, Y X/G,max, Y X/A,max, m G, m A, etc.) are taken into account the parameter values are fairly reasonable (Table 4). Preferably, these parameters should have been estimated directly from the experiments which—unfortunately—was not possible because of the properties of the E. coli YYC202 ldhA::Kan used. For example, cell growth is realized by the simultaneous consumption of the C sources glucose and acetate, thus making independent growth-parameter estimation in fed-batch experiments impossible. Nevertheless, a simple, easy-to-use, and robust model has successfully been identified, which can be used for typical process engineering applications such as process optimization and design. This model (model 9) and the estimated parameter are the first step in the quantitative understanding of the basic cellular characteristics of the genetically modified production strain used in this study.

4.2 Validation of the models

Kinetic modeling is still a difficult task for bioprocesses, where a succession of decisions must be made in respect of which systematic methodologies are still lacking. Therefore it is important to validate a model by using data other than those applied to identify parameters. In order to further validate the selected models (models 5, 7, and 9), further fed-batch experiments were performed at constant volumetric flow rate of glucose of q VG=20 and 30 mL h−1 as mentioned before. For model validation experimental data were compared to data obtained by model simulation for biomass and pyruvate concentrations (Figs. 2 and 3).

The good fitting quality of the model 9 to the biomass concentration and especially to the pyruvate concentration is remarkable for both volumetric glucose flow rates. The models 5 and 7 were not able to predict biomass and pyruvate concentrations with acceptable accuracy. Pyruvate levels predicted by those models were approximately 50% higher than experimental observations.

Unfortunately, all developed models were not able to predict glucose concentration profiles adequately especially when high glucose feeds were used (q VG=30 mL h−1). Rising glucose levels were suggested by all models for the pyruvate production period starting after 20 h fermentation time (data not shown). This was in accordance with the primary assumption that glucose is significantly used for biomass growth, which obviously did not occur during the pyruvate production. However, no glucose accumulation was observed experimentally indicating that a higher amount of glucose was consumed for pyruvate production, by-product formation and maintenance as primarily expected. In this regard, the working hypothesis is remarkable in that high extracellular pyruvate concentrations (which occur during the production phase) could cause futile cycling owing to active pyruvate export and diffusive pyruvate import (see [2]). Obviously, as yet, this significant impact has not been covered by the models, nor has the fact that approximately 50% of the acetate used is converted into CO2 production. Both aspects would necessitate a structured model formulation and could not be covered by the current modeling approaches used.

4.3 Sensitivity analysis

In order to qualify the sensitivity of model predictions with respect to estimated parameter errors, a sensitivity analysis has been performed [29]. The parameter values of model 9 (Table 4) were taken as reference values. The solutions of the model equations are then calculated with relative parameter errors ranging from −50% to +50% of the reference values. The final concentration of biomass c X and the final pyruvate concentration c P were than compared to the reference concentrations. All simulations were performed for the period of 35 h. The results are shown in Fig. 4.

Fig. 4
figure 4

Sensitivity analysis for the kinetic parameters of the model. The changes in the final value of the biomass (solid line) and pyruvate (dashed line) concentrations are represented with respect to the deviation of the nominal value of considered parameter

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Both, calculated biomass and pyruvate concentration, are sensitive to yield coefficients, Y X/G, Y X/A, Y P/G. This effect was expected because decreasing yields thus indicate reduced final titers provided that the same amount of substrate is used. Interestingly, there was almost no effect on the investigated concentrations when yield coefficients are higher than their nominal value.

The maintenance coefficient for growth on acetate, m A, was found to be the most sensitive parameter for the achievable biomass concentration. If m A was higher than the reference value, the final biomass concentration reduced linearly. On the contrary, there was no effect on the biomass titer if m A values were lower than the reference. m A also significantly affects the pyruvate concentrations (as shown in Fig. 4). The non-growth-associated product formation coefficient β was found to be most important parameter for product formation. Increasing β caused linear pyruvate concentration rises.

Both, calculated biomass and pyruvate concentrations are slightly sensitive to maintenance coefficient for growth on glucose m G and growth associated product formation coefficient α.

Finally, the model predictions are almost insensitive to the parameters μ max, K S G, K S A, and K P. Unfortunately, this clearly indicates the necessity to improve the parameter accuracy in further studies. For this, the optimal experimental design method focusing on continuous (or fed-batch) experiments should be applied [11, 13].

5 Conclusions

Ten simple, unstructured models have been developed for the bioconversion of glucose to pyruvate in a fed-batch process by genetically modified E. coli strain. All models were identified by the least-square fit and qualified by the MSC thus allowing the estimation of corresponding model parameters. It is noteworthy that the model identification was based on a different data set than the model validation. Within the investigated experimental range the model, which combines growth inhibition by pyruvate (Jerusalimsky approach), and pyruvate inhibited product formation (described by modified Luedeking–Piret/Levenspiel term) should be favored. Using this modeling approach, an acceptable model prediction for cell growth and pyruvate formation was achieved, which are both essential variables to model process alternatives and scale-ups. In the case of glucose, further studies must be performed to increase the model predictive quality.

In principle, even the favored model 9 represents only the first step in the modeling of the pyruvate production process, strictly speaking, by uncovering the limits of the unstructured model used. The need to incorporate additional aspects of energy demand or by-product formation was identified with necessities to extend the existing model by structured modeling terms. Nevertheless, the identified model 9 should be qualified as a promising tool for modeling studies as well as for further, more detailed, kinetic modeling approaches.