Previously stored In the fedbatch culture,
concentrations of glucose, NH
Fermentation was carried
out using a 5 l BioFlo 310 type fermentor (New Brunswick Scientific Co., USA) equipped with pH, dissolved oxygen (DO) (Mettler Toledo Co., Switzerland)
and temperature sensors. BioProfile 300A Biochemical analyzer together with the feedback control box (Nova Biomedical Co., USA) was used for off-line (for
batch culture) and on-line glucose, NH
Following dynamic equations were used in the present study:
Where, S is the glucose concentration (g/l); X is the cell
concentration (g/l); X is the maximum cell concentration (g/l); _{m}k is the saturation constant (g/l);_{S} q_{S }_{(}g/g/h), q (mmol/g/h) and _{N}q (mmol/g/h) are the
specific consumption rates of glucose, NH_{P}_{4}^{+}, and PO_{4}^{3-}; m is the maintenance coefficient defined by the amount of
glucose consumed per gram of cells per hour for maintenance (g/g/h); _{S}Y is the cell yield from glucose (g/g); _{X/S}Y is the theoretical
cell yield from glucose (g/g); _{G}Y is the cell yield from NH_{X/N}_{4}^{+} (g/mmol); Y is the cell yield from PO_{X/P}_{4}^{3-} (g/mmol).
Mass balance equations were used as follows:
Where,
The initial parameter
values of X,_{m} Y,_{G} Y, and_{X/N} Y were calculated from the
experimental data. The initial values of _{X/P}k and _{S}m were estimated to be 1 and theoretically calculated as shown in next
section, respectively. Subsequently, genetic algorithm (GA) (Goldberg, 1989) was used to refine the
parameter values. Small spans were given around the above parameter values,
within which GA was used to search the optimal parameter values with the least
sum of square errors (_{S}Err) as described in Equation [9], between
the model prediction and the experimental data.
Where, S´,_{i} N´ and_{i} P´are the
measured _{i}ith concentrations of the cells, glucose, NH_{4}^{+} and PO_{4}^{3-}, respectively; X,_{i, }S_{i} N and_{i} P are the model predicted values
corresponding to _{i}X_{i}^{´}, S´, N´and_{i} P´,
respectively. The denominator of each item in Equation [9] ensures an
equal contribution from each item to _{i}Err.In application of GA in
parameter optimization (Figure 1), a population containing
Figure 2 shows that the model prediction was quite accurate, confirming the GA efficiency in the model parameter identification. GA is an optimization algorithm developed by imitating the evolution of a biological population, and it is efficient especially in optimizing nonlinear and sophisticated systems (Goldberg 1989). The perfect fits of the model prediction with the experimental data also indicated that the developed mathematical model was reasonable (Figure 2), and it can be used in the analysis of the fermentation process. Table 1 shows that the values of Y optimized using GA are 0.013 g/g/h and 0.32 g/g, respectively. According to Equation [1], the averaged value of _{G}µ was 0.28 1/h. Equation [10] can be obtained from Equation [2]. Accordingly, Y was calculated to be 0.29 g/g from Equation [10] using the above values. _{X/S}
Equation [10] indicates that the consumed glucose is composed of two parts; one part is used
in the real growth, and the other one part is used in the maintenance. The
typical values of maintenance coefficient for ATP ( m in the range of 0.01~0.02 g/g/h for eukaryotic cells in the case of aerobic catabolism of glucose through tricarboxylic acid (TCA) cycle with P/O of 3. Using the values of _{S}m and averaged value of _{S}µ, the item of m/_{s}µ in Equation [10] was calculated between 0.036 and 0.071 g/g. The consumed glucose in the real growth also includes two parts. One part is converted to cellular building blocks (ΔS),
and the other part is used to produce the energy (Δ_{Cell}S) for polymerization of the building blocks into macromolecules.
For the glucose converted to Δ_{ATP}S,
about 91.3% of the glucose can be converted for the decarboxylation occurred in
the pathways of Δ_{Cell}S synthesis (Heijnen 1992). An empirical
molecular formula for _{Cell}Candida utilis cell composition, CH_{1.84}O_{0.56}N_{0.2}P_{0.01} with the molecular weight (M) of 25.8, was modified besed
on the previous study (Heijnen 1992) and
used in the calculations. The total amount of the produced cells was 26.2 g cells/l, containing 1.02 mole carbon/l (26.2/_{w_cell}M), and it required 33.52 g glucose/l (1.02/91.3% x _{w_cell}M/6, _{w_gluc}M the molecular weight of glucose) for Δ_{w_gluc}S in syntheses of cellular building blocks. On the other hand,
energy is required in the polymerization of cellular building blocks into
macromolecules, which is defined by_{Cell} Y with the typical
value of 10 g cells /mole ATP. The produced 26.2 g cells/l required 2.62 mol ATP/l. When glucose is metabolized by eukaryotic cells under the aerobic condition through TCA cycle, 36 mole ATP is produced when consuming
1 mole glucose. The synthesis of 26.2 g cells/l consumed 13.1 g glucose/l for the ATP production (Δ_{ATP}S)
((2.62/36) x _{ATP}M). Totally, 46.62 g glucose/l (Δ_{w_gluc}S + Δ_{ATP}S) was
needed for the real growth of 26.2 g cells/l. Therefore, _{Cell}Y was calculated to be 0.56 g/g according to Equation [11]. Then, _{G}Y can be theoretically calculated between 0.54
and 0.55 g/g, which is much larger than 0.29 g/g obtained from the model predicted parameters using Equation [10]. The reason is that the above assumption that the glucose consumed for Δ_{X/S}S and _{ATP}m were fully oxidized under the aerobic condition
through TCA cycle is not true. In fact, NaOH was added to control the pH during
the fermentation, indicating that the organic acid was produced and the glucose
was partially metabolized through the fermentation. The results showed that the
fermentation process could be further improved in order to increase_{S} Y.
For example, for a more efficient use of glucose, glucose metabolism can be
directed to TCA cycle through glucose concentration control and DO control._{X/S}
Repeated fedbatch culture was performed as described in the section of Materials and Methods. Fedbatch culture is the most commonly used method in substrate concentration control. If high concentrated feeding solution is used and the volume change of the fermentation broth can be neglected or compensated by evaporation, the mathematical model developed in last section can also be used in the repeated fedbatch culture without any change. The simulation of repeated fedbatch culture was performed, fulfilled by resetting the initial values of the differential equations to the step changed concentrations at the start of the pulse feeding and solve the equations from this time point until the start of the next pulse feeding, like a new batch culture after another. This method made the simulation of the repeated fedbatch culture easier. The measured time course
data were used to re-optimize the model parameter values of the fedbatch
culture using GA. The optimized values were Y 0.11 g/mmol,_{X/N} Y 1.5 g/mmol and the other values were the same with Table 1. By the computer simulation, Figure 3 shows that the glucose concentration was well controlled
with small fluctuations, NH_{X/P}_{4}^{+} concentration was well
controlled but with larger fluctuations, while PO_{4}^{3-} was
depleted from 10 to 16 hrs resulted from the late in the start of the feeding
of PO_{4}^{3-} solution. The obtained maximum cell
concentration was 71 g/l, which was more than doubled compared with that of the
batch culture of 26 g/l.According to Equation
[10], the portion of the glucose consumed on maintenance is increased when Y in repeated fedbatch
culture, we carried out the computer simulation (Figure 3). The time
courses of _{X/S}µ of the batch and repeated fedbatch cultures were compared,
and Figure 3 shows that µ of the
repeated fedbatch culture was higher than that of the batch culture resulted
from the substrate feeding. As a result, Y was maintained at
a relatively high level by using repeated fedbatch culture (Figure 3),
indicating the usefulness of repeated fedbatch culture in process optimization.
Compared with the continuous feeding, repeated fedbatch culture has several
advantages, such as ease of operation and less contamination. Therefore, it is
often used in industry. The results of fermentation operations can be predicted
through using computer simulation with the mathematical model, which is
efficient, labor-saving and cost effective. _{X/S}In the present study, we
developed a mathematical model for the production of
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