## 3 Bioprocess and Bioreactor Dynamic Model
### The Structured Hybrid Kinetic Model
[79], modified by [PERSON] [11; 44], with the following modifications: (i) PEP (from glycolysis) is the substrate of TRP synthesis, and the node coupling this synthesis with the glycolysis module; (ii) a novel model for the TRP-synthesis inhibition was proposed and identified from experiments. The model rate constants are estimated by [PERSON] [11] based on the experimental data of [PERSON] [72] (Table 1 and Figures 6, 8, 9 and 10) collected from a FBR using a modified _E. Coli_ T5 strain. The initial estimate of the rate constants is those of [PERSON] [11]. Species abbreviations are given in the abbreviation list. above, performs well, and it is suitable for further engineering calculations. The estimated rate constants were validated by comparison with the literature data [11; 43; 44; 54]. A comparison, presented by [PERSON] [11], of the model-estimated rate constants for the modified T5 _E. coli_ strain using the FBR experimental data presented by [PERSON] [72] with those of the same model, but estimated for the 'wild' _E. coli_ strain by [PERSON] et al. [55], (not presented here), revealed that most of the estimated rate constants present similar values. However, due to the mentioned modifications of the kinetic model for the used _E. coli_ T5 strain of this paper, some important differences are reported for the following: (i) the rate expression and parameters of the GLC import system (V1 in Tables 2 and 3), (ii) the biomass growing dynamics (Table 2), and (iii) the TRP-synthesis module [c] rate constants and expressions (Table 4). The bioreactor ideal model (Table 2, and Equations (1)-(4)) main hypotheses are the following [34]: (i) isothermal, iso-pH, iso-DO operation; (ii) it is self-evident that nutrients, additives, antibiotics, and pH-control compounds are added initially and during FBR operation to ensure the optimal grow of the biomass [72]; (iii) aeration in excess (even with pure oxygen [72]) over the batch to ensure an optimal biomass maintenance, and to contribute to the liquid homogeneity; (iv) perfectly mixed liquid phase (with no concentration gradients), of a volume increasing according to the liquid feed flow rate constant or time-varying strategy; (v) the limits of the liquid feed flow rate (\(F_{L,j}\) in Equation (12)) are adjusted so that they do not exceed the maximum bioreactor capacity (\(V_{L}\)) in Equation (12); (vi) negligible mass resistance to the transport of oxygen and compounds into the liquid and biomass flocks; (vii) GLC-substrate is initially added in the bioreactor and during the batch according to an optimal feeding mode to be determined; (viii) the feed flow rate during the batch \(F_{L,j}\) is varied according to an optimal feeding strategy to be determined for every 'time-arc' index 'j' in Equation (5) (that is an interval in which the batch time is divided). From a mathematical point of view, in a general form, the FBR HSMDM (Tables 2-4) translates to a set of 12 differential mass balances (ODE set) written for the key species of the FBR, as follows:  
Species in the bulk phase:  
\[\frac{dc_{i}}{dt}\ =\ \frac{F_{L,j}}{V_{L}}\Big{(}c_{inlet,i,j}-c_{i}\Big{)} \pm\ r_{i}(\boldsymbol{c}(t),\ \boldsymbol{c_{0}},\ \boldsymbol{k})\ ;\ \boldsymbol{c_{i,0}} =\boldsymbol{c_{i}}\ (t\ =\ 0) \tag{1}\]  
\"i\" denotes species present in the FBR bulk (GLC, X, TRP, PYR); \"j\" denotes the FBR feeding time-arcs; j = 1, \(\ldots\), N\({}_{\rm div}\). Key-species inside cells:  
\[\frac{dc_{i}}{dt}\ =\ \pm\ r_{i}(\boldsymbol{c}(t),\ \boldsymbol{c_{0}}, \ \boldsymbol{k})-\mu c_{i}\ \ ;\ \ \boldsymbol{c_{i,0}} =\boldsymbol{c_{i}}\ (t\ =\ 0) \tag{2}\]  
In the Equations (1) and (2), index \"i\" denotes the species inside cells; that is, (GLC, F6P, FDP, PEP, PYR, ATP) for the glycolysis, and (OR, MRNA, E, TRP) for the TRP-operon expression.