## 3 Bioprocess and Bioreactor Dynamic Model
### The Structured Hybrid Kinetic Model
[73], the PEP precursor is the limiting factor for the TRP synthesis. For such a reason, intense efforts have been made to increase its production by glycolysis intensification. This can be realized by optimizing the FBR operating strategy (as is carried out in this paper), and/or by using the modified _E. coli_ T5 strain culture of [PERSON] et al. [47; 72] (also carried out in this paper). The operation of the FBR. A simple ideal model of FBR was adopted to describe the key-species dynamics during the batch at a macroscopic level (GLC, X, TRP, and PYR in bulk), but linked to the cell species dynamics [ that is, (GLC, F6P, FDP, PEP, PYR, ATP) for the glycolysis, and (OR, MRNA, E, TRP) for the TRP-operon expression] (Tables 2-4) [11]. In total, the developed HSMDM includes 49 rate constants to be estimated from the experimental kinetic curves of four observed species (GLC, TRP, PYR, X), with each species time-trajectory including 17 quasi-uniformly distributed recorded points (Figures 6, 8, 9 and 10). This estimation problem is equivalent to the high-difficulty nonlinear programming problem (NLP) of [11; 48] due to its dimension, and the high nonlinearity of the model and constraints. Details about the used sequential estimation approach to avoid unfeasible local estimates are given by [PERSON] [11]. The estimated rate constants using the experimental data of [PERSON] [72], and the glycolysis initial estimation of [PERSON] [54] are given in (Tables 2-4). A standard weighted least-square estimator (WLSQ) was employed, as it was the most suitable statistical estimator for this case study, due to the different standard deviations and units of the observed species [11; 48]. As proven by the plotted results given in (Figures 6, 8, 9 and 10), the species time-trajectories (the blue curves) predicted by the HSMDM fit the experimental data fairly well (the blue circles). Consequently, it can be concluded that the HSMDM of the FBR, adopted \begin{table}
\begin{tabular}{l l} \hline \hline \multicolumn{1}{c}{**Rate Expression**} & \multicolumn{1}{c}{**Kinetic Model Parameters**} \\  & **(Units in mM, \(\mu\)M, min)** \\ \hline  & \(k_{1}\)= 59.062, 1/(min\(\cdot\)mM) \\ \(\frac{d_{G\alpha}}{dl}=k_{1}c_{OT}\)C\({}_{1}(c_{rp})-k_{d1}c_{OR}-\mu c_{OR}\) & \(k_{d1}\)= 0.5443, 1/min \\ \(\frac{d_{G\alpha}}{dl}=k_{2}c_{OR}\)C\({}_{2}(c_{rp})-k_{d2}c_{MRNA}-\) & \(k_{2}\)= 17.796, 1/min \\ \(\frac{d_{C\alpha}}{dl}=k_{3}c_{MRNA}-\mu c_{E}\) & \(k_{3}\)= 1.157, 1/min \\ \(\frac{d_{C}}{dl}=k_{3}c_{MRNA}-\mu c_{E}\) & \(K_{i,1}\)= 3.55, 1/M \\ \(C_{1}(c_{rp})=\frac{K_{i}^{\mu}}{K_{i,1}^{\mu}+c_{rp}^{\mu}};\) & \(C_{2}(c_{rp})=\frac{K_{i}^{1/2}}{K_{i,2}^{1/2}+c_{rp}^{1/2}}\) & \(n_{H}\)= 1.92 \\  & \(K_{i,2}\)= 0.04, 1/M \\  & (see Footnote 1) \\ \hline \(\frac{d_{c_{rp}}}{dl}=D\Big{(}c_{rpj}^{fead}-c_{rp}\Big{)}+r_{rpj};\)\(c_{rpj}^{fead}=0;\) & \(g=-0.32\) \\ \(r_{rpj}=\big{(}c_{rpp}c_{E}\big{)}^{8}\frac{H^{\mathrm{G}c_{rp}}}{(\mu c\exp( \mu_{T}))^{\mathrm{G}c_{rp}}}-\mu c_{rpj}.\) & \(\mu_{T}\)= 0.36365, 1/min \\  & \(n_{T}\)= 3.9923 \\  & \(b_{T}\)= 0.017153, 1/min \\  & \(N_{T}\)= 0.071515 \\ \hline \hline \end{tabular}  
* Footnote: (a) The adopted modification for the TRP-synthesis inhibition replaces the C3 variable of the [PERSON] et al. [9] model (not displayed here) with a modified [PERSON] model [85], including a power-law inhibition with TRP growth at the denominator. (b) The nitrogen source in the TRP synthesis is considered in excess and included in the \(\mu_{T}\) constant. In order to connect this module to the to the glycolysis kinetic model, the PEP species dynamics, generated by the glycolysis was explicitly included in the TRP-synthesis rate as a substrate. (c) The initial concentrations of the TRP-operon species (OR, MRNA, E, TRP) are given in Table 1.  
\end{table} 
Table 4: Species’ mass balance in the _TRP_-operon expression kinetic model of [PERSON] et al.