## 3 Bioprocess and Bioreactor Dynamic Model
### The Structured Hybrid Kinetic Model
Biomass in the bulk phase:  
\[\frac{dX}{dt}\ =\ r_{X}(\boldsymbol{c}(t),\ \boldsymbol{c_{X,0}},\ \boldsymbol{k})\ \ ;\ \ \boldsymbol{c_{X,0}} =\boldsymbol{c_{X}}\ (t\ =\ 0) \tag{3}\]  
Liquid volume dynamics:  
\[\frac{dV_{L}}{dt}\ =\ \ F_{L,j}\ \ ;\ \ \boldsymbol{V_{L,0}} =\boldsymbol{V_{L}}\ (t\ =\ 0) \tag{4}\]  
In the Equation (4), index \"j\" denotes the FBR feeding time-arcs; j = 1, \(\ldots\), N\({}_{\rm div}\). In Equation (1), \(c_{inlet,i,j}\) refers to the concentration of the species i in the feeding solution, constant over the time interval j ( j = 1, \(\ldots\), N\({}_{\rm div}\)). In the present case only, GLC is fed in the FBR during the batch. The reaction rate r\({}_{1}\) expressions, together with the associated rate constants and other details, are given in (Tables 2-4). In Equations (1)-(3), \(c\) = vector of species concentrations; \(c_{\theta}\) = initial value of \(c\) (at time t = 0) given in (Table 1); \(k\) = vector of the model rate constants. The reactor content dilution in Equation (1), determined by the increasing V\({}_{\rm L}\) in Equation (4), is due to the continuously added F\({}_{\rm L}\) term. In Equation (1), GLC and F\({}_{\rm L}\) are the control variables. The optimal \(F_{L,j}\) can be constant, or it can be variable. In the last case, it has to be determined for every time-step-wise interval, over j = 1,..., N\({}_{\rm div}\) time-arcs. For instance, for the adopted N\({}_{\rm div}\) = 5, the j = 1,..., and N\({}_{\rm div}\) time-arc switching points given in Equation (5) are the following: T1 = \(t_{f}/\)N\({}_{\rm div}\) (12.5 h); T2 = 2 \(\times\) T1 (25 h); T3 = 3 \(\times\) T1 (37.5 h); T4 = 4 \(\times\) T1 (50 h); \(t_{f}\) = 63 h. More specifically:  
Feed flow rate strategy in the time-step-wise operation of the FBR:  
\[F_{L,j}\ =\ \left\{ \begin{array}{ll}F_{L,0}&\mbox{if}\ \ 0\ \leq\ t<T1\\ F_{L,1}&\mbox{if}\ \ T1\ \leq\ t<T2\\ F_{L,2}&\mbox{if}\ \ T2\ \leq\ t<T3\\ F_{L,3}&\mbox{if}\ \ T3\ \leq\ t<T4\\ F_{L,4}&\mbox{if}\ \ T4\ \leq\ t<t_{f}\end{array} \right. \tag{5}\]  
Similarly, if a variable feeding strategy was chosen, for the adopted N\({}_{\rm div}\) = 5 equal time-arcs, the GLC solution concentration feeding strategy in a time-step-wise operation of the FBR is the following:  
\[c_{glc,j}^{feed}\ =\ \ \left\{ \begin{array}{ll}c_{glc,0}^{feed}&\mbox{if}\ \ 0\ \leq\ t<T1\\ c_{glc,1}^{feed}&\mbox{if}\ \ T1\ \leq\ t<T2\\ c_{glc,2}^{feed}&\mbox{if}\ \ T2\ \leq\ t<T3\\ c_{glc,3}^{feed}&\mbox{if}\ \ T3\ \leq\ t<T4\\ c_{glc,4}^{feed}&\mbox{if}\ \ T4\ \leq\ t<t_{f}\end{array} \right. \tag{6}\]  
Ways to intensify the TRP production in the FBR: As revealed by the literature [11; 43; 44; 72], the cell internal and external factors determine whether the TRP production increases strongly, as follows: (1) \"The glycolysis intensity (GLC uptake flux, and the average levels of the glycolytic species), transmitted to the TRP-synthesis module [c] via the shared PEP intermediate, and (2) The glycolysis dynamics (QSS, or oscillatory behaviour). On the other hand, the glycolysis intensity is controlled by several cell internal and external factors, well reviewed by [PERSON] [43; 45];\" (3) The optimal operation type of the FBR--while the factors of (1) and (2) can be influenced by modifying the _E. coli_ characteristics, by developing a GMO [47; 72], factor (3) can be solved by applying an in silico, off-line engineering analysis (detailed in this paper).