## 3 Bioprocess and Bioreactor Dynamic Model
### The Structured Hybrid Kinetic Model
\begin{table}
\begin{tabular}{l l l} \hline \hline \multicolumn{1}{c}{**Reactions**} & \multicolumn{1}{c}{**Rate Expressions**} & \multicolumn{1}{c}{**Estimated Rate Constants**} \\  & & & **(Units in mM, min)** \\ \hline GLC import system & & & \(r_{uptake}^{max}=1.0902\) (mM/min) \\ glc + pep \(\rightarrow\) f6p + pyr & Modification for the T5 strain & & \(K_{PTS,41}=3581.8\) (mM) \\ pyr + atp \(\rightarrow\) pep + adp + h & \(V_{1}\ =\ r_{uptake}\ =\ \frac{r_{uptake}^{max}+r_{\mathrm{eff}}^{max}}{\left(K_{PTS,41}+c_{ \mathrm{eff}}^{min}\right)}\) & & \(K_{PTS,42}=0\) \\ glc + atp \(\rightarrow\) f6p + adp + h & & & \(K_{PTS,43}=0\) \\ \hline \hline \multirow{4}{*}{f6p + atp \(\rightarrow\) fdp + adp + h} & \multirow{2}{*}{\(V_{2}\ =\ r_{PFK}\ =\ \frac{(V_{1}/V_{2m})_{cl}^{d}}{\left(K_{\mathrm{Lm}}^{d}+K_{\mathrm{Lm}}^{d} \left[\frac{K_{\mathrm{Lm}}^{d}}{x_{\mathrm{eff}}^{d}}\right]^{d}\left(\frac{ c_{\mathrm{eff}}}{x_{\mathrm{eff}}^{d}}\right)^{n}+c_{dp}^{d}\right)}\). & \(K_{\mathrm{Zm}}=6.16423\) (mM) \\  & & & \(K_{\mathrm{R}}^{\mathrm{LP}}=25\)\(\mu\)M \\  & & & \(K_{\mathrm{R}}^{\mathrm{LP}}=60\)\(\mu\)M \\ \hline \hline \multirow{4}{*}{fdp + 2 adp (+ 2 nad + 2 p) \(\leftrightarrow\) 2 pep + 2 atp (+ 2 nad + 2 h 2o)} & & \(k_{3}=4602.3\) (1/min) \\  & & & \(k_{3p}=31.917\) (1/min) \\
2 apt (+ 2 nadh + 2 h + 2 h2o) & & \(\alpha=0.05\) \\  & & & \(\beta=3\) \\ \hline \hline \multirow{4}{*}{pep + adp + h \(\rightarrow\) pyr + atp} & \multirow{2}{*}{\(V_{4}\ =\ r_{PK}\ =\ \frac{(V_{1}/V_{4m})_{cl}^{d}}{\left(K_{\mathrm{Lm}}^{d}+K_{\mathrm{Lm}}^{d} \left[\frac{x_{\mathrm{eff}}^{d}}{x_{\mathrm{eff}}^{d}}\right]^{d}\left(\frac {c_{\mathrm{eff}}}{x_{\mathrm{eff}}^{d}}\right)^{n}+c_{dp}^{d}\right)}\). & \(\gamma=1.331879\) \\  & & & \(m=4\) \\ \cline{1-1}  & & & \(V_{4m}=1.333655\) (mM/min) \\ \cline{1-1}  & & & \(K_{4m}=1.146443\) (mM) \\ \cline{1-1}  & & & \(K_{K_{\mathrm{R}}^{ddp}}=0.2\) (mM) \\ \cline{1-1}  & & & \(K_{\mathrm{T},PK}^{dpp}=9.3\) (mM) \\ \hline \hline \multirow{4}{*}{pyr \(\rightarrow\) products (accoa, cit, succ, lac, etc,...)} & \multirow{2}{*}{\(V_{5}\ =\ \frac{k_{5}c_{symm,py}^{clean}+c_{py}}{K_{compm,py}+c_{py}}\).} & \(k_{5}=9913.5\) (1/min) \\ \cline{1-1}  & & & \(K_{\mathrm{Consum,pyr}}=395.525\) (mM) \\ \cline{1-1}  & & & \(n_{\mathrm{Consum,pyr}}=2.6814\) \\ \hline \hline \multirow{2}{*}{2 adp \(\rightarrow\) atp + amp} & \multirow{2}{*}{\(V_{6}\ =\ k_{6}c_{alp}\).} & \multirow{2}{*}{\(k_{6}=552.38\) (1/min)} \\ \cline{1-1}  & & & \(c_{alp}c_{amp}\ =\ K_{alp}^{2}\). & \(K=1\) \\ \cline{1-1} \cline{2-2} \cline{4-4} \cline{1-1}  & & & (i) Observations by [PERSON] and [PERSON] [82; 83] indicated experimental evidence of a very fast reversible reaction catalyzed by AKase, with the equilibrium being reached quickly. (ii) The k6 constant takes values according to the micro-organism phenotype (related to the gene encoding the enzyme ATPase that catalyzes this reaction). (iii) \(c_{amp}+c_{alp}+c_{alp}\ =\ c_{amdtp}\ =\) constant [82; 83]; (iv) \(c_{alp}\) results from solving the thermodynamic equilibrium relationship: \(c_{amp}c_{alp}\ =\ K_{alp}^{2}\) that is: \(c_{alp}^{2}\frac{K_{\mathrm{c}}}{c_{alp}}+c_{alp}-c_{amdtp}+c_{alp}\ =\ 0\). \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Reaction rate expressions V1-V6 of the HSMDM model of (Table 2), describing the dynamics of the cellular glycolytic species according to the kinetic model by [PERSON] [11; 54], and by [PERSON] et al. [55]. In the present study, this glycolysis kinetic model was modified by replacing the PTS system (V1 flux) for the GLC uptake in the “wild” _E. coli_ with those of the mutant T5 _E. coli_ strain tested in this paper, and by [PERSON] [11]. The model rate constants were estimated by [PERSON] [11] to fit the experimental data of [PERSON] [72] presented in (Table 1 and Figures 6, 8, 9 and 10). Species abbreviations are given in the abbreviation list. in (Table 4). More details on this module construction are given by [PERSON] and [PERSON] [32]. In fact, as remarked by [PERSON] et al.