## Materials and Methods  
### Mathematical Model of the Biotechnological Process  
In this study, the biotechnological process was simulated using the mathematical model of the _Escherichia coli_ BL21 strain. This strain also harbored a pBR322 plasmid derivative and was cultivated in a recombinant fed-batch process. Mainly because this recombinant protein is used in typical bioreactor-scale processes. The cultivation of the selected recombinant protein can be described as a two-phase process. The target of the first phase is to accumulate the bioreactor with a sufficient biomass. The specific growth rate is usually relatively high for this phase. During phase two, the recombinant protein is produced. During the described two phases, different optimal temperatures were defined and maintained for each phase. In the simulated process, the broth temperature was maintained at 37 \({}^{\circ}\)C during the first phase thus maximizing the biomass growth to an optimal level. At the start of the production process, the temperature was reduced to 32 \({}^{\circ}\)C. The substrate feeding rate profiles and the induction time also have a significant influence on the process performance. Therefore, they are also subject to model-based optimization. The induction time (8 h) for the investigated process of a given total duration was determined in [17] using model-based optimization techniques. During the model-based optimization of the process [17], the process performance (productivity) index equal to the total target protein amount at the end of the process was used. The off-line measurement techniques applied for biomass and target protein analysis in the investigated process are described in detail in [17]. For the online measurements of the biomass concentrations, alternative techniques based on turbidity (optical density), permittivity measurements or off-gas analysis can be used. For protein measurement, a promising alternative to the applied techniques are model-based approaches (application of soft-sensors, generic estimators, e.g., [18]) that would enable the online monitoring of the key process variables. A mathematical model presented in [17] was used to simulate the described biotechnological process. According to other studies, pre-optimized feeding rate profiles are used to control the biomass growth and protein production [17]. This type of open-loop system can be controlled effectively only then when no considerable process condition deviations or equipment disturbances/malfunctions take place. Failure to do so may lead to deviations in the specific growth rate thus reducing the process productivity. Therefore, the SGR was selected as the main controlled variable of this research that would be maintained by controlling the substrate feeding rate. In this study, an adaptive closed-loop control algorithm was used for keeping the optimal SGRs trajectory on track. The discussed process can be described using the bellow provided differential equations. \[\frac{dx}{dt}=\mu(s,T)x-u\frac{x}{w} \tag{1}\]  
\[\frac{ds}{dt}=-q_{s}(s,T)x+u\frac{s_{f}-s}{w} \tag{2}\]  
\[\frac{dp_{x}}{dt}=q_{px}(\mu,p_{x}) \tag{3}\]  
\[\frac{dw}{dt}=u+F_{smp} \tag{4}\]  
where \(x\) (g/kg) describes the biomass concentration. \(\mu\) (1/h) is the biomass specific growth rate that is a function of the glucose concentrations (g/kg) and culture broth temperature \(T\) (\({}^{\circ}\)C). Here, \(w\) (kg) is the culture broth weight, \(q_{s}\) (g/(gh)) is the glucose specific consumption rate. The glucose concentration in the feeding solution is noted as \(s_{f}\) (g/kg) while \(p_{x}\) (U/(g biomass)) is the specific protein activity. \(q_{px}\) (U/(gh)) is the specific protein accumulation rate; \(u\) (kg/h) and \(F_{smp}\) (kg/h) are the substrate feeding and the sampling rates, respectively. SGR is modeled using the Haldane-type model [19]:  
\[\mu(s,T)=\mu_{max}\frac{s}{K_{s}+s}\frac{K_{i}}{K_{i}+s}\textit{exp}\big{(} \alpha\big{(}T-T_{ref}\big{)}\big{)} \tag{5}\]  
where \(\mu_{max}\) (1/h) is a maximal specific growth rate parameter.