## 2 Materials and Methods
### Model Evaluation and Validation  
To evaluate the kinetic model for each medium, root-mean-square-error (RMSE), mean-absolute-error (MAE), regression coefficient (\(R^{2}\)), slope, bias factor (BF), and accuracy factor (AF) were used as follows: The RMSE (Equation (32)) has been applied to be a standard statistical metric to evaluate model performance in biotechnological processes or other processes and the MAE (Equation (33)) is also commonly used measure in model assessments [39]. After obtaining the predicted data using kinetic models, they were plotted versus experimental data and thus the \(R^{2}\) values were determined.  
\[RMSE = \sqrt{\left[\sum_{t=1}^{n}\left(x_{t}-y_{t}\right)^{2}\right]/n}, \hskip 28.452756 ptt = \ 1,\ 2,\ \ldots\ldots\ldots\ldots\ldots\ldots,\ n \tag{32}\]  
\[MAE = \frac{1}{n}\sum_{t=1}^{n}\left|x_{t}-y_{t}\right|,\hskip 28.452756 ptt = \ 1,\ 2,\ \ldots\ldots\ldots\ldots\ldots\ldots\ldots,\ n. \tag{33}\]  
Additionally, the BF (Equation (34)) and AF (Equation (35)) have been suggested to validate the kinetic model [40]. A kinetic model would ideally have AF = BF = 1 in the case of no structural deviation, illustrating the exact match between actual observation and model predictions. If \(1\leq\text{AF}<1.20\), the model is \"good\", if \(1.20\leq\text{AF}\leq 1.30\), the model is \"acceptable\", and if \(\text{AF}>1.30\), the model is \"unacceptable\". If \(0.95\leq\text{BF}\leq 1.11\), the model is \"good\", if \(0.87\leq\text{BF}<0.95\) or \(1.11<\text{BF}\leq 1.43\), the model is \"acceptable\", and if BF < 0.87 or BF > 1.43, the model is \"unacceptable\" [41].  
\[BF = 10^{\sum_{t=1}^{n}-\frac{\log\left(x_{t}/y_{t}\right)}{n}},t = \ 1,\ 2,\ \ldots\ldots\ldots\ldots\ldots\ldots\ldots,\ n \tag{34}\] \[AF = 10^{\sum_{t=1}^{n}\frac{\log\left(y_{t}/x_{t}\right)}{n}},t = \ 1,\ 2,\ \ldots\ldots\ldots\ldots\ldots\ldots\ldots,\ n \tag{35}\]  
where \(n\) is the number of observations (n = 7) and \(x_{t}\) and \(y_{t}\) represent the experimental and predicted data at time \"\(t\)\" (g/L), respectively.  
In order to assess the fitting accuracy of the kinetic models, the objective function (\(\varPhi\)-value, Equation (36)) was used. The parameters of kinetic models have been calculated to minimize \(\varPhi\)-value:  
Figure 2: A growth curve, where \(A_{0}\) is the lower asymptote, \(A_{m}\) is the upper asymptote (\(A_{m}=\ln N/N_{0}\)), \(Q\) is the maximum growth rate (g/L/h), lag-time (\(\lambda\)) is \(x\)-axis intercept of the tangent in the inflection point, ln (\(N/N_{0}\)) is the logarithm of the relative population size.  
\[\Phi-\text{value}\ =\ \frac{1}{\left(X_{\text{max}}\right)^{2}}\sum\limits_{All\ \text{times}}\left(X_{\text{exp}}-X_{\text{pre}}\right)^{2}+\frac{1}{\left(P_{ \text{max}}\right)^{2}}\sum\limits_{All\ \text{times}}\left(P_{\text{exp}}-P_{\text{pre}}\right)^{2} \tag{36}\] \[+\frac{1}{\left(S_{\text{max}}\right)^{2}}\sum\limits_{All\ \text{times}}\left(S_{\text{exp}}-S_{\text{pre}}\right)^{2}\]  
where \(X_{\text{max}}\), \(P_{\text{max}}\) and \(S_{\text{max}}\) are the maximum observations of the relevant measurements and subscripts \"exp\" and \"pre\" denote the experimental and calculated values, respectively. Since the statistic is not dependent on the size of the measurements, the \(\Phi\)-value was used. Predictions for model parameters were attained by minimizing the \(\Phi\)-value [42].