## 5 Discussion  
It is verified in Figure 2 that the Markov Chains showed a tendency to converge to the equilibrium distribution for the constant \(q\) of the different evaluated models. For this parameter, less than 200 states were needed for the Markov Chain to converge to the value of approximately 1.402. It is observed in Table 4 that models 1, 2, and 3 presented values very close to each other. However, it is noted that for model 3, most parameters presented results closer to the reference values, when compared to models 1 and 2. When comparing the value of the reference yield coefficient, \(Y_{\text{cell/Fe}}\)\({}^{2+}\) = 2.25 \(\times\) 10\({}^{9}\) cells/(gFe\({}^{2+}\)), with the results obtained for models 1, 2, and 3, it is observed that they are in the range of 2.24 \(\times\) 10\({}^{9}\) to 2.34 \(\times\) 10\({}^{9}\) cells/ (gFe\({}^{2+}\)). Previous studies reported values of 2.5 \(\times\) 10\({}^{10}\) cells/ (gFe\({}^{2+}\)) [47], 2.4 \(\times\) 10\({}^{10}\) cells/ (gFe\({}^{2+}\)) [48], 2.23 \(\times\) 10\({}^{10}\) cells/ (g Fe\({}^{2+}\)) [49], 1.7 \(\times\) 10\({}^{10}\) cells/ (gFe\({}^{2+}\)) [50], and 2.3 \(\times\) 10\({}^{10}\) cells/ (gFe\({}^{2+}\)) [51]. \(Y_{\text{cell/Fe}}\)\({}^{2+}\) represents the  
Figure 3: Comparison of estimated state variables with experimental measurements [19]: (**a**) case 1 of model 1; (**b**) case 2 of model 2; (**c**) case 3 of model 3; and (**d**) case 4 of model 3. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**State Variables** & **Model 1** & **Model 2** & **Model 3** \\ \hline
[Cell] & 0.9617 & 0.9630 & 0.9559 \\
[U] & 0.9978 & 0.9963 & 0.9941 \\
[Fe\({}^{2+}\)] & 0.9904 & 0.9925 & 0.9934 \\
[Fe\({}^{3+}\)] & 0.9629 & 0.9574 & 0.9624 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Values of determination coefficients (R\({}^{2}\)) for case 3 of models 1, 2, and 3.  
yield of cells produced per gram of Fe\({}^{2+}\) oxidized, reflecting the efficiency of bacterial metabolism [52]. The estimated values for this parameter show that bacterial growth is favored by the higher availability of Fe\({}^{2+}\) as an energy source, as expected. The values of the present work differ from those reported in the literature due to the different experimental conditions, although present the same order of magnitude of the values. Comparing the maximum rate of specific bacterial growth, \(\mu^{max}\) = 0.086 h\({}^{-1}\), with the results obtained for the different models, it is observed that they are in the range of 0.093 to 0.098 h\({}^{-1}\). The literature reports results in the same order of magnitude as 1.78 h\({}^{-1}\)[23; 53], and 0.047 h\({}^{-1}\)[54]. \(\mu^{max}\) represents the speed at which the cells divide, directly affecting the kinetics of the bioleaching process [55]. The estimated values for this parameter show that bacterial growth is limited by the low concentration of Fe\({}^{2+}\), as reported in [PERSON] et al. [56]. Similarly, it was also verified for the substrate saturation constant, \(k_{s}\) = 2.49 \(\times\) 10\({}^{3}\) g/m\({}^{3}\), where results for models 1, 2, and 3 are in the range of 2.79 \(\times\) 10\({}^{3}\) to 3.06 \(\times\) 10\({}^{3}\) g/m\({}^{3}\), which have the same order of magnitude as the values presented in the literature, which are 0.3 \(\times\) 10\({}^{3}\) g/m\({}^{3}\)[57]. In general, it was observed that the results obtained from the parameter estimates for models 1, 2, and 3, presented the same order of magnitude as the values found in the literature, with some differences due to the experimental conditions. This parameter represents the concentration of Fe\({}^{2+}\) at which the bacterial growth rate is half of the maximum rate, indicating the sensitivity of the bacteria to the substrate [58]. The estimated values for this parameter show that the bacteria are able to grow at low concentrations of Fe\({}^{2+}\), which is an advantage for the bioleaching process, as it reduces the need for nutrient addition [59].