## 2 Direct Model  
Considering the phenomenon based on the bioleaching mechanism, a mathematical model was established to describe the bioleaching process of uranium by _Acidithiobacillus ferrooxidans_, where a phenomenon can occur directly in which the microorganism attacks the metal, or indirectly, when the byproduct of the metabolism of the microorganism oxidizes the metal, solubilizing it [15; 16]. However, for the present model, the indirect bioleaching mechanism was considered to evaluate the phenomenon in which microorganisms do not maintain direct contact with the mineral during the process; that is, the agents are created by microbes that will oxidize the ore, as illustrated in Figure 1[17; 18; 19]. In Equation (1), the metabolic reaction for bioleaching in the indirect mechanism is presented, which represents the global stoichiometry for Fe(II) oxidation [20]. \[\text{Fe}^{2+}+0.2223\text{O}_{2}(\text{g})+0.0278\text{CO}_{2}(\text{g})+0.0056 \text{NH}^{4+}+0.9944\text{H}^{+}\rightarrow\text{Fe}^{3+}+0.0056\text{C}_{2} \text{H}_{7}\text{O}_{2}\text{N}+0.4889\text{H}_{2}\text{O} \tag{1}\]  
According to [PERSON] and [PERSON] [21], when bacteria derive energy exclusively from the oxidation of [Fe\({}^{2+}\)] and allocate all energy towards bacterial growth, inhibitory processes affect the rate of cell growth. According to [PERSON] et al. [22], the inhibitory effect is denoted by blocking the cell multiplication of the microorganism by reversibly binding to the ribosomes, inhibiting protein synthesis, resulting in the slow death of the microbial population. When exposed to conditions favorable to bacterial growth, microorganisms reproduce again, and the population grows [16]. Equation (2) represents the cellular process without inhibitory effects. \[r_{cell}=\frac{d[cell]}{dt}=\mu[cell]=\mu^{\text{max}}\frac{\left[\text{Fe}^{ 2+}\right]}{k_{s}+\left[\text{Fe}^{2+}\right]}[cell] \tag{2}\]  
where [_cell_] is the biomass concentration, \(\mu\) is the specific coefficient of bacterial growth in iron oxidation, \(\mu^{\text{max}}\) is the maximum specific rate of bacterial growth, [Fe\({}^{2+}\)] is the ferrous ion concentration, and \(k_{s}\) is the substrate saturation constant. For the present work, it was considered that the ferrous ion is the limiting substrate. The inhibition process of Fe(II) oxidation affects cell growth, caused both by the [Fe(III)] reaction product, by a certain concentration of Fe(II), or even by the substrate itself [23; 24]. The inhibitory influence of substrate [Fe\({}^{2+}\)], product [Fe\({}^{3+}\)], bacterial cells, and concentrations of heavy metals on the Fe(II) oxidation rate and bacterial growth are represented in Equation (3). \[\mu=\mu^{\text{max}}\frac{\left[\text{Fe}^{2+}\right]}{k_{s}\left(1+\frac{ \left[\text{Fe}^{3+}\right]}{k_{p}}\right)+\left[\text{Fe}^{2+}\right]} \tag{3}\]  
where [Fe\({}^{3+}\)] is the ferric ion concentration, and \(k_{p}\) is the product inhibition constant. Assuming that substrate inhibition can be represented by Equation (4):  
\[\mu=\mu^{\text{max}}\frac{\left[\text{Fe}^{2+}\right]}{k_{s}+\left[\text{Fe}^ {2+}\right]+\frac{\left[\text{Fe}^{2+}\right]^{2}}{k_{s_{i}}}} \tag{4}\]  
where \(k_{s_{i}}\) is the substrate inhibition constant. According to [PERSON] et al. [25], metal ions present different inhibitory processes, which affect microbial physiology. This inhibitory effect depends on the metallic concentration to which the microorganisms were exposed. If the concentration is not enough, they enter a phase of prolonged latency until their complete adaptation to the conditions is satisfactory. Figure 1: Scheme of the indirect mechanism developed in the model. In the construction of the mathematical model, the oxidation of uraninite by oxidants other than Fe\({}^{3+}\) and the effect of contaminants on the leaching kinetics are assumed to be insignificant. The integration of immediate iron species concentrations accounts for the effect of Fe impurities. According to [PERSON] et al.