## 2 Materials and Methods
### Mathematical Model
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Experiment Number** & **N Source** & **C Source** & **K Source** \\ \hline Carbon Source & \(C\) & kg m\({}^{-3}\) & \(ML^{-3}\) \\ Concentration & & \(N\) & kg m\({}^{-3}\) & \(ML^{-3}\) \\ Nitrogen source & & \(K\) & kg m\({}^{-3}\) & \(ML^{-3}\) \\ concentration & & \(\mathrm{Trace}\) concentration & \(T\) & kg m\({}^{-3}\) & \(ML^{-3}\) \\ Agitation & \(A\) & \(\mathrm{rad}\) s\({}^{-1}\) & \(T^{-1}\) \\ Air flow for aeration & \(Q\) & \(\mathrm{m}^{3}\) s\({}^{-1}\) & \(L^{3}T^{-1}\) \\ Wavelength & \(\lambda\)\(\mathrm{I}\) & \(\mathrm{m}\) & \(\mathrm{L}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Variables and dimensions used in the application of [PERSON]’s Pi theorem. \[M^{0}*L^{0}*T^{0}=\left(\frac{M}{L^{3}}\right)^{a_{2}}*\left(\frac{L^{3}}{T}\right) ^{b_{2}}*\left(\frac{1}{T}\right)^{c_{2}}*\frac{M}{L^{3}} \tag{9}\]

\[0=a_{2}+1\qquad a_{2}=-1\] \[0=-3a_{2}+3b_{2}-3\qquad b_{2}=0 \tag{10}\] \[0=-b_{2}-c_{2}\qquad c_{2}=0\]

\[\Pi_{2}=\frac{N}{C} \tag{11}\]

Analogous to the parameter \(\Pi_{2}\), the dimensionless parameters \(\Pi_{3}\) and \(\Pi_{4}\) are obtained, as shown in Equations (12) and (13). \[\Pi_{3}=\frac{C}{K} \tag{12}\]

\[\Pi_{4}=\frac{T}{C} \tag{13}\]

In many biotechnological processes, the relationship between the amount of carbon and other nutrients such as nitrogen has been reported as a relevant variable to characterize the nutritional balance for microorganisms, especially in fermentation processes; previous studies show how the _C/N_ ratio must be close to 42 to optimize the production of organic acids or for bioremediation processes [34; 35]. With the results obtained from the dimensional analysis, the functional relationship [36] is presented in the form of Equation (14). \[\sqrt[3]{\frac{A}{Q}}*\lambda_{1}=f\left(\frac{C}{N},\frac{C}{K},\frac{T}{C}\right) \tag{14}\]

### Experimental Results Obtained from the Correlation of the Mathematical Model

Using the results of the experimental design that were presented in Table 2 and Figure 1, in which it was found that the source of carbon, nitrogen, potassium, and the presence of trace elements are relevant variables for the bioleaching of gold at a wavelength of 1060 cm\({}^{-1}\) after obtaining \(\pi\) dimensionless parameters, the effect of the _C/N_, _C/K_, and _T/C_ ratios was analyzed using multivariate statistical analysis. The data are presented in Table 6.

The Pearson Product-Moment correlation matrix was estimated to find dependency relationships between the dimensionless parameters _C/N_, _C/K_, _T/C_, and the absorbance at 1060 cm\({}^{-1}\). Of the parameters evaluated, only _C/N_ and _C/K_ have a correlation coefficient greater than 0.60, which reflects a representative functional relationship between these relationships and absorbance. Since the dimensionless parameter _T/C_ presented a comparatively lower Pearson coefficient, it was not considered for the subsequent fit of a mathematical model. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Experiment** & _C/N_ **Ratio** & _C/K_ **Ratio** & _T/C_ **Ratio** &
\begin{tabular}{c} **Absorbance** \\ **(1060 cm\({}^{-1}\))** \\ \end{tabular} \\ \hline
1 & 160.00 & 937.50 & 0.00 & 0.0952 \\
2 & 93.46 & 50.00 & 0.00 & 0.0656 \\
3 & 280.37 & 150.00 & 208.33 & 0.0981 \\
4 & 53.33 & 50.00 & 625.00 & 0.0532 \\
5 & 280.37 & 937.50 & 208.33 & 0.0961 \\
6 & 53.33 & 312.50 & 625.00 & 0.0630 \\
7 & 160.00 & 150.00 & 0.00 & 0.0662 \\
8 & 93.46 & 312.50 & 0.00 & 0.0681 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Fermentation data. Independent variables: _C/N_ and _C/K_ ratios; dependent variable: absorbance at 1060 cm\({}^{-1}\). The analysis of the Pearson Product-Moment correlations is a bivariate statistical technique that aims to determine the degree of relationship or joint variation existing between two continuous variables, and the correlation index is expressed in the interval between \(-1\) to \(+1\). As the relationship approaches one, the relationship between the variables will be strong.