## 2 Materials and Methods
### Mathematical Model
The Pearson Product-Moment correlations are shown in Figure 4. Independent variables: C/N ratio, C/K ratio; dependent variable: absorbance at 1060 cm\({}^{-1}\). Although the design of experiments allows establishing a functional relationship between the studied factors, these relationships, also known as combined effects, are second and third-order polynomial models whose topology limits the mathematical representation of complex or highly non-linear relationships between the independent variables and the response variable. Additionally, [PERSON]'s Pi theorem reduces the number of model parameters by formulating dimensionless parameters \(\Pi\), as indicated in Equation (12), which allows a mathematical model to be reconstructed with greater precision between said parameters. ### Results of the Mathematical Model

To establish the mathematical model for the functional relationship between the dimensionless parameter C/N and C/K at a wavelength of 1060 cm\({}^{-1}\), a calculation routine to fit models using multilinear regression in Python was used. The sum of the squares of the absolute errors was the objective function. The model is shown in Equation (15), which has a total absolute error of 5.03 \(\times\) 10\({}^{-5}\) and a correlation coefficient R\({}^{2}\) of 0.9889. \[Absorbance=a+b(C/N)+\frac{c}{Ln(C/K)}+d(C/N)^{2}+\frac{f}{[Ln(C/K)]^{2}}+g \frac{(C/N)}{Ln(C/K)} \tag{15}\]

This model shows the non-linearity existing between the absorbance and the operational conditions that were used to produce the fermented broth acting on the mineral. The C/N and C/K ratios present in the culture medium affect the composition of the fermented broth and, therefore, the potential to leach metallic species present in the minerals. The adjustment coefficients obtained for the model are summarized in Table 7.

Figure 4: Pearson Product-Moment Correlations. Independent variables: C/N ratio, C/K ratio; dependent variable: absorbance at 1060 cm\({}^{-1}\). The method of regression analysis estimated the relationship between each dimensionless variable or variables, referring to the value of the regression coefficient (R\({}^{2}\)). Commonly, the data with R\({}^{2}\) values above 0.90 can be accepted as accurate predictions, and the effectiveness of the mathematical model can be assumed [16]. Figures 5 and 6 show the fit of the data, represented as black dots, to the response surface generated by the mathematical model. The proximity between these points and the surface makes it possible to accurately predict the effect that the C/N ratio and the C/K ratio will have on the absorbance of the fermented broth after treatment with the mineral. Figure 7 shows, in detail, the contour lines of the response surface in a two-dimensional plane, with lines of constant absorbance in the form of envelopes that validate the results of the dimensional analysis and the bivariate Pearson correlations, where the C/N and C/K ratios, as dimensionless parameters, have a significant effect on the bioleaching of gold using fermented broths. The highest absorbance of the evaluated region was 0.096, which is achieved by producing a fermented broth with a C/N ratio close to 50 and in the vicinity of a C/K ratio close to 800. A similar result was obtained by [35], in which the C/N ratio presented a statistically significant effect in the optimization of a medium for the production of organic acids in submerged fermentation of _A. niger_.