## 2 Materials and Methods
### Mathematical Model
Where q is the number of unknown variables
3 Take the coefficients of the linear equation as:

Do for k = 1 to n

Do for j = 1 to n + 1

Read a[k][j]

End for j

End for k

4 Do for k = 1 to q - 1

Do for i = k + 1 to n

Do for j = k + 1 to q + 1

a[i][j] = a[i][j] - a[i][k]/a[k][k] * a[k][j]

End for j

End for i

End for k

5 Compute x[q] = a[q][q + 1]/a[q][q]

6 Do for k = q - 1 to 1

sum = 0

Do for j = k + 1 to q

sum = sum + a[k][j] * x[j]

End for j

x[k] = 1/a[k][k] * (a[k][q + 1] - sum)

End for k

7 Display the result x[k]

8 Stop

## 3 Results and Discussion

### Bioprocess

In this work, \(2^{5-2}\) fractional factorial design of experiments were applied to evaluate the relevant factors in the fermentation stage. The use of design of experiments as statistical methods for the optimization of culture media in the fermentation process can overcome the limitations of the classical methodology, which evaluates one factor at a time and can be a valuable tool for the optimization of the production of metabolites. [PERSON] proposed a basic theory of experimental design that shows how changing more than one component in the medium at a time can be more efficient than changing only one factor at a time [2]. Although there are different types of design of experiments, in this study the fractional factorial is applied, and not the full factorial. The first design was composed of five factors, equivalent to eight trials plus the replicates. In contrast, in the second design, 25 trials plus the replicates were evaluated. Statistical analysis for both cases was similar. The response surface design was preferably not used in this study since more than three variables and qualitative factors were used in the experiments [20]. The factorial experiment design method is an appropriate technique to study the influence of the main bioprocess parameters on the response factors by significantly reducing the number of experiments and, henceforth, saving time, energy, and money [21]. The response variables that were monitored as a control in the fermented broths due to their relevance in the bioleaching processes were pH, the presence/absence of proteins, and organic acids. Organic acids are released to reduce the environmental pH, which promotes the growth of fungi and inhibits the growth of some bacteria [22]. Similarly, the catalytic properties of the enzyme molecule are determined by its amino acid sequence, presenting a bioleaching effect on minerals [23]. Table 2 presents the design matrix, with the results, the control variables, and the response variables. This table shows the eight trials without the replicas because they were reproducible; therefore their behavior over time was similar in each of the tests and their corresponding replication. In fermentation, the bio acid in the leach solution is formed in situ in the presence of the metal-laden solid. The nutrients added to the mixture can be any suitable carbon source, such as sucrose or molasses. In the in situ process, the metabolism of nutrients leads to the growth of the organism and the production of the organic bio acid metabolite in parallel with the chemical dissolution of the metal, that is, leaching [24]. Another type of fermentation occurs when the bio acid is generated ex situ in the absence of the metal-loaded solid in a microbial or synthetic way. The bio acid produced in this ex situ process is then contacted with the solid [24]. The nature and amount of organic acids excreted by fungi are influenced by the pH of the medium. Maintaining a low pH is essential for maximum citric acid production. The initial pH required depends on the carbon source used. Generally, a pH below 2.0 is required for optimal production. A low initial pH has the advantage of controlling contamination and inhibiting the formation of oxalic acid [25]. At a pH of four or higher, the formation of oxalic acid accelerates due to the high buffering capacity of the medium [26].