## 2 Materials and Methods
### Biosorption Assays  
All biosorption assays were performed at 27 \({}^{\circ}\)C and 200 rpm in duplicate in an SI-600R Orbital (Lab Companion, Chalgrove, Oxfordshire, UK); 100 mL flasks were used with a working volume of 50 mL of Cu(II) solution at a concentration of 50 mg/L. Samples obtained in the assays were filtered with 0.22 \(\upmu\)m PES filters and analyzed by atomic absorption spectroscopy (AAS) in a Perkin Elmer AAnalyst 800 instrument (Midland, ON, Canada) using Equation (2). \[q=\frac{\left(C_{i}-C_{f}\right)V}{m} \tag{2}\]  
where \(C_{i}\) and \(C_{f}\) are the initial and final concentrations of Ag(I) in mg/L, respectively; \(V\) is the volume of metal solution (0.05 L), and \(m\) is the dry mass of biomass (g). Response surface methodology (RSM) and a rotatable central composite design were used to determine the optimal operating conditions (pH and biomass concentration). The equilibrium tests to obtain the biosorption isotherms were carried out under the optimal conditions obtained in the experimental design and in the concentration range between 10 and 50 mg/L, with a contact time of 4 days to ensure that the process, generally rapid, was completed. In order to fit the experimental data, the models shown in Table 1 have been used. The Langmuir model is well known [13]. It assumes monolayer adsorption on a homogeneous adsorbent. The Freundlich model is an empirical equation that assumes that the adsorbent increases its adsorption area as the adsorbate concentration increases and can then be applied to multilayers and heterogeneous surfaces [14]. The Sips model incorporates the basic principles of [PERSON] and Freundlich and can be used over a wide range of concentrations [15]. Finally, the Redlich-Peterson model is an empirical model that can also be used over a wide range of concentrations [16]. When \(\beta\) equals one, we have the Langmuir isotherm, and when \(\beta\) equals zero, we have [PERSON]'s law. IBM SPSS Statistics v.27 was also used to fit the experimental data. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model** & **Equation** & \multicolumn{2}{c}{**Parameter**} \\ \hline \multirow{3}{*}{Langmuir [13]} & & qm & 48.14 \\  & \(q_{e}=\frac{q_{m}\,b\,C_{e}}{1+B\,C_{e}}\) & b & 0.349 \\  & & R2 & 0.987 \\  & & \(\Sigma(\mathrm{q}-\mathrm{q}_{\mathrm{cal}})^{2}\) & 0.876 \\ \hline \multirow{3}{*}{Freundlich [14]} & & K\({}_{\mathrm{F}}\) & 22.884 \\  & \(q_{e}=K_{F}\,C_{e}^{\frac{1}{2}}\) & n & 5.27 \\  & & R2 & 0.928 \\  & & \(\Sigma(\mathrm{q}-\mathrm{q}_{\mathrm{cal}})^{2}\) & 4.984 \\ \hline \multirow{3}{*}{[PERSON] [15]} & & K\({}_{\mathrm{s}}\) & 16.530 \\  & \(q_{e}=\frac{K_{e}\,C_{e}^{\frac{1}{2}}}{1+a_{e}\,C_{e}^{\frac{1}{2}}}\) & a\({}_{\mathrm{s}}\) & 0.344 \\  & \(\mathrm{n}\) & 0.987 \\  & \(\mathrm{R}^{2}\) & 0.987 \\  & \(\mathrm{\Sigma}(\mathrm{q}-\mathrm{q}_{\mathrm{cal}})^{2}\) & 1.313 \\ \hline \multirow{3}{*}{[PERSON]–Peterson [16]} & & K\({}_{\mathrm{RP}}\) & 17.149 \\  & \(q_{e}=\frac{K_{RP}\,C_{e}}{1+a_{RP}\,C_{e}^{\frac{1}{2}}}\) & \(\mathrm{\beta}\) & 0.363 \\ \cline{1-1}  & \(\mathrm{R}^{2}\) & 0.995 \\ \cline{1-1}  & \(\mathrm{\Sigma}(\mathrm{q}-\mathrm{q}_{\mathrm{cal}})^{2}\) & 1.310 \\ \hline \hline \end{tabular}  
\(\mathrm{q}_{e}\): biosorption capacity (mg/g) at equilibrium; \(\mathrm{q}_{m}\): maximum biosorption capacity (mg/g); b: Langmuir biosorption equilibrium constant (L/mg); \(\mathrm{C}_{e}\): equilibrium concentrations of metal (mg/L); \(\mathrm{K}_{F}\): characteristic constant related to the biosorption capacity; n: characteristic constant related to the biosorption intensity; \(\mathrm{K}_{s}\) and \(\mathrm{a}_{\mathrm{s}}\): Sips isotherm parameters; \(\mathrm{K}_{RP}\), \(\mathrm{app}\), and \(\mathrm{g}\): Redlich–Peterson parameters, and \(\mathrm{\beta}\) varies between 0 and 1; \(\mathrm{R}^{2}\): coefficient of determination; \(\mathrm{\Sigma}(\mathrm{q}-\mathrm{q}_{\mathrm{cal}})^{2}\): sum of the errors squared.  
\end{table}