## 4 Estimation of Time in the Linear Approximation  
One of the important parameters that should be examined in this theoretical model is the time required for the protein toxin drift to be achieved. This time can only be estimated by solving through the Poisson Nernst Planck equations (PNP Equations) in the linear \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Toxin 1** & **Toxin 2** & **Toxin 3** & **Toxin 4** \\  & _Staphylococcus aureus_ & _Staphylococcus aureus_ & _Staphylococcus aureus_ & _E. coli_ **Shige Toxin 2** \\  & **Alpha-Hemolysin** & **Toxic Shock Syndrome** & **Enterotoxin Type A** & **(One Subunit A and Five Subunits B)** \\ \hline \(C_{max}\) (mol/m\({}^{3}\)) & 10.43 & 14.46 & 13.0 & 4.88 \\ \hline \(C_{M}\) (mol/m\({}^{3}\)) & \(56\times 10^{-9}\) & \(56\times 10^{-9}\) & \(56\times 10^{-9}\) & \(56\times 10^{-9}\) \\ \hline \(\varphi_{s}^{max}\) (V) & 0.21 & 0.22 & 0.41 & 0.062 \\ \hline \(\varphi_{max}(0)\) (V) & 0.24 & 0.26 & 0.41 & 0.082 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Calculation of \(\varphi_{s}^{max}\) (the potential at the outer Helmholtz plane (OHP) when the concentration on it is \(C_{max}\) ) and \(\varphi_{max}(0)\) (the corresponding potential applied to one side of the duct) per toxin. \(C_{max}\): the maximum concentration that can be achieved due to steric effects and \(C_{M}\): the toxin concentration at the middle of the duct after the electric potential application. regime, following the methodology developed in Ref. [16]. The linear approximation applies only when the applied potential is very small, i.e., when  
\[\left|z\frac{\mathrm{F}}{\mathrm{RT}}\varphi(0)\right|<1\ \mathrm{or}\ z\ \varphi_{l}^{max}(0)=0.026\ \mathrm{V} \tag{21}\]  
where \(\varphi_{l}^{max}\ (0)\) is the maximum potential for which the linear approximation applicable to the four ionized toxins is calculated in Table 4.  
Surface charge density in the linear regime (\(\left|\sigma_{l}\right|\)) according to Ref. [16] is given by the relationship  
\[\left|\sigma_{l}\right|=\varphi(0)\frac{\varepsilon}{(\lambda_{S}+\kappa^{-1})} \tag{22}\]  
where  
\[\kappa^{-1}=\lambda_{D}=\sqrt{\frac{\varepsilon RT}{2z^{2}C_{M}F^{2}}} \tag{23}\]  
From Equation (22), the schematic representation of surface charge density as a function of \(C_{M}\),\(\left|\sigma_{l}\right|=f(C_{M}))\) in the linear regime is shown in Figure 5 with solid lines. We observe that the results of both the linear model (green line) and the exact calculation with the Stern model (blue points) coincide, as expected for low potentials, and diverge in higher potentials. This result confirms the correctness of our analysis since we expect the linear model to be valid only in low potentials and the Stern model analysis is valid and in higher potentials, as demonstrated in Section 3.3. Regarding the estimation of time, the time constant is given by the relationship [16]  
\[\tau=\frac{L\Big{(}1+\frac{2\lambda_{s}}{L}\Big{)}}{2D\kappa\Big{(}\lambda_{s }\kappa+\tanh\Big{(}\kappa\frac{L}{2}\Big{)}\Big{)}} \tag{24}\]  
In our case and for the values of \(\lambda_{s}\) and \(L\) that we are interested, shown in Table 4 (\(L\) is set less or equal \(10^{-2}\) m, because according to Figures 7-10 at this width we have the best efficiency of the model),  
\[\frac{\lambda_{s}}{L}\leq 3.5\times 10^{-5} \tag{25}\]  
and  
\[\tanh\Big{(}\kappa\frac{L}{2}\Big{)}\approx 1\]