## 3 Results
### Super-Linear Response Regime  
To reproduce the super-linear behavior exhibited by the current-voltage (I-V) experiments obtained with a CAFM technique, at increasing values of an applied positive-voltage [12], the microscopic model makes use of a sequential tunneling mechanism of charge transfer between different nodes that are assumed to be separated by an energy barrier \(\Phi\). Accordingly, the link resistivity, \(\rho_{j,k}\) is chosen to depend on the local voltage drop \(V_{j,k}\) between the couple of \(j,k\) amino acids involved in the link, as:  
\[\rho_{j,k}=\left\{ \begin{array}{c}\rho_{max}\,\ if\ eV_{j,k}\leq\ \Phi\\ \rho_{max}\ \left(\frac{\Phi}{eV_{j,k}}\right)+\rho_{min}\ \left(1-\frac{\Phi}{eV_{j,k}} \right),\ if\ eV_{j,k}\geq\ \Phi\end{array} \right. \tag{3}\]  
Specifically, the current response is simulated by using a stochastic Monte Carlo procedure to allow the charge transfer channels to reduce their initial resistivity from a \(\rho_{max}\) value to a \(\rho_{min}\) value at increasing of the potential drop between nodes [14; 25; 26; 27; 28; 29]. This is on the wake of the well-known Simmons model for the charge injection in an electronic junction [30]. Indeed, the model describes two different tunneling regimes: at low bias it envisages a direct tunneling mechanism, at high bias overtaken by an injection tunneling mechanism. At low bias, the condition \(eV_{j,k}\leq\ \Phi\) holds for most of the channels that take the same \(\rho_{max}\) value, thus the global response of the protein is like that of a high resistance Ohmic conductor. At increasing bias, \(eV_{j,k}\geq\ \Phi\), an abrupt drop of the resistivity to the minimal value, \(\rho_{min}\), occurs for the given channel and the global resistance of the  
Figure 5: [PERSON] (on the left) and Bode (on the right) plots of the bRC normalized impedance. Calculations refer to the single protein in the up-down orientation in dark, with pH = 6.5 (continuous line) and pH = 10 (dashed line). Calculations have been performed using \(\rho\) = 2.7 \(\times\) 10\({}^{12}\)\(\Omega\) Å and \(D\) = 10.5 Å.  
protein will decrease accordingly. At further increasing of the bias, most of the channels will take the \(\rho_{min}\) value and the global response of the protein will be like that of a low resistance Ohmic conductor. The tunneling transition probabilities including direct, \(\mathcal{P}_{j,k}\)\({}^{D}\), and injection, or Fowler Nordheim (FN), mechanisms, \(\mathcal{P}_{j,k}\)\({}^{FN}\), reads:  
\[\mathcal{P}_{j,k}\)\({}^{D}=\exp\left[-\left(\frac{2l_{j,k}\sqrt{2m}}{\hbar}\right)\sqrt{\left( \Phi-\frac{eV_{j,k}}{2}\right)}\right]\)_if eV_j,k\(\leq\)\(\Phi,\)_  
\[\mathcal{P}_{j,k}\)\({}^{FN}=\exp\left[-\left(\frac{2l_{j,k}\sqrt{2m}}{\hbar}\right)\frac{\Phi^{3 /2}}{\sqrt{2}eV_{j,k}}\right]\)_if eV_j,k\(\leq\)\(\Phi,\)_  
(5)  
where \(\hbar\) is the reduced Planck constant, \(e\) is the unit charge, and \(m\) is the electron effective mass, here taken the same of the bare value. Some more details about the procedure can be found in the Appendix A (Figure A1). Current-voltage characteristics under different conditions are reported in the following Figures 6-8. By using up-down and transverse orientations, Figure 6 compares the \(I\)-\(V\) characteristics of b\(RC\) in dark, at pH = 6.5 and 10. Furthermore, in the up-down orientation, the \(I\)-\(V\) characteristics of b\(RC\) in dark and light at pH = 6.5 are compared. The current response of the up-down orientation is found to be of about 2 orders of magnitude higher than that of the transverse orientation. The sensitivity to the increase of pH is small for the transverse orientation, while it goes up to about 50% for the case of the up-down orientation (at 1 Volt). Calculations use the following parameters: \(\rho_{max}\) = 2.7 \(\times\) 10\({}^{12}\)\(\Omega\) A, \(\rho_{min}\) = 2.7 \(\times\) 10\({}^{4}\)\(\Omega\) A and \(\Phi\) = 0.219 eV. These data confirm that the primary role of contact position does not disappear in the super-ohmic regime.