## 3 Results
### Linear Response Regime  
Under linear response regime all the parameters entering the local impedance in Equation (1) do not depend on the applied bias. Accordingly, Figure 3 reports the normalized resistance of the single protein in dark, R (#)/R, i.e., the static (DC, i.e., \(\omega\) = 0) global resistance R (#) of the protein, normalized to the benchmark value R, in dark, as a function of the cut-off distance between neighborhood nodes, D. In particular, the protein at pH = 10 (PDB code 2 uwj) is compared with the protein at pH = 6.5 (PDB code 2 uww). Calculations have been performed by using two different contact configurations as reported in Figure 1.  
From the above comparison we conclude that the value of the pH plays a significant role only in the up-down orientation, where the value of the resistance stays well below that of the benchmark. By using the up-down orientation, Figure 4 reports the normalized resistance of the single protein, R (#)/R, calculated at different values of pH. A higher sensitivity is observed at increasing values of pH. In particular, the best resolution is observed for the value \(D\) = 10.5 A of the cut-off distance, that compares well with the value \(D\) = 6 A found for the case of bacteriorhodopsin (bR), a protein like bRC but with a smaller number of amino acids for about a factor of 4 [14]. In the inset of Figure 4 the resistance of the protein in light at pH 6.5 is compared with the benchmark resistance. In this case, the maximal resolution also shows up for a cut-off value \(D\) = 10.5 A, with a reduction of resistance in fair agreement with experiments [12]. Figure 2: Differential contact maps of the bRC in dark with two different values of pH. Circles (squares) describe the contacts which are specific for the configuration. Data are calculated using \(D\) = 6 Å. Finally, making use of the up-down orientation in dark, and taking \(\rho\) = 2.7 \(\times\) 10\({}^{12}\)\(\Omega\) A we evaluate the complete impedance spectra for pH = 6.5 and 10 respectively, in the frequency range (10\({}^{-2}\)-10\({}^{4}\) Hz) which covers the full experimental range. In this range, experiments detect the solution effects, mainly at the highest frequencies, and the electrode resistance, at the lowest frequencies. Otherwise, all the cell elements contribute to the global impedance response in the full frequency range [20,21]. The impedance model may be complex to account for several effects that could be detected by experiments. Anyway, the main contribution is given by Equation (1) and accounts for both the polarization and some leakage currents which are usually present in real dielectrics [9,14,21,25]. In an electrochemical cell conduction there is both an electronic and ionic origin, i.e., charge may travel through both organic/inorganic matter and the electrolytic solution. Furthermore, charge may accumulate between different solid/liquid or solid/solid interfaces [21]. Equation (1) describes an elemental circuit consisting of a resistance (R) and a capacitor (C) in parallel, R//C, and its graphical representations in terms of frequency are called Bode plots. The imaginary vs. the real part of impedance is represented by the so-called Nyquist plot. Figure 4: The normalized resistance R (#)/R of b\(RC\) in dark, as a function of the cut-off distance between neighborhood nodes, \(D\), at different values of pH. Calculations refer to up-down orientation and three different pH values. The inset reports the normalized resistance R (#)/R in light for pH = 6.5. Figure 3: The normalized -resistance R (#)/R of b\(RC\) at pH = 10, as a function of the cut-off distance \(D\). Data are obtained using different contact orientations: the up-down in which the protein is assumed to be with the chain H (but helix 1) on the electrode (see Figure 1a), and the transverse one in which the protein lays on its back on the electrode (see Figure 1b). Data refer to structures in dark.