## 3 Results  
The analysis of specificity of protein-receptor interaction has been presented here with the example of interaction between leptin and leptin receptors. As mentioned above, leptin acts on leptin receptors in the hypothalamus to regulate appetite. In addition, leptin also binds neuropeptide Y in NPY neurons by decreasing the activity of these neurons, while leptin receptor activation inhibits neuropeptide Y. Here, we analyzed all three groups of proteins involved in the regulation of hunger: leptin, leptin receptor, and neuropeptide Y [10]. To find out the RRM characteristic frequency relevant to the activity of leptin proteins, i.e., regulation of hunger and energy balance, we initially compared leptin proteins from the UniProt database, as listed in the Materials section. The single common RRM frequency was found at f1 = 0.2764 \(\pm\) 0.0103, as presented in Figure 2. According to the relationship between calculated RRM frequencies and corresponding electromagnetic radiation wavelengths, as described in the Methods section, frequency f1 represents an electromagnetic radiation wavelength of 727 nm, which is within the red-light spectrum. To check if frequency f1 is characterized by recognition between leptons and leptin receptors, we compared, using the RRM model, leptons and leptin receptors from the UniProt database, as listed in the Materials section. The same common RRM frequency of f1 = 0.2764 \(\pm\) 0.0103 (727 nm) was found, as presented in Figure 3. This result confirms indeed that frequency f1 is characterized by recognition between leptins and leptin receptors, and thus can be proposed to be crucial for control of hunger and energy balance. As it has been shown that leptons and activated leptin receptors interact with neuropeptide Y [10], we compared, using the RRM model, leptons, leptin receptors, and neuropeptides Y from the UniProt database, as listed in the Materials section. The same characteristic RRM frequency of f1 = 0.2764 \(\pm\) 0.0103 appeared to be common for all three groups of proteins, as presented in Figure 4. This result shows that the whole cascade of leptin-related interactions is characterized by only one RRM frequency of f1 = 0.2764 \(\pm\) 0.0103 (727 nm), indicating that this frequency f1 is crucial for leptin-related hunger control and energy balance within the body. Figure 3: RRM cross-spectrum of twenty leptins and five leptin receptors. The common characteristic frequency for these proteins is at f1 = 0.2764 \(\pm\) 0.0103, which represents an electromagnetic radiation wavelength of 727 nm. X-axis represents numerical RRM frequency, which could reach a maximum of 0.5, and Y-axis represents percentages of maximum peak, which could reach a maximum of 100%. Figure 2: RRM cross-spectrum of twenty leptins. The common characteristic frequency is at f1 = 0.2764 \(\pm\) 0.0103, which represents an electromagnetic radiation wavelength of 727 nm. X-axis represents numerical RRM frequency, which could reach a maximum of 0.5, and Y-axis represents percentages of maximum peak, which could reach a maximum of 100%. For protein and receptor recognition, apart from common frequency, it is necessary that the phase at this frequency is opposite to that of the protein-receptor pair, as explained in the Methods section and presented in detail within the Supplementary Material. As an example, we analyzed here phases at RRM characteristic frequency f1 = 0.2764 \(\pm\) 0.0103 for human leptin and human leptin receptors. It was found that, for human leptin at frequency f1, the phase was +0.96 rad, while, for the same frequency f1, the phase for human leptin receptor was \(-\)2.10 rad, as presented in Figure 5 within the phase circle. This means that the phase difference between human leptin and leptin receptor is 3.06 rad, which is very close to 3.14 rad, supporting again the RRM approach that protein and protein receptors should have opposite phases at RRM frequency characterizing their recognition and interaction.