## 3 Results  
We have analysed parathyroid hormones and related proteins using the RRM model to find out their characteristic wavelengths (frequencies), which we propose to be related to their ability to initiate stem cells differentiation. When parathyroid hormone proteins from the UniProt Database, as listed above, were analysed using the RRM, the prominent common frequency was found at f1 = 0.4072 \(\pm\) 0.0108, as shown in Figure 1. According to the RRM model, the frequency f1 is characterising parathyroid hormone biological function, as well as representing an electromagnetic radiation wavelength of about 502 nm, which is within the blue spectrum of visible light. To find out the characteristic frequency of recognition/interaction between parathyroid hormone and its receptor, these proteins have been analysed using the RRM model. The prominent common frequency has been found at f2 = 0.1885 \(\pm\) 0.0108, as shown in Figure 2. According to RRM model the frequency f2 represents an electromagnetic radiation wavelength of about 1066 nm, which is within the infra-red spectrum. The f2 frequency is proposed to characterise recognition/interaction between parathyroid hormone and parathyroid receptor1. In addition, we have analysed active fragments of the human parathyroid hormone [21]:  
* Fragment PTH 1-34 that binds to the receptor and exhibits parathyroid hormone function (agonist), as shown in Figure 3; and
* fragment PTH 7-34, which only binds to the receptor but does not exhibit parathyroid hormone function (antagonist), as shown in Figure 4.  
Figure 1: Resonant Recognition Model (RRM) cross-spectrum of parathyroid hormone proteins. The prominent common characteristic frequency for these proteins is at f1 = 0.4072 \(\pm\) 0.0108, which represents an electromagnetic radiation wavelength of about 502 nm. Figure 2: RRM cross-spectrum of parathyroid hormone proteins and parathyroid hormone receptor1 proteins. The prominent common characteristic frequency for these proteins is at f2 = 0.1885 \(\pm\) 0.0108, which represents an electromagnetic radiation wavelength of about 1066 nm. It is important to note that both fragments have peak frequencies at both parathyroid hormone functional frequency f1, as well as receptor recognition frequency f2. However, when phases at these two frequencies have been calculated for peptide fragments, it was found that phases at frequency f2 for receptor recognition were similar and opposite to phase for receptor at frequency f1, confirming that both fragments are binding to the receptor. In addition, phases at frequency f1, characterising parathyroid hormone function for fragment PTH 1-34, is close to phases for PTH, while phases for fragment PTH 7-34 was opposite to phases for PTH. This means that both fragments have the ability to interact with receptors at frequency f2, while fragment PTH 1-34 has the same function as PTH due to similar phases at frequency f1, and fragment PTH 7-34 does not have the same function as PTH due to opposite phases at frequency f1. Phases for the parathyroid hormone protein, fragments and receptor at both frequencies are presented in Table 1, as well as on phase circles shown in Figure 5. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Protein/Frequency** & **f2 = 0.1885** & **f1 = 0.4072** \\ \hline PTH & \(-\)1.98 & \(+\)0.58 \\ PTH 1-34 & \(-\)1.94 & \(+\)1.26 \\ PTH 7-34 & \(-\)1.38 & \(-\)1.88 \\ PTH receptor & \(+\)0.28 & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Phases for parathyroid hormone protein, fragments and receptor. The colour of each protein corresponds to the colour of phases in Figure 5.  
Figure 4: RRM cross-spectrum of parathyroid hormone protein fragment 7-34. Figure 3: RRM cross-spectrum of parathyroid hormone protein fragment 1-34. These results confirm that frequency f2 = 0.1885 \(\pm\) 0.0108 is characterising PTH recognition to its receptor, while frequency f1 = 0.4072 \(\pm\) 0.0108 is characterising PTH function which could be related to stem cell differentiation into osteoblasts.