## 3 Results and Discussion  
### All-Atom Model  
The 1000 heavy atoms of the lysozyme have their temperature factors (B-factors) available in the PDB file. These experimental values were used as the reference benchmark to fix the value of the adopted axial rigidity \(EA\). This procedure was performed to establish a quantitative comparison between the calculated and experimental values of the B-factors. Note that the resulting axial rigidity \(EA\) does not necessarily have a physical meaning since it only represents the mean rigidity of all the elastic connections simulating the complex chemo-physical interactions among the atoms. Although shorter links can be associated with covalent bonds, the longer ones are related to weaker interactions due to electrostatic and Van der Waals effects. These are not easily quantifiable with the use of a single-parameter harmonic potential; therefore, the value of the axial rigidity set here is only understood as the average strength of the generated elastic connections. Nevertheless, the choice of the \(EA\) value has a certain influence on the obtained vibrational frequencies. Moreover, choosing the value of the axial rigidity to have consistency between the mean values of the numerical and experimental B-factors leads to some consequences in the relationships between the different cutoffs. Table 1 shows the obtained axial rigidity and average stiffness for the considered cutoff values concerning the all-atom models. These values demonstrate that the mean rigidity decreases as the number of connections increases. Therefore, increasing the network's connectivity leads to the necessity of considering weaker elastic connections to have the same average thermal fluctuations. As shown in Table 1, the model with a cutoff value of 4 A presents a very low number of connections with respect to the other cases, which can generate some problems considering uniaxial elements. For this reason, this model is found to require a very high value of axial rigidity (\(\sim\)1360 N/m) due to the low number of generated connections. Such configuration is not tied enough to allow a suitable truss structure. On the other side, the model with a cutoff value of 30 A constitutes more than half of the maximum body length (46 A) and it creates a massive number of connections. However, the axial stiffness is proportional to the inverse of the connection length, making most of the longer links less critical for larger cutoffs. The results from the analysis involving the B-factors for all heavy atoms are displayed in Figure 2. Three sets of plots are arranged to facilitate the visualization. In Figure 2a, the normalized B-factors arising from the lower cutoffs, i.e., 4, 6, and 8 A, are shown. Figure 2b shows the ones with the medium cutoff values, i.e., 10, 12, and 15 A. In Figure 2c, the ones with the higher cutoffs, i.e., 20, 25, and 30 A. The vertical lines reported in each graph indicate the location of the 129 C\({}^{\alpha}\) atoms along the sequence of all heavy atoms. It was observed that the B-factors at the C\({}^{\alpha}\) locations do not necessarily reach their highest value for the considered amino acid. As briefly remarked in Section 2.3, this interesting feature led us to address a change in the node position for the construction of the coarse-grained model (see Section 3.2) and consider the CM of the amino acid as the representative node. \begin{table}
\begin{tabular}{c c c c} \hline
**Cutoff (Å)** & **Axial Rigidity \(EA\) (N)** & **N\({}^{\circ}\) of Connections** & **Average Stiffness \(k\) (N/m)** \\ \hline
4 & 3471.7 \(\times\) 10\({}^{-10}\) & 5979 & 1.36 \(\times\) 10\({}^{3}\) \\
6 & 0.6470 \(\times\) 10\({}^{-10}\) & 19,151 & 0.1675 \\
8 & 0.2436 \(\times\) 10\({}  
By observing Figure 2a, it is evident that the curve related to the model with the cutoff of 4 A presents exceptionally high peaks in four positions. This effect occurs because these protein regions are too loose due to the very short-range connections, leading to tremendously localized flexibility.