## 2 Methodology
### The Truss-Like Structure, the Connectivity Assumption and Modal Analysis  
This work focuses on the all-atom and coarse-grained elastic schemes for the protein structure. The assumptions are the same for both cases, with small variants between them. The all-atom model considers each heavy atom as the node of the truss, whereas the coarse-grained takes one point for each amino acid (residue) as a point mass. In the real structure of proteins, peptide bonds connect various amino acids in different patterns, forming secondary structures like \(\alpha\)-helices and \(\beta\)-sheets. Figure 1a displays the tertiary structure of the lysozyme, describing the closely packed domain. However, these structures are not taken into account directly for the generation of the numerical models. The criterion of geometrical proximity generates the connections between the nodes, i.e., a cutoff value sets a spherical horizon from every designated node. In Figure 1c, the sphere of influence is shown graphically. Links are created from the central node connecting it to the surrounding ones lying below the selected cutoff value. The connections are defined as massless elements, having only axial stiffness \(k_{ij}\) and no bending, shear, or torsional rigidity. In this type of analysis, only the axial rigidity \(E_{ij}A_{ij}\), which is defined by the Young's Elastic Modulus \(E_{ij}\) and cross-sectional area \(A_{ij}\), is essential [44]. The 3D representation of the 6 \(\times\) 6 stiffness matrix of the elastic element connecting nodes \(i\) and \(j\) is shown in Equation (1), where \(x\), \(y\), and \(z\) are the global coordinates of the node in the 3D space, and \(L_{ij}\) the length of the elastic connection. \[k_{ij}=\frac{E_{ij}A_{ij}}{L_{ij}}\left[ \begin{array}{ccccc}C_{x}^{2}&C_{x}C _{y}&C_{x}C_{z}&-C_{x}^{2}&-C_{x}C_{y}&-C_{x}C_{z}\\ C_{x}C_{y}&C_{y}^{2}&C_{y}C_{z}&-C_{x}C_{y}&-C_{y}^{2}&-C_{y}C_{z}\\ C_{x}C_{z}&C_{y}C_{z}&C_{z}^{2}&-C_{x}C_{z}&-C_{y}C_{z}&-C_{z}^{2}\\ -C_{x}^{2}&-C_{x}C_{y}&-C_{x}C_{z}&C_{x}^{2}&C_{x}C_{y}&C_{x}C_{z}\\ -C_{x}C_{y}&-C_{y}^{2}&-C_{y}C_{z}&C_{x}C_{y}&C_{y}^{2}&C_{y}C_{z}\\ -C_{x}C_{z}&-C_{y}C_{z}&-C_{z}^{2}&C_{x}C_{z}&C_{y}C_{z}&C_{z}^{2}\\ \end{array} \right] \tag{1}\]  
It has to be noted that both in the all-atom and coarse-grained model used in this work, a unique value of the axial rigidity \(EA\) has been set for all the elastic connections, consistently with some previous works [15; 32; 34], i.e., \(E_{ij}A_{ij}=EA\). In this way, the axial  
Figure 1: (**a**) The ribbon structure of the hen-egg white lysozyme (PBD code: 4 YM8); (**b**) all carbon, nitrogen, oxygen, and sulfur atoms and their covalent bonds; (**c**) the 129 nodes and the indication of the cutoff (sphere of influence) which is used to generate the elastic connections in the coarse-grained model. stiffness of each connection \(k_{ij}\) turns out to be inversely proportional to the length of the connection itself \(L_{ij}\). The definition of nodal locations is discussed in the next section, depending on the considered model (all-atom or coarse-grained). The masses are concentrated at the nodes; therefore, the lumped mass assumption can be set, originating diagonal mass matrices and facilitating the numerical solution. The all-atom model uses the atomic mass of each atom at the specific position as provided in the PDB file (PDB code: 4 YMS) [36]. On the other hand, for the coarse-grained structure scheme, the amino acid masses are condensed in a single representative point. A linear elastic structure assumes a direct superposition process to obtain the global stiffness \(\mathbf{K}\) and mass \(\mathbf{M}\) matrix. The multi-degree of freedom (MDOF) system, in our case, is addressed as a free MDOF structure where no damping \(\zeta\) is considered in the equations of motion. The solution for Equation (2) involves sinusoidal displacement functions, where the amplitude vector \(a\) and frequency vector \(\omega\) are associated with each DOF.