# PNMAVis: Visual Analysis Tool of Protein Normal Mode for Understanding Cavity Dynamics
## 4 Data Preparation
### Normal Mode Calculation  
The root-mean-square fluctuation data and normal modes (eigenvectors) of amino acid atoms were obtained byvia normal mode calculation (**T1**). This vector contains the possible motion directions of each amino acid atom in different modes. Normal mode calculation can be carried out with different granularity, including all atom calculations and coarse-grained (CG) normal mode calculation. In this paper, we mainly focus on the CG method, in which only one atom is considered for each amino acid (the \(\alpha\)-carbon of each amino acid), because it is possible to study larger systems. The normal modes for all proteins were computed using the WEBnm@[35] tool. WEBnm@ employs the Elastic Network Model (ENM) with the C\(\alpha\) force field developed by [PERSON] et al. In the elastic network model, the mass of each amino acid is represented by the position of its C\(\alpha\) atom.  
Figure 3: The pipeline of our data preparation. (**a**) A PDB file. (**b**) Normal Mode Calculation. (**c**) Cavity Processing. (**d**) Dimension Reduction & Clustering.  
The force between atoms is described as an interaction matrix. By solving the eigenvalue decomposition of the matrix, 3\(N\) normal modes are obtained, where \(N\) is the number of \(\alpha\)-carbons in the model. Each mode \(k\) contains a set of \(N\) three-dimensional vector sets \(\mathbf{V}_{k}\), which describes the independent periodic motion of the whole system around the initial position \(\mathbf{r}_{0}\). All possible motions of each protein particle \(i\) can be reconstructed by the equation:  
\[\mathbf{r}_{i}(t)=\mathbf{r}_{i}^{0}+\frac{1}{\sqrt{m_{i}}}\sum_{k=1}^{3N}\mathbf{v}_{ki}a_ {k}\cos(2\pi f_{k}t+\delta_{k}) \tag{1}\]  
with \(a_{k}\) denoting the thermal amplitude per mode \(k\), \(f_{k}\) being its frequency (linked to the eigenvalue from the original eigenproblem), \(\delta_{k}\) the phase shift, and \(m_{i}\) the mass of the considered particle \(i\). It is important to note that thermal amplitude \(a_{k}\) is a function of the given normal mode frequency, highlighting the fact that the lowest-frequency modes are also displaying the largest motions [10]:  
\[a_{k}=\frac{\sqrt{2k_{B}T}}{2\pi f_{k}} \tag{2}\]  
with \(k_{B}\) is the Boltzmann constant, and the temperature \(T\) is set to 300. It describes and normalizes the amplitude of a normal mode \(k\) by informing only about the particles' position extrema. The root-mean-square fluctuations of the atom \((\Delta R_{i})^{2}\) are formulated as follows:  
\[(\Delta R_{i})^{2}=\frac{k_{B}T}{m_{i}}\sum_{j=1}^{m}\frac{a_{ij}^{2}}{w_{j}^{ 2}} \tag{3}\]  
In Formula (3), \(m_{i}\) is the considered the mass of particle \(i\), \(w_{i}\) represents the frequency of each mode \(j\), and \(a_{ij}\) represents the coordinates of atom \(i\) on the simple vibration mode \(j\).  
In addition, the first six modes (also called trivial modes) are rigid-body transformations, and therefore, are not of interest. When domain experts analyze normal modes, they usually consider only the lowest frequency modes, also known as the slow modes, as they have been shown to experimentally and computationally relate to protein function [36; 37]. Therefore, we used six low-frequency modes for visual analysis and used all modes for data processing.