## 2 Overview of the Workflow
Once the protein dataset is chosen (**a**), dynamics-based alignment is performed between all protein pairs (**b**); the resulting similarity scores (**c**) are used to perform a clustering and to identify one representative protein for each cluster and one for the whole dataset (**d**). All the cluster representatives are dynamically aligned with respect to the latter (**e**), and their normal modes are interpolated on a cubic lattice (**f**). Once orthonormalized and ordered, the latter are used to construct the final basis set. order to identify groups of dynamics-related proteins (Figure 1d). The optimal number of clusters is identified from the interplay between _resolution_ and _relevance_[53; 54; 55; 56; 57]. These two quantities are entropies that are related to each other and depend on the clusterization procedure adopted. We exploited them to select the number of clusters to retain, by considering the smallest number of clusters (hence, the lowest resolution) that gives the highest relevance (Figure 2). Specifically, given a labeling \(\hat{s}:=(s_{1},\ldots,s_{\eta})\) (e.g., a clustering) to a sparse dataset made by \(N\geq\eta\) data points (in our case, the single proteins in the dataset), the resolution is defined as an entropy \(\hat{H}_{res}\) representing the relative amount of information loss in the process:  
\[\hat{H}_{res}[\hat{s}]:=-\sum_{s}p_{s}\cdot\log_{2}(p_{s})\quad p_{s}:=\frac{ k_{s}}{N} \tag{4}\]  
where \(k_{s}\) is the number of data points that fall into the same cluster \(s\). It has been proven [54] that \(\hat{H}_{res}\) increases monotonically with the number of clusters, in accordance with the idea that the coarser is our clustering, the more information we lose. On the other hand, the relevance \(\hat{H}_{rel}\) is defined as:  
\[\hat{H}_{rel}[\hat{k}]:=-\sum_{k}\frac{k\cdot m_{k}}{N}\cdot\log_{2}\left( \frac{k\cdot m_{k}}{N}\right) \tag{5}\]  
where \(m_{k}\) is the number of clusters containing the same amount \(k=0,\ldots,N\) of data points, for a given clustering process. By choosing the lowest resolution value corresponding to the largest relevance (Figure 2), we can rely on the most compact clusterization (thus increasing the statistics within each cluster) that preserves the highest empirical information content. Once the optimal number of clusters is derived, protein representatives of each cluster are identified as the cluster centroids, namely the proteins with the shortest distance to every other protein of the cluster itself. In addition, a representative for the whole dataset is selected as the protein with the most characteristic dynamics, expressed in terms of the lowest distance with respect to all the other dataset members. The other protein structures are then dynamically aligned to this one with ALADYN, so as to have a consistent orientation in space (Figure 1e). From an ENM representation of each of these newly oriented structures, normal modes are computed. In order to facilitate the comparison between modes belonging to proteins with a different sequence length, the first five reoriented normal modes of the cluster representatives are placed on a cubic lattice and interpolated on the grid points so as to obtain a smooth vector field (Figure 1f). In this way, we move from comparing the \(3N\)-dimensional modes of different proteins (where \(N\) is the number of residues, different for each protein), to comparing vector fields defined on identical 3D lattices having the  
Figure 2: Resolution–relevance curve used to determine the optimal number of clusters in the dynamics-based clusterization of the protein dataset. Each point corresponds to a different number of clusters. The optimal subdivision, indicated with an orange star, corresponds to 9 clusters. same dimension. More details on the lattice construction and interpolation are given in Section 3.