## 4 Results and Discussion
### Comparison with MD Simulations  
In order to better assess the ability of the basis to reproduce the general dynamics of chymotrypsin-like proteases, we performed MD simulations of four proteins belonging to the same family and compared the per-residue fluctuations emerging from the simulations with those obtained by filtering the trajectory along the vectors of the basis; a good agreement would be indicative of the ability of the basis to describe the large-scale dynamics of the protein. Two of the proteins used as a test-case belong to the dataset; these are 1 EKB [84] and 1 NPM [85], eukaryotic proteases belonging to the S1A subfamily. The other two proteins, 4 YOG [86] and 3W94 [87], are external to the dataset and, as such, have not been used to define the basis. 4 YOG is a C30 protease from the bat coronavirus HKU4, while 3W94 is an S1A enteropeptidatase. These two proteins have been included here in order to test the generality of the identified basis for the description of the dynamics of the PA clan, independently of the specific members of the initial dataset.  
For each of the four proteins we compared the root-mean-squared fluctuations (RMSF) as computed from the simulation and as computed from the same trajectory filtered along the \"modes\" given by the backmapping of the protein structure on the basis vectors. The comparison shows a good qualitative agreement (Figures 7 and S7), in particular in correspondence with all the secondary structure elements. In the unstructured regions, the comparison is slightly less accurate; this is particularly true for long loops, which are  
Figure 6: (**a**) Root-mean-squared inner product (RMSIP) between the subspaces spanned by the first 5 modes of each protein and the first \(n\) basis vectors, as a function of the basis size \(n\). Each line corresponds to one protein of the dataset. The histogram on the right represents the distribution of the RMSIP values attained when the full basis is used. The RMSIP shows a good overlap of the subspaces (RMSIP \(>\) 0.5) for \(\approx\)94% of the proteins. (**b**) RMSIP normalized with respect to the value attained from the use of the full basis. For each dataset member, as few as 15 basis components are sufficient to reproduce 80% of the dynamics that would be attained with the use of the full basis set.  
more sensitive to the limitations of the ENM and of the NMA employed to define the modes of the basis, since both assume small-amplitude fluctuations from a well-defined reference structure. From the two sets of trajectories, namely the original MD simulations and the filtered ones, we also computed the dynamic cross-correlation matrices (Figures S8 and S9), which give a measure of the degree of correlation between each pair of C\({}_{\alpha}\) atoms in terms of fluctuations from their average position. When comparing the original and filtered trajectories, the intensity of the resulting correlations are different, with higher correlations/anti-correlations emerging from the trajectory filtered on the basis; however, the patterns of correlation are strikingly similar between the two trajectories for all four proteins. In addition, we computed the RMSIP between the first \(n\) modes obtained from the PCA of the MD simulation and of the filtered trajectory, where \(n\) is the number of components that capture 80% of the variance in the original simulation (Table S2); in all cases, the results show a good overlap of the two subspaces, with RMSIP > 0.5. Therefore, the basis set appears to be able to describe the relevant large-scale dynamics of the considered protein systems.