## 2 ENMs Overview and the Surprising Accuracy of Shape-Encoded Harmonic Motions  
A central problem in the study of protein dynamics has always been the difficulty of capturing motion, i.e., fully sampling conformational spaces. Protein flexibility is challenging to trap, describe, and predict, both experimentally and computationally [26]. Despite advances in hardware and algorithm parallelization, fully atomistic Molecular Dynamics (MD) simulations are still only feasible for ns-us timescales and middle-sized proteins. To gain insight into the mechanisms of bigger sub-mesoscopic systems or the slow large-scale transitions associated with biological function, the physical description needs to change accordingly to lower-resolution Coarse-Grained (CG) models. Among the plethora of CG methods to model the dynamics of proteins, ENMs stand out as possibly the most simple and powerful, considering the balance between their minimal computational cost and striking predictive power. ENMs can be described as the CG flavor of Normal Mode Analysis (NMA), a classical mechanics technique used since the 1940s-1950s to analyze the vibrational spectra of simple molecules [27; 28]. Soon after the first MD simulations, in 1982-1983 [29; 30; 31; 32; 33], NMA was applied for the first time to proteins to gain insight into their near-equilibrium dynamics. Instead of numerically solving Newton's equations as MD does, NMA assumes the harmonicity of the system around an energy minimum and, thus, through diagonalization of the mass-weighted Hessian matrix, allows the computation of a unique analytical solution, i.e., a set of linearly independent Normal Nodes (NMs) (see details in [21; 34]). NMs are a series of eigenvectors (\(\
u_{\rm i}\)) ordered by their eigenvalues or frequencies (\(\lambda_{\rm i}\)), that describe the natural motions of the system. Importantly, the first 5-10 ones, the so-called lowest frequency, \"soft\" or \"slow\" modes, capture the largest amplitude, more collective, and energetically \"easiest\" movements, which usually coincide with the experimentally and biologically relevant ones, as we will discuss below. Despite its simplicity versus MD, NMA was still computationally heavy for large systems, as it required energy minimization and significant memory resources for matrix diagonalization. Inspired by early \"random networks\" and \"beads-and-springs\" polymer models developed by [PERSON] and [PERSON] [35; 36], ENMs took the simplification of NMA one step further, replacing detailed physical force fields with a minimalist representation of proteins as networks of residue nodes connected with elastic springs, devoid of chemical or sequence information. Moreover, the system was assumed to be already at a minimum, skipping energy minimization. The first ENM [37] was still an all-atom model but with a simple pairwise Hookean potential: the native structure was defined as the minimum, and detailed interactions were replaced with a squared potential and a uniform constant within a cutoff. Shortly after, [PERSON]'s one-dimensional Gaussian Network Model (GNM) [38] introduced the coarse-graining of structures to the C\(\alpha\) trace, and finally, the Anisotropic Network Model (ANM) [39] combined [PERSON]'s 3D-model with GNM coarse-graining, becoming the basis for most ENM methods nowadays [22; 40]. The similarity of the motions described using coarse-grained ENMs with the atomistic [PERSON]'s model, and of [PERSON]'s with classical NMA based on accurate molecular potentials, was initially puzzling. How can such minimal one- or two-parameter models reproduce the vibrational properties of a complex macromolecule? The answer lies in the fact that soft modes involve coherent motions of large groups of atoms, and thus are mostly defined by the overall mass/domain architecture. For that matter, CG and atomistic mappings are nearly equivalent.