## 2 ENMs Overview and the Surprising Accuracy of Shape-Encoded Harmonic Motions
ENM-NMA can have apparent simplicity--with \"toy\" ad hoc force fields and the naive assumption that structures are in an energy minimum--but it is often unsurpassed in the prediction of experimentally observed large-scale conformational changes (Figure 1, center). There have been endless studies comparing ENMs with functional transitions between bound/unbound, active/inactive and open/closed pairs derived from X-ray conformers, NMR ensembles, etc., which show that the lowest-frequency modes are indeed both biologically and functionally relevant [41; 42; 43; 44] and can unravel complex allosteric mechanisms [45], even for subtle transitions such as those seen in GPCRs [46; 47; 48]. Protein conformational changes often involve large rigid-body motions, e.g., domain swapping, hinge-bending, or shear movements, which are strikingly well described via a small number of ENM modes [49; 50; 51]. An early study on the first database of molecular motions, MolMov [52], determined that 95% of experimentally observed transitions can be described using just a couple of soft ENM modes. Further benchmark studies have confirmed that large-scale motions also coincide with the collective modes extracted from MD simulations or experimental ensembles [53; 54; 55; 56; 57] via Principal Components Analysis (PCA, see [58; 59; 60]). Systematic comparison with MD of representative meta-folds in the MODEL database as well as with experimental data [61; 62] confirmed that ENMs are extremely robust to spring definitions and perform exceedingly well in predicting large-scale transitions, occasionally surpassing MD simulations. Nevertheless, as often happens with CG models, a major weakness of ENMs is the lack of a consistent and universal consensus on force-field parameterization, i.e., the functions used to determine the \"springs\" connecting different residues or \"beads\". This has both positive and negative aspects. On one hand, although ENMs can predict the preferred directions for conformational change, the time and length scales of the motions (i.e., the magnitudes of the eigenvalues) are usually arbitrary. On the other, and paradoxically, this weakness reflects their major strength: ENMs are determined by protein shape, topology, and local packing density, and are thus insensitive to fine details. Despite these shortcomings and their dramatic simplicity, soft ENM modes are surprisingly accurate at predicting anharmonic, far-from-equilibrium transitions [20; 40]. Together with the lack of a solvent and thus damping, this was initially a major point of controversy, questioning the validity of both NMA and its CG approximation [63]. What is the time and length scale of NMs? How can harmonic NMs capture anharmonic, damped and slow transitions over high energy barriers? It has been argued that proteins oscillate around the equilibrium, with energy increasing as they stretch along NMs' directions. This could elegantly agree with a dynamical systems perspective, as the Kolmogorov Arnold Moser (KAM) theorem assures the persistence of quasi-periodic motions under small perturbations [21; 64]. Under this view, NMs would define major directions around a potential well, that hold relatively far from equilibrium. Following these, the high energy states reached would be further stretched and stabilized by different ligands or signals capable of \"tipping\" the free energy landscape (the so-called pre-existing equilibrium model [65; 66]), experimentally observedin enzymes [16]). Already in the 1990s, MD studies showed that indeed, the energy surface probed via simulations is well-approximated by a rescaled version of the harmonic potential [67; 68]. Recent work has related anharmonicity to mode collectivity: low-frequency modes that are collective enough, remain harmonic even for large displacements and better correlate with experimental transitions [69]. The power of ENMs to explore the boundaries of free energy minima is thus being more and more recognized, to the point that they are now used to enhance sampling via MD [70].