## 3 Lowest-Frequency Modes and Evolution
There are two central questions to be addressed: Is it function that primarily drives the conservation of dynamics? Or is it due to physical constraints such as stability, topology, local packing, etc., or properties like mode energies or robustness? What about evolutionary constraints such as population sizes, mutational rates or bias? In other words: are soft modes conserved because they are functional or because they are energetically \"easy\" and robust? Probably, the truth is in the middle. Evidence for direct evolutionary pressure on normal modes is still scarce, as quantitative comparisons of functional dynamics are relatively recent [78]. It has been proposed that there is negative selection against the divergence of functionally important modes, while other studies suggest that they are conserved just because they are more robust to mutational perturbations (Figure 1a,b). Soon after ENMs were developed, it became evident that proteins with similar architecture shared similar motions [90]. Early studies on the evolution of soft modes, led by [PERSON] and colleagues, focused on how structural cores modify their shape across homologous proteins [91; 92; 93]. These pioneering works revealed significant similarity in the conformational ensembles explored within a superfamily and the soft modes, i.e., proteins seem to evolutionarily diverge along soft modes or, vice versa, protein topology constrains evolutionary divergence. In parallel, [PERSON] also showed that the lowest-frequency modes are conserved in homologous proteins [94], and there is a significant correlation between mode collectivity and its conservation [95]. The conservation of lowest-frequency modes is apparent in residue fluctuation patterns, which can be easily aligned for homologous proteins [96]. Some studies have also pointed out that protein sites evolve at different rates depending on properties such as their solvent accessibility, packing density, and flexibility [97; 98]. In general, there is an inverse relation between local flexibility and evolutionary rates [99] i.e., exposed and flexible loops are less conserved than cores or rigid regions [100], which can act as hinges for global motions. Consequently, ENM analyses show clear correlations between sequence evolution and structural dynamics, especially relevant for hinge regions [100; 101]. These rigid regions are so critical that hinge migration has been proposed as a mechanism for protein evolution [102]. Moreover, cancer and disease-related mutations tend to focus on hinge-like areas [103; 104]. Therefore, ENM dynamics is a key predictor of functional impact for point mutations [105; 106] as well as for insertions and deletions [107], further discussed below. Importantly, even in the case of random mutations, structural changes correlate with the lowest frequency modes [108], as happens also for ensembles of the same protein determined in different experimental conditions [109]. Perturbative ENMs indicate that the con servation of soft modes might arise precisely from their robustness against mutations [110] and, conversely, structural divergence is proportional to mutational sensitivity [73]. Only mutations targeting critical regions such as rigid hinges could thus have the potential to change ENM mode patterns and function, causing either disease or driving evolution. The majority of changes would have no effect due to mode robustness, which would be the primary factor for evolutionary conservation. Apart from mode robustness, protein modularity and size also contribute to the overlap between the NMs and evolutionary modes and explain their low dimensionality, according to recent studies [111]. Altogether, these studies point out that biophysical properties are key for mode conservation. Nevertheless, the functional motions observed experimentally seem to correlate with the soft modes more than expected based on just their amplitude and energies, indicating that selection plays a central role [112].