## 3 Lowest-Frequency Modes and Evolution  
At the macroscopic level, we can easily appreciate how form, biological motion, and function evolve together under the laws of physics, shaping animal and plant morphologies [76]. Evolution seems to select the shapes best suited to perform functional motions. In the molecular world, if we assume the structure-motion-function paradigm, i.e., from motion comes function, it just follows to wonder whether evolution is selecting dynamics and resulting function rather than sequence or shape. Is there evidence of direct evolutionary pressure on protein motion? It is in this arena--where molecular evolution meets protein biophysics--that conformational dynamics becomes central [77]. Lowest-frequency modes allow for quantitative comparisons of the dynamics linked to function between similar cores [78], which are shedding new light on these questions. Back in the 1980s, as soon as enough structures accumulated in the Protein Data Bank, it emerged that homologous proteins share similar folds, but this similarity wanes with increasing evolutionary distance [79; 80]. Still, in practice, proteins with sequence similarities as low as 20% can display identical cores. The space of protein sequences is  
Figure 2: **A closer look at CPA exchangers’ “elevator modes” conserved from bacteria to mammals. ** (**a**) **Left**: Core alignment between a mammalian exchanger, NHE9 (black) and distant bacterial homologs NapA, PanNhaP and MjNhaP1 (sequence identity \(\approx\) 20%). The first principal component (PC1) of this ensemble of n = 8 structures renders the well-known “elevator-like” motion that distinguishes outward and inward states. **Right**: Projections onto PC1 of the experimental ensemble track the conformational inward-to-outward pathway and assigns the conformational status of the solved structures along it. (**b**) ENM of the mammalian NHE9 structure and derived “elevator-like” NM. (**c**) Similarity between NMs, PC1 and the prokaryotic NapA transition are all above 70%, despite the low sequence identities. Overlaps between vectorial spaces shaded in gradient; note that overlaps around 20% are considered random and from 40–50% significant. Adapted from figures and data by [PERSON] in Ref. [74], under the Creative Commons Attribution 4.0 License. known to be much larger than that of structures, close to optimal [81] and restrained by the length, stability, and topology of each fold [82]. Importantly, from this fact, it also follows that structural folds, i.e., protein shapes, are highly robust against mutations. What about conformational spaces? ENMs have revealed that each structure preferentially samples a limited set of elemental motions; the shape determines the conformations/motions, and the motions define the function. Being defined by global shape, soft modes are also incredibly robust to perturbations like mutations [83] or local structural features, and therefore hold across protein families and even remote homologs. Hence, when two sequences have low but sizeable sequence similarity, they often share a common core, motions, and probably function [84]. Moreover, proteins sharing one similar conformation often share other conformations, i.e., their conformational spaces are conserved, a concept exploited to predict new conformers or model conformational changes [85]. Therefore, we could argue that, in the same way the sequence space is bigger than the structure space, the structure space is bigger than the motion space--and this inversely relates to fold and function robustness. Based on mounting evidence from ENMs [20] and parallel studies on residue flexibility [86], protein global dynamics has been suggested to be maximally conserved versus sequence and structure. Nevertheless, the degree of conservation of conformational spaces as well as the contributing factors are unclear. Due to the entanglement of function, motion, and shape, together with protein biophysical and evolutionary constraints, the issue is intensely debated [87; 88; 89].