The level of rigidity of protein structures can be estimated by the variable Delta-u (see Eqs. 3 and 5), the value of which is expected to be equal to zero for atom pairs that behave as a rigid body. Obviously, this occurs when the two atoms are covalently bound and very close to each other, while Delta-u values larger than zero are expected for atoms very distant from each other.
Actually, Delta-u values are observed to increase progressively if the interatomic distance increases, either when the interatomic distance is the Euclidean distance (Fig. 1a) or the number of covalent bonds intercalated between the two atoms (Fig. 1b).
However, the dependence of Delta-u on Euclidean distance is probably a consequence of the fact Euclidean distance depends on covalent separation (Fig. 1c). In fact, as it is shown in Table 1, Delta-u is rather independent of Euclidean distance at each value of covalent separation—each line in the table. This suggests that protein rigidity is largely due to its covalent structure and less to non-bonding interactions amongst moieties far from each other along the sequence. Certainly, covalent connections between atoms separated by numerous backbone covalent bonds can exist, for example disulfide bonds or contacts mediated by metal cations, and they contribute to confer some rigidity to the protein. However, most of the contacts between atoms separated by numerous backbone covalent bonds involve van der Waals interactions, which apparently do not confer much rigidity to the protein despite the high protein packing efficiency. Further studies are nevertheless necessary to reach a deeper understanding of this phenomenon.
At large distances, the Delta-u approaches the upper value close to 0.06–0.07 Å, computed with isotropic B-factors (Eq. 5), which is considerably larger than the upper value close to 0.015–0.02 Å, computed with anisotropic B-factors (Eq. 3). This clearly indicates that protein flexibility is enormously overestimates by isotropic B-factors.
These Delta-u values are nevertheless considerably small. This is quite surprising since globular proteins are known to be quite flexible, even if they are compact. For example, water molecules buried into the protein core easily exchange with bulk solvent by opening transient channels that allow the entrance/exit of water [24, 25]. Also, aromatic side-chains are known to flip, with 180° rotation, with high flip rates [26].
All these processes require atomic displacements that are considerably larger than the upper Delta-u limits observed in the present communication.
It can be hypothesized that these considerable local deformations, which allow water molecules to enter in and exit from the protein core and that allow aromatic ring flipping, are due to conformational transitions that do not depend on progressive rigidity loss. For example, it is possible to imagine side-chains that pass from a stable, rotameric conformation to another one, both being relatively rigid; or it is possible to imagine a rearrangement of the hydrogen bond network, with stable hydrogen bonds being broken and being replaced by equally stable, new hydrogen bonds. The classic hinge motions of rigid structural moieties might also disconnected from B-factors [27].
Therefore, even if B-factors are known since long time to monitor conformational strain [28], which larger B-factor being associated with dihedral angles far from their stable values, it is possible to hypothesize that B-factors cannot provide information about transitions from a stable structure to a similarly stable but different conformation, which are often referred to as conformational sub-states [29,30,31].
A metaphor for this phenomenon can be an auditorium, all the seats of which are occupied by spectators that can exchange their seats: before and after the exchange, the ensemble of spectators is rather compact and rigid, while a large flexibility is observed when the spectators move from a one seat to another, exchanging their position.
Interestingly, this trend seems to be independent of protein dimension, type of fold, secondary structure composition or biochemical function. As an example, Fig. 2 shows the relationship between Delta-u and covalent separation for three proteins, two of which are enzymes (human aldose reductase, 1us0, and human parvulin, a small peptidyl-prolyl isomerase, 3ui4) and one of which is not (Trichoderma reesei hydrophoibin, a small fungal protein that spontaneously forms amphiphilic monolayers). They adopt different fold types, a TIM-barrel for 1us0, essentially a β-barrel for 2b97, and a α-β-α roll for 3ui4, and one of them, 1us0, is much larger than the others. These proteins show similar trends and there are no enormous differences between them; furthermore, the difference between the two enzymes is comparable to their difference from hydrophoibin, and the largest protein (1us0) is intermediate between the other two.
Crystallographic B-factors are largely unable to monitor transitions amongst conformational sub-states. This has been observed, implicitly, in some previous studies. For example, according to a recent study, protein conformational entropy, defined as the movements of certain groups in proteins, is not monitored quantitatively by crystallographic B-factors [32]. Also, it was observed that crystallographic B-factors underestimate the positional heterogeneity in protein crystals [33].
These observations can be explicated as it follows. Crystal structures show the dominating and most stable protein conformation while alternative sub-states remain undetected, especially at low resolution. Some conformational disorder can be observed and refined experimentally only at high resolution [7,8,9,10]. B-factors therefore describe the positional scattering around one conformation and do not reflect the more complex conformational flexibility of proteins. Moreover, B-factors do not monitor only the atomic oscillations around equilibrium positions but depend also on crystal heterogeneity in spaced and time. Crystal structures are in effect representations of the electron density maps of the asymmetric unit, which are the average electron density maps computed (1) on all the asymmetric units present in the crystal and (2) with diffraction data measured over a certain time lapse.
As a consequence, B-factors can be computed quite successfully in—very—small molecule crystals, independently of diffraction data, where B-factors monitor quite effectively atomic fluctuations. The vibrational component of the atomic displacement parameter can be computed with quantum chemistry computations in crystals with very small asymmetric units. For example, density functional theory (DFT)-based methods were used for crystalline l-alanine and crystalline urea [34], and density functional perturbation theory was applied to stishovite and quartz [35]. Recently, B-factors have been computed from ab initio phonon frequencies and displacements for elemental crystals of magnesium, ruthenium, cadmium and silicon [36].
On the contrary, protein crystallographic B-factors are affected by too many non-vibrational components and cannot be predicted by computing the energy of the environment of the atoms by means of quantum chemistry approaches, though it has been shown that protein B-factors are somehow correlated to packing density [37]. At this regard, it is noteworthy that B-factors have also been used to estimate atomic coordinate errors [38, 39], based on the diffraction precision index of Cruickshank [40]. Consequently, they cannot be reproduced reliably in silico, independently of diffraction data.
It must be remembered too that most of protein crystal structure information is being produced at low temperature—100 K—and that a different flexibility might be detected at room temperature or at physiological temperature [41]. However, cryo-crystallography is the predominant form of macromolecular crystallography, given its advantages in reducing radiation damage, especially in modern, high brilliance synchrotron beam lines [42,43,44].
The above discussion does not imply that crystallographic B-factors are of limited value and disconnected from the physicochemical nature of proteins. For example, information about local flexibility can be extracted from B-factor analyses, for example for protein-DNA complexes [45], cold adaptation of psychrophilic enzymes has been shown to be closely related to B-factors [46, 47], and a procedure called B-Fit has been proposed for increasing the thermostability of enzymes and allows their use in chemistry and biotechnology [19]. More in general, protein regions characterized by large B-factors can be considered to be very mobile, though not necessarily rigid; it clearly appears that protein flexibility is not fully described by B-factors, which capture only partially the wide range of distortions that proteins can afford.