Effective harmonic potentials: insights into the internal cooperativity and sequence-specificity of protein dynamics

Abstract

The proper biological functioning of proteins often relies on the occurrence of coordinated fluctuations around their native structure, or on their ability to perform wider and sometimes highly elaborated motions. Hence, there is considerable interest in the definition of accurate coarse-grained descriptions of protein dynamics, as an alternative to more computationally expensive approaches. In particular, the elastic network model, in which residue motions are subjected to pairwise harmonic potentials, is known to capture essential aspects of conformational dynamics in proteins, but has so far remained mostly phenomenological, and unable to account for the chemical specificities of amino acids. We propose, for the first time, a method to derive residue- and distance-specific effective harmonic potentials from the statistical analysis of an extensive dataset of NMR conformational ensembles. These potentials constitute dynamical counterparts to the mean-force statistical potentials commonly used for static analyses of protein structures. In the context of the elastic network model, they yield a strongly improved description of the cooperative aspects of residue motions, and give the opportunity to systematically explore the influence of sequence details on protein dynamics.

Figure 1. Schematic illustration of the apparent stiffness .

A simple model containing 8 beads connected by elastic springs was subjected to integration steps under Gaussian noise. Selected values of , and are given in arbitrary units. Individually, the pairs A–B and C–D would be identical, but they experience differently the influence of the other beads. As a result, the C–D pair is effectively more rigid than A–B (). In both cases, the motions are somewhat correlated, as the apparent stiffness is larger than what is expected from the knowledge of their individual motions (). Beads A and E do not interact directly but the effect of the network on their relative motions is captured by the values of and .

Figure 2. Comparison of the experimental and predicted values of the apparent stiffness .

Experimental values of (continuous black) and (dashed black), extracted from the dataset of 1500 NMR ensembles. Values of predicted on the same dataset by the (dashed red); (continuous red); (dashed blue); (continuous blue).

Figure 3. Comparison of the experimental and predicted values of the apparent stiffness .

For each amino acid, the median value of over the 20 possible partners is given in units of , along with the maximal, minimal, and quartile values. Outliers from these distributions are depicted as circles. (A) Experimental values of , extracted from the dataset of 1500 NMR ensembles. (B) Experimental values of , extracted from the same dataset. (C) Values of predicted by the , on the same dataset. (D) Values of predicted by the , on the same dataset.

Figure 4. Effective harmonic potentials.

(A) Spring constants of the , for the 210 amino acid pairs. (B) Spring constants of the dENM. The dashed line corresponds to . (C) Spring constants of the sdENM for 3 amino-acid pairs. The error bars in panels B–C correspond to the bootstrap estimates of the 90% confidence intervals (see Methods). All values are given in Tables S2, S3, S4, S5, and in Dataset S1.

Figure 5. Performances of the sdENM in a mean protein environment.

(A) Experimental and predicted values of , in the dataset of 1500 NMR ensembles. The Pearson correlation coefficient between predictions and experimental data is equal to 0.95 (). See also Figures S3 and S5. (B) Experimental (continuous) and predicted (dashed) values of , in the dataset of 1500 NMR ensembles. See also Figure S2.

Figure 6. Performances of the sdENM on individual proteins.

The accuracy of the estimation of pairwise residue fluctuations by different ENM variants is illustrated on the basis of two individual proteins. For each protein, 20 randomly selected residue pairs (10 with Å, and 10 with Å) are connected by solid lines. A green line indicates that the amplitude of the fluctuations of the interresidue distance is well estimated by the model. A red (blue) line indicates that the amplitude of the fluctuations of the interresidue distance is largely overestimated (underestimated) by the model. Values larger than 100% or lower than −100% are assimilated to 100% and −100%, respectively. In addition, for each protein and each ENM variant, we report the error on the estimation of pairwise fluctuations (eq. 15), which accounts for all pairs of residues in the protein. (A,C,E) High quality structural ensemble of ubiquitin, obtained by combining NMR information with molecular dynamics simulations (PDB: 1xqq) . (B,D,F) NMR structural ensemble of periplasmic chaperone FimC (PDB: 1bf8). The relatively rigid orientation of the two domains is ensured by specific interdomain interactions . (A–B) . (C–D) . (E–F) .