Characterizing conformation changes in proteins through the torsional elastic response

Why are large conformational changes well described by harmonic normal modes?

Low-frequency normal modes generated by elastic network models tend to correlate strongly with large conformational changes of proteins, despite their reliance on the harmonic approximation, which is only valid in close proximity of the native structure. We consider 12 variants of the torsional network model (TNM), an elastic network model in torsion angle space, that adopt different sets of torsion angles as degrees of freedom and reproduce with similar quality the thermal fluctuations of proteins but present drastic differences in their agreement with conformational changes. We show that these differences are related to the extent of the deviations from the harmonic approximation, assessed through an anharmonic energy function whose harmonic approximation coincides with the TNM. Our results indicate that mode anharmonicity is more strongly related to its collectivity, i.e., the number of atoms displaced by the mode, than to its amplitude; low-frequency modes can remain harmonic even at large amplitudes, provided they are sufficiently collective. Finally, we assess the potential benefits of different strategies to minimize the impact of anharmonicity. The reduction of the number of degrees of freedom or their regularization by a torsional harmonic potential significantly improves the collectivity and harmonicity of normal modes and the agreement with conformational changes. In contrast, the correction of normal mode frequencies to partially account for anharmonicity does not yield substantial benefits. The TNM program is freely available at https://github.com/ugobas/tnm.

Predicted dynamical couplings of protein residues characterize catalysis, transport and allostery

MOTIVATION

Protein function is intrinsically linked to native dynamics, but the systematic characterization of functionally relevant dynamics remains elusive besides specific examples. Here we exhaustively characterize three types of dynamical couplings between protein residues: co-directionality (moving along collinear directions), coordination (small fluctuations of the interatomic distance) and deformation (the extent by which perturbations applied at one residue modify the local structure of the other one), which we analytically compute through the torsional network model.

RESULTS

We find that ligand binding sites are characterized by large within-site coordination and co-directionality, much larger than expected for generic sets of residues with equivalent sequence distances. In addition, catalytic sites are characterized by high coordination couplings with other residues in the protein, supporting the view that the overall protein structure facilitates the catalytic dynamics. The binding sites of allosteric effectors are characterized by comparably smaller coordination and higher within-site deformation than other ligands, which supports their dynamic nature. Allosteric inhibitors are coupled to the active site more frequently through deformation than through coordination, while the contrary holds for activators. We characterize the dynamical couplings of the sodium-dependent Leucine transporter protein (LeuT). The couplings between and within sites progress consistently along the transport cycle, providing a mechanistic description of the coupling between the uptake and release of ions and substrate, and they highlight qualitative differences between the wild-type and a mutant for which chloride is necessary for transport.

AVAILABILITY AND IMPLEMENTATION

The program tnm is freely available at https://github.com/ugobas/tnm.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Can Conformational Changes of Proteins Be Represented in Torsion Angle Space? A Study with Rescaled Ridge Regression

Torsion angles are the natural degrees of freedom of protein structures. The ability to determine torsional variations corresponding to observed changes in Cartesian coordinates is highly valuable, notably to investigate the mechanisms of functional conformational changes or to develop computational models of protein dynamics. This issue is far from trivial in practice since the impact of modifying one torsion angle strongly depends on all other angles, and the compounding effects of small variations in bond lengths and valence angles can completely disrupt a protein fold. We demonstrate that naive strategies, such as directly comparing torsion angles between structures without correcting for variations in bond lengths and valence angles or fitting torsional variations without a proper regularization scheme, fail at producing an adequate representation of conformational changes in internal coordinates. In contrast, rescaled ridge regression, a method recently introduced to regularize multidimensional regressions with correlated explanatory variables, is shown to consistently identify a minimal set of torsion angles variations that closely reproduce changes in Cartesian coordinates. This torsional representation of conformational changes is shown to be robust to the choice of experimental structures. It also provides a better agreement with theoretical models of protein dynamics than the Cartesian representation, regarding notably the predominance of low-frequency normal modes in functional motions and the presence, in predicted equilibrium dynamics, of hints of natural selection for specific functional motions. The software is available at https://github.com/ugobas/tnm .

The Molecular Clock in the Evolution of Protein Structures

The molecular clock hypothesis, which states that substitutions accumulate in protein sequences at a constant rate, plays a fundamental role in molecular evolution but it is violated when selective or mutational processes vary with time. Such violations of the molecular clock have been widely investigated for protein sequences, but not yet for protein structures. Here, we introduce a novel statistical test (Significant Clock Violations) and perform a large scale assessment of the molecular clock in the evolution of both protein sequences and structures in three large superfamilies. After validating our method with computer simulations, we find that clock violations are generally consistent in sequence and structure evolution, but they tend to be larger and more significant in structure evolution. Moreover, changes of function assessed through Gene Ontology and InterPro terms are associated with large and significant clock violations in structure evolution. We found that almost one third of significant clock violations are significant in structure evolution but not in sequence evolution, highlighting the advantage to use structure information for assessing accelerated evolution and gathering hints of positive selection. Clock violations between closely related pairs are frequently significant in sequence evolution, consistent with the observed time dependence of the substitution rate attributed to segregation of neutral and slightly deleterious polymorphisms, but not in structure evolution, suggesting that these substitutions do not affect protein structure although they may affect stability. These results are consistent with the view that natural selection, both negative and positive, constrains more strongly protein structures than protein sequences. Our code for computing clock violations is freely available at https://github.com/ugobas/Molecular_clock.

The maximum penalty criterion for ridge regression: application to the calibration of the force constant in elastic network models

Tikhonov regularization, or ridge regression, is a popular technique to deal with collinearity in multivariate regression. We unveil a formal analogy between ridge regression and statistical mechanics, where the objective function is comparable to a free energy, and the ridge parameter plays the role of temperature. This analogy suggests two novel criteria for selecting a suitable ridge parameter: specific-heat (C_v) and maximum penalty (MP). We apply these fits to evaluate the relative contributions of rigid-body and internal fluctuations, which are typically highly collinear, to crystallographic B-factors. This issue is particularly important for computational models of protein dynamics, such as the elastic network model (ENM), since the amplitude of the predicted internal motion is commonly calibrated using B-factor data. After validation on simulated datasets, our results indicate that rigid-body motions account on average for more than 80% of the amplitude of B-factors. Furthermore, we evaluate the ability of different fits to reproduce the amplitudes of internal fluctuations in X-ray ensembles from the B-factors in the corresponding single X-ray structures. The new ridge criteria are shown to be markedly superior to the commonly used two-parameter fit that neglects rigid-body rotations and to the full fits regularized under generalized cross-validation. In conclusion, the proposed fits ensure a more robust calibration of the ENM force constant and should prove valuable in other applications.

Detecting selection on protein stability through statistical mechanical models of folding and evolution

The properties of biomolecules depend both on physics and on the evolutionary process that formed them. These two points of view produce a powerful synergism. Physics sets the stage and the constraints that molecular evolution has to obey, and evolutionary theory helps in rationalizing the physical properties of biomolecules, including protein folding thermodynamics. To complete the parallelism, protein thermodynamics is founded on the statistical mechanics in the space of protein structures, and molecular evolution can be viewed as statistical mechanics in the space of protein sequences. In this review, we will integrate both points of view, applying them to detecting selection on the stability of the folded state of proteins. We will start discussing positive design, which strengthens the stability of the folded against the unfolded state of proteins. Positive design justifies why statistical potentials for protein folding can be obtained from the frequencies of structural motifs. Stability against unfolding is easier to achieve for longer proteins. On the contrary, negative design, which consists in destabilizing frequently formed misfolded conformations, is more difficult to achieve for longer proteins. The folding rate can be enhanced by strengthening short-range native interactions, but this requirement contrasts with negative design, and evolution has to trade-off between them. Finally, selection can accelerate functional movements by favoring low frequency normal modes of the dynamics of the native state that strongly correlate with the functional conformation change.