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

Riemannian geometry for efficient analysis of protein dynamics data

An increasingly common viewpoint is that protein dynamics datasets reside in a nonlinear subspace of low conformational energy. Ideal data analysis tools should therefore account for such nonlinear geometry. The Riemannian geometry setting can be suitable for a variety of reasons. First, it comes with a rich mathematical structure to account for a wide range of geometries that can be modeled after an energy landscape. Second, many standard data analysis tools developed for data in Euclidean space can be generalized to Riemannian manifolds. In the context of protein dynamics, a conceptual challenge comes from the lack of guidelines for constructing a smooth Riemannian structure based on an energy landscape. In addition, computational feasibility in computing geodesics and related mappings poses a major challenge. This work considers these challenges. The first part of the paper develops a local approximation technique for computing geodesics and related mappings on Riemannian manifolds in a computationally feasible manner. The second part constructs a smooth manifold and a Riemannian structure that is based on an energy landscape for protein conformations. The resulting Riemannian geometry is tested on several data analysis tasks relevant for protein dynamics data. In particular, the geodesics with given start- and end-points approximately recover corresponding molecular dynamics trajectories for proteins that undergo relatively ordered transitions with medium-sized deformations. The Riemannian protein geometry also gives physically realistic summary statistics and retrieves the underlying dimension even for large-sized deformations within seconds on a laptop.

Identification of a Cryptic Pocket in Methionine Aminopeptidase-II Using Adaptive Bandit Molecular Dynamics Simulations and Markov State Models

Methionine aminopeptidase-II (MetAP-II) is a metalloprotease, primarily responsible for the cotranslational removal of the N-terminal initiator methionine from the nascent polypeptide chain during protein synthesis. MetAP-II has been implicated in angiogenesis and endothelial cell proliferation and is therefore considered a validated target for cancer therapeutics. However, there is no effective drug available against MetAP-II. In this study, we employ Adaptive Bandit molecular dynamics simulations to investigate the structural dynamics of the apo and ligand-bound MetAP-II. Our results focus on the dynamic behavior of the disordered loop that is not resolved in most of the crystal structures. Further analysis of the conformational flexibility of the disordered loop reveals a hidden cryptic pocket that is predicted to be potentially druggable. The network analysis indicates that the disordered loop region has a direct signaling route to the active site. These findings highlight a new way to target MetAP-II by designing inhibitors for the allosteric site within this disordered loop region.

Molecular mechanism underlying high-affinity terephthalate binding and conformational change of TBP from Ideonella sakaiensis

Ideonella sakaiensis is the bacterium that can survive by degrading polyethylene terephthalate (PET) plastic, and terephthalic acid (TPA) binding protein (IsTBP) is an essential periplasmic protein for uptake of TPA into the cytosol for complete degradation of PET. Here, we demonstrated that IsTBP has remarkably high specificity for TPA among 33 monophenolic compounds and two 1,6-dicarboxylic acids tested. Structural comparisons with 6-carboxylic acid binding protein (RpAdpC) and TBP from Comamonas sp. E6 (CsTphC) revealed the key structural features that contribute to high TPA specificity and affinity of IsTBP. We also elucidated the molecular mechanism underlying the conformational change upon TPA binding. In addition, we developed the IsTBP variant with enhanced TPA sensitivity, which can be expanded for the use of TBP as a biosensor for PET degradation.

Local Normal Mode Analysis for Fast Loop Conformational Sampling

We propose and validate a novel method to efficiently explore local protein loop conformations based on a new formalism for constrained normal mode analysis (NMA) in internal coordinates. The manifold of possible loop configurations imposed by the position and orientation of the fixed loop ends is reduced to an orthogonal set of motions (or modes) encoding concerted rotations of all the backbone dihedral angles. We validate the sampling power on a set of protein loops with highly variable experimental structures and demonstrate that our approach can efficiently explore the conformational space of closed loops. We also show an acceptable resemblance of the ensembles around equilibrium conformations generated by long molecular simulations and constrained NMA on a set of exposed and diverse loops. In comparison with other methods, the main advantage is the lack of restrictions on the number of dihedrals that can be altered simultaneously. Furthermore, the method is computationally efficient since it only requires the diagonalization of a tiny matrix, and the modes of motions are energetically contextualized by the elastic network model, which includes both the loop and the neighboring residues.

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.

Parameterizing elastic network models to capture the dynamics of proteins

Coarse-grained normal mode analyses of protein dynamics rely on the idea that the geometry of a protein structure contains enough information for computing its fluctuations around its equilibrium conformation. This geometry is captured in the form of an elastic network (EN), namely a network of edges between its residues. The normal modes of a protein are then identified with the normal modes of its EN. Different approaches have been proposed to construct ENs, focusing on the choice of the edges that they are comprised of, and on their parameterizations by the force constants associated with those edges. Here we propose new tools to guide choices on these two facets of EN. We study first different geometric models for ENs. We compare cutoff-based ENs, whose edges have lengths that are smaller than a cutoff distance, with Delaunay-based ENs and find that the latter provide better representations of the geometry of protein structures. We then derive an analytical method for the parameterization of the EN such that its dynamics leads to atomic fluctuations that agree with experimental B-factors. To limit overfitting, we attach a parameter referred to as flexibility constant to each atom instead of to each edge in the EN. The parameterization is expressed as a non-linear optimization problem whose parameters describe both rigid-body and internal motions. We show that this parameterization leads to improved ENs, whose dynamics mimic MD simulations better than ENs with uniform force constants, and reduces the number of normal modes needed to reproduce functional conformational changes.

Mathematical Model of SARS-Cov-2 Propagation Versus ACE2 Fits COVID-19 Lethality Across Age and Sex and Predicts That of SARS

The fatality rate of Covid-19 escalates with age and is larger in men than women. I show that these variations correlate strongly with the level of the viral receptor protein ACE2 in rat lungs, which is consistent with the still limited data on human ACE2. Surprisingly, lower receptor levels correlate with higher fatality. I propose two possible explanations of this negative correlation: First, a previous mathematical model predicts that the velocity of viral progression in the organism as a function of the receptor level has a maximum and declines for abundant receptor. Secondly, degradation of ACE2 by the virus may cause the runaway inflammatory response that characterizes severe CoViD-19. I present here a mathematical model that predicts the lethality as a function of ACE2 protein level based on the two above hypothesis. The model fits Covid-19 fatality rate across age and sex in three countries with high accuracy (r 2 > 0.9 ) under the hypothesis that the speed of viral progression in the infected organism is a decreasing function of the ACE2 level. Moreover, rescaling the fitted parameters by the ratio of the binding rates of the spike proteins of SARS-CoV and SARS-CoV-2 allows predicting the fatality rate of SARS-CoV across age and sex, thus linking the molecular and epidemiological levels.

HOPMA: Boosting Protein Functional Dynamics with Colored Contact Maps

In light of the recent very rapid progress in protein structure prediction, accessing the multitude of functional protein states is becoming more central than ever before. Indeed, proteins are flexible macromolecules, and they often perform their function by switching between different conformations. However, high-resolution experimental techniques such as X-ray crystallography and cryogenic electron microscopy can catch relatively few protein functional states. Many others are only accessible under physiological conditions in solution. Therefore, there is a pressing need to fill this gap with computational approaches. We present HOPMA, a novel method to predict protein functional states and transitions by using a modified elastic network model. The method exploits patterns in a protein contact map, taking its 3D structure as input, and excludes some disconnected patches from the elastic network. Combined with nonlinear normal mode analysis, this strategy boosts the protein conformational space exploration, especially when the input structure is highly constrained, as we demonstrate on a set of more than 400 transitions. Our results let us envision the discovery of new functional conformations, which were unreachable previously, starting from the experimentally known protein structures. The method is computationally efficient and available at https://github.com/elolaine/HOPMA and https://team.inria.fr/nano-d/software/nolb-normal-modes.