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Extracting dynamical pairwise correlations and identifying key residues from large molecular dynamics trajectories or normal-mode analysis of coarse-grained models are important for explaining various processes like ligand binding, mutational effects, and long-distance interactions. Efficient and flexible tools to perform this task can provide new insights about residues involved in allosteric regulation and protein function. In addition, combining and comparing dynamical coupling information with sequence coevolution data can help to understand better protein function. To this aim, we developed a Python package called correlationplus to calculate, visualize, and analyze pairwise correlations. In this way, the package aids to identify key residues and interactions in proteins. The source code of correlationplus is available under LGPL version 3 at https://github.com/tekpinar/correlationplus. The current version of the package (0.2.0) can be installed with common installation methods like conda or pip in addition to source code installation. Moreover, docker images are also available for usage of the code without installation.

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., (© 2021 Wiley Periodicals LLC.)

We present an extension of the Poisson-Boltzmann model in which the solute of interest is immersed in an assembly of self-orienting Langevin water dipoles, anions, cations, and hydrophobic molecules, all of variable densities. Interactions between charges are controlled by electrostatics, while hydrophobic interactions are modeled with a Yukawa potential. We impose steric constraints by assuming that the system is represented on a cubic lattice. We also assume incompressibility; i.e., all sites of the lattice are occupied. This model, which we refer to as the Hydrophobic Dipolar Poisson-Boltzmann Langevin (HDPBL) model, leads to a system of two equations whose solutions give the water dipole, salt, and hydrophobic molecule densities, all of them in the presence of the others in a self-consistent way. We use those to study the organization of the ions, cosolvent, and solvent molecules around proteins. In particular, peaks of densities are expected to reveal, simultaneously, the presence of compatible binding sites of different kinds on a protein. We have tested and validated the ability of HDPBL to detect pockets in proteins that bind to hydrophobic ligands, polar ligands, and charged small probes as well as to characterize the binding sites of lipids for membrane proteins.

Residues in proteins that are in close spatial proximity are more prone to covariate as their interactions are likely to be preserved due to structural and evolutionary constraints. If we can detect and quantify such covariation, physical contacts may then be predicted in the structure of a protein solely from the sequences that decorate it. To carry out such predictions, and following the work of others, we have implemented a multivariate Gaussian model to analyze correlation in multiple sequence alignments. We have explored and tested several numerical encodings of amino acids within this model. We have shown that 1D encodings based on amino acid biochemical and biophysical properties, as well as higher dimensional encodings computed from the principal components of experimentally derived mutation/substitution matrices, do not perform as well as a simple twenty dimensional encoding with each amino acid represented with a vector of one along its own dimension and zero elsewhere. The optimum obtained from representations based on substitution matrices is reached by using 10 to 12 principal components; the corresponding performance is less than the performance obtained with the 20-dimensional binary encoding. We highlight also the importance of the prior when constructing the multivariate Gaussian model of a multiple sequence alignment.

We present an extension of the self-consistent mean field theory for protein side-chain modeling in which solvation effects are included based on the Poisson-Boltzmann (PB) theory. In this approach, the protein is represented with multiple copies of its side chains. Each copy is assigned a weight that is refined iteratively based on the mean field energy generated by the rest of the protein, until self-consistency is reached. At each cycle, the variational free energy of the multi-copy system is computed; this free energy includes the internal energy of the protein that accounts for vdW and electrostatics interactions and a solvation free energy term that is computed using the PB equation. The method converges in only a few cycles and takes only minutes of central processing unit time on a commodity personal computer. The predicted conformation of each residue is then set to be its copy with the highest weight after convergence. We have tested this method on a database of hundred highly refined NMR structures to circumvent the problems of crystal packing inherent to x-ray structures. The use of the PB-derived solvation free energy significantly improves prediction accuracy for surface side chains. For example, the prediction accuracies for χ(1) for surface cysteine, serine, and threonine residues improve from 68%, 35%, and 43% to 80%, 53%, and 57%, respectively. A comparison with other side-chain prediction algorithms demonstrates that our approach is consistently better in predicting the conformations of exposed side chains.

We present an extension to the Poisson-Boltzmann model in which the solvent is modeled as an assembly of self-orienting dipoles of variable densities. Interactions between these dipoles are included implicitly using a Yukawa potential field. This model leads to a set of equations whose solutions give the dipole densities; we use the latter to study the organization of water around biomolecules. The computed water density profiles resemble those derived from molecular dynamics simulations. We also derive an excess free energy that discriminates correct from incorrect conformations of proteins.

The non-linear problem of simulating the structural transition between two known forms of a macromolecule still remains a challenge in structural biology. The problem is usually addressed in an approximate way using 'morphing' techniques, which are linear interpolations of either the Cartesian or the internal coordinates between the initial and end states, followed by energy minimization. Here we describe a web tool that implements a new method to calculate the most probable trajectory that is exact for harmonic potentials; as an illustration of the method, the classical Calpha-based Elastic Network Model (ENM) is used both for the initial and the final states but other variants of the ENM are also possible. The Langevin equation under this potential is solved analytically using the Onsager and Machlup action minimization formalism on each side of the transition, thus replacing the original non-linear problem by a pair of linear differential equations joined by a non-linear boundary matching condition. The crossover between the two multidimensional energy curves around each state is found numerically using an iterative approach, producing the most probable trajectory and fully characterizing the transition state and its energy. Jobs calculating such trajectories can be submitted on-line at: http://lorentz.dynstr.pasteur.fr/joel/index.php.

We describe a new way to calculate the electrostatic properties of macromolecules which eliminates the assumption of a constant dielectric value in the solvent region, resulting in a Generalized Poisson-Boltzmann-Langevin equation (GPBLE). We have implemented a web server (http://lorentz.immstr.pasteur.fr/pdb_hydro.php) that both numerically solves this equation and uses the resulting water density profiles to place water molecules at preferred sites of hydration. Surface atoms with high or low hydration preference can be easily displayed using a simple PyMol script, allowing for the tentative prediction of the dimerization interface in homodimeric proteins, or lipid binding regions in membrane proteins. The web site includes options that permit mutations in the sequence as well as reconstruction of missing side chain and/or main chain atoms. These tools are accessible independently from the electrostatics calculation, and can be used for other modeling purposes. We expect this web server to be useful to structural biologists, as the knowledge of solvent density should prove useful to get better fits at low resolution for X-ray diffraction data and to computational biologists, for whom these profiles could improve the calculation of interaction energies in water between ligands and receptors in docking simulations.

Prediction of structural changes resulting from complex formation, both in ligands and receptors, is an important and unsolved problem in structural biology. In this work, we use all-atom normal modes calculated with the Elastic Network Model as a basis set to model structural flexibility during formation of macromolecular complexes and refine the non-bonded intermolecular energy between the two partners (protein-ligand or protein-DNA) along 5-10 of the lowest frequency normal mode directions. The method handles motions unrelated to the docking transparently by first applying the modes that improve non-bonded energy most and optionally restraining amplitudes; in addition, the method can correct small errors in the ligand position when the first six rigid-body modes are switched on. For a test set of six protein receptors that show an open-to-close transition when binding small ligands, our refinement scheme reduces the protein coordinate cRMS by 0.3-3.2 A. For two test cases of DNA structures interacting with proteins, the program correctly refines the docked B-DNA starting form into the expected bent DNA, reducing the DNA cRMS from 8.4 to 4.8 A and from 8.7 to 5.4 A, respectively. A public web server implementation of the refinement method is available at http://lorentz.immstr.pasteur.fr.

As more and more structures of macromolecular complexes get solved in different conditions, it has become apparent that flexibility is an inherent part of their biological function. Normal mode analysis using simplified models of proteins such as the elastic network model has proved very effective in showing that many of the structural transitions derived from a survey of the Protein Data Bank can be explained by just a few of the lowest-frequency normal modes. In this work, normal modes are used to carry out medium- or low-resolution structural refinement, enforcing collective and large-amplitude movements that are beyond the reach of existing methods. Refinement is carried out in reciprocal space with respect to the normal mode amplitudes, by using standard conjugate-gradient minimization. Several tests on synthetic diffraction data whose mode concentration follows the one of real movements observed in the Protein Data Bank have shown that the radius of convergence is larger than the one of rigid-body refinement. Tests with experimental diffraction data for the same protein in different environments also led to refined structural models showing drastic reduction of the rms deviation with the target model. Because the structural transition is described by very few parameters, over-fitting of real experimental data is easily detected by using a cross-validation test. The method has also been applied to the refinement of atomic models into molecular envelopes and could readily be used to fit large macromolecular complex rearrangements into cryo-electron microscopy-reconstructed images as well as small-angle x-ray scattering-derived envelopes.

Simulated-annealing methods have been used to resolve the phase-ambiguity problem in the centrosymmetric P¿bar¿1¿ space group. First, an energy function based on the Sayre equation is introduced and a formal comparison with classical spin systems is drawn. The energy landscape is studied in detail and the validity of several energy criteria thoroughly tested. Classical Monte Carlo methods proved to be successful using a multistart optimization of the Sayre score, along with the additional monitoring of other energetic criteria. These involved the Terwilliger map quality index in reciprocal space in the absence of envelope information, and an envelope score if the shape of the molecule is known. The inherent phase-ambiguity problem of the P¿bar¿1¿ space group was therefore technically solved by Monte Carlo methods. The method should also work to resolve phase ambiguity in the SIR method of protein crystallography.

The goal of the inverse folding problem is to supply a list of sequences compatible with a known protein structure. If two-body interactions are taken into account in energy calculations, an exhaustive exploration of the energy landscape in sequence space cannot be achieved because of the huge number of possible combinations. To circumvent this problem, we propose a method in which multiple copies corresponding to every possible side-chain type are attached to each C alpha position in the protein. The weights of each copy (stored in the sequence matrix SM) are refined using mean field theory: each side-chain copy interacts with the mean field generated by all possible side-chain copies at neighbouring positions, weighted by their respective probabilities. The potential energy is simply taken to be amino acid pair potentials of mean force. The method converges in a few cycles to a self-consistent solution. The refined matrix does not depend on the starting point; therefore the method succeeds in removing memory effects. Starting solely from the backbone of the known structure, and without information from the initial sequence, the final sequence matrix SM is shown to be able to retrieve significant sequence information, as observed through a series of structure-recognizes-sequence(s) computer experiments. The issue of specificity is discussed in detail.

Globular proteins have highly compact structures and the corresponding packing interactions are widely considered as the principal determinant of the native structure. It is therefore important that theoretical approaches to protein design explicitly take in account packing, which requires that a full atomic representation of the designed protein is maintained. As a first step towards this goal, we have developed in this report an inverse folding algorithm with the aim of specifically designing amino acid sequences which optimise sidechain packing for a given protein fold. The design is performed by a global Monte Carlo optimisation in sequence space, with constant amino acid composition and a full-atom representation of the various protein models. Packing is defined by a Lennard-Jones potential. The program was tested by designing stable sequence variants for the chymotrypsin inhibitor fold. The final protein models showed an increase in intramolecular atomic contacts and a decrease in the overall volume compared to the native structure. Starting from the backbone only of the target structure, the algorithm did gradually retrieve reliable though limited sequence information. Higher compatibility might be achieved by improving the potential, however our results suggest that packing interactions are an essential element of a yet-to-be-defined successful energy function for protein design.

Atomic contact potentials are derived by statistical analysis of atomic surface contact areas versus atom type in a database of non-homologous protein structures. The atomic environment is characterized by the surface area accessible to solvent and the surface of contacts with polar and non-polar atoms. Four types of atoms are considered, namely neutral polar atoms from protein backbones and from protein side-chains, non-polar atoms and charged atoms. Potential energies delta Ej(E) are defined from the preference for an atom of type j to be in a given environment E compared to the expected value if everything was random; Boltzmann's law is then used to transform these preferences into energies. These new potentials very clearly discriminate misfolded from correct structural models. The performance of these potentials are critically assessed by monitoring the recognition of the native fold among a large number of alternative structural folding types (the hide-and-seek procedure), as well as by testing if the native sequence can be recovered from a large number of randomly shuffled sequences for a given 3D fold (a procedure similar to the inverse folding problem). We suggest that these potentials reflect the atomic short range non-local interactions in proteins. To characterise atomic solvation alone, similar potentials were derived as a function of the percentage of solvent-accessible area alone. These energies were found to agree reasonably well with the solvation formalism of Eisenberg and McLachlan.

Understanding the relations between the conformation of the side-chains and the backbone geometry is crucial for structure prediction as well as for homology modelling. To attempt to unravel these rules, we have developed a method which allows us to predict the position of the side-chains from the co-ordinates of the main-chain atoms. This method is based on a rotamer library and refines iteratively a conformational matrix of the side-chains of a protein, CM, such that its current element at each cycle CM (ij) gives the probability that side-chain i of the protein adopts the conformation of its possible rotamer j. Each residue feels the average of all possible environments, weighted by their respective probabilities. The method converges in only a few cycles, thereby deserving the name of self consistent mean field method. Using the rotamer with the highest probability in the optimized conformational matrix to define the conformation of the side-chain leads to the result that on average 72% of chi 1, 75% of chi 2 and 62% of chi 1 + 2 are correctly predicted for a set of 30 proteins. Tests with six pairs of homologous proteins have shown that the method is quite successful even when the protein backbone deviates from the correct conformation. The second application of the optimized conformational matrix was to provide estimates of the conformational entropy of the side-chains in the folded state of the protein. The relevance of this entropy is discussed.

The crystal structure of the catalytic core of murine terminal deoxynucleotidyltransferase (TdT) at 2.35 Å resolution reveals a typical DNA polymerase β-like fold locked in a closed form. In addition, the structures of two different binary complexes, one with an oligonucleotide primer and the other with an incoming ddATP-Co2+ complex, show that the substrates and the two divalent ions in the catalytic site are positioned in TdT in a manner similar to that described for the human DNA polymerase β ternary complex, suggesting a common two metal ions mechanism of nucleotidyl transfer in these two proteins. The inability of TdT to accommodate a template strand can be explained by steric hindrance at the catalytic site caused by a long lariat-like loop, which is absent in DNA polymerase β. However, displacement of this discriminating loop would be sufficient to unmask a number of evolutionarily conserved residues, which could then interact with a template DNA strand. The present structure can be used to model the recently discovered human polymerase μ, with which it shares 43% sequence identity. [ABSTRACT FROM AUTHOR]