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Special Issue Computational Methods and Optimization for Numerical Analysis; International audience; The 3D Zernike polynomials form an orthonormal basis of the unit ball. The associated 3D Zernike moments have been successfully applied for 3D shape recognition; they are popular in structural biology for comparing protein structures and properties. Many algorithms have been proposed for computing those moments, starting from a voxel-based representation or from a surface based geometric mesh of the shape. As the order of the 3D Zernike moments increases, however, those algorithms suffer from decrease in computational efficiency and more importantly from numerical accuracy. In this paper, new algorithms are proposed to compute the 3D Zernike moments of a homogeneous shape defined by an unstructured triangulation of its surface that remove those numerical inaccuracies. These algorithms rely on the analytical integration of the moments on tetrahedra defined by the surface triangles and a central point and on a set of novel recurrent relationships between the corresponding integrals. The mathematical basis and implementation details of the algorithms are presented and their numerical stability is evaluated.

International audience; Originally defined for the optimal allocation of resources, optimal transport (OT) has found many theoretical and practical applications in multiple domains of science and physics. In this Letter we develop a new method for solving the discrete version of this problem using techniques derived from statistical physics. We derive a strongly concave free energy function that captures the constraints of the OT problem at a finite temperature. Its maximum defines an optimal transport plan, or registration between the two discrete probability measures that are compared, as well as a pseudodistance between those measures that satisfies the triangular inequalities. The computation of this pseudodistance is fast and numerically stable. The temperature dependent OT pseudodistance is shown to decrease monotonically with respect to the inverse of the temperature and to converge to the standard OT distance at zero temperature, providing a robust framework for temperature annealing. We illustrate applications of this framework to the problem of image comparison.

International audience; Optimal transport (OT) has become a discipline by itself that offers solutions to a wide range of theoretical problems in probability and mathematics. Despite its appealing theoretical properties, solving the OT problem involves the resolution of a linear program whose computational cost can quickly become prohibitive whenever the size of the problem exceeds a few hundred points. The recent introduction of entropy regularization, however, has led to the development of fast algorithms for solving an approximate OT problem. The successes of those algorithms have resulted in a popularization of the applications of OT in several applied fields such as imaging sciences and machine learning, and in data sciences in general. Problems remain, however, as to the numerical convergence of those regularized approximations towards the actual OT solution. In addition, the physical meaning of this regularization is unclear. In this paper, we propose an approach to solving the discrete OT problem using techniques adapted from statistical physics. Our first contribution is to fully describe this formalism, including all the proofs of its main claims. In particular we derive a strongly concave effective free energy function that captures the constraints of the optimal transport problem at a finite temperature. Its maximum defines a pseudo distance between the two set of weighted points that are compared, which satisfies the triangular inequalities. The temperature dependent OT pseudo distance decreases monotonically to the standard OT distance, providing a robust framework for temperature annealing. Our second contribution is to show that the implementation of this formalism has the same properties as the regularized OT algorithms in time complexity, making it a competitive approach to solving the OT problem. We illustrate applications of the framework to the problem of protein fold recognition based on sequence information only.

International audience; Understanding the dynamics of biomolecules is the key to understanding their biological activities. Computational methods ranging from all-atom molecular dynamics simulations to coarse-grained normal-mode analyses based on simplified elastic networks provide a general framework to studying these dynamics. Despite recent successes in studying very large systems with up to a 100,000,000 atoms, those methods are currently limited to studying small- to medium-sized molecular systems due to computational limitations. One solution to circumvent these limitations is to reduce the size of the system under study. In this paper, we argue that coarse-graining, the standard approach to such size reduction, must define a hierarchy of models of decreasing sizes that are consistent with each other, i.e., that each model contains the information of the dynamics of its predecessor. We propose a new method, Decimate, for generating such a hierarchy within the context of elastic networks for normal-mode analysis. This method is based on the concept of the renormalization group developed in statistical physics. We highlight the details of its implementation, with a special focus on its scalability to large systems of up to millions of atoms. We illustrate its application on two large systems, the capsid of a virus and the ribosome translation complex. We show that highly decimated representations of those systems, containing down to 1% of their original number of atoms, still capture qualitatively and quantitatively their dynamics. Decimate is available as an OpenSource resource.

International audience; Connecting the dots among the amino acid sequence of a protein, its structure, and its function remains a central theme in molecular biology, as it would have many applications in the treatment of illnesses related to misfolding or protein instability. As a result of high-throughput sequencing methods, biologists currently live in a protein sequence-rich world. However, our knowledge of protein structure based on experimental data remains comparatively limited. As a consequence, protein structure prediction has established itself as a very active field of research to fill in this gap. This field, once thought to be reserved for theoretical biophysicists, is constantly reinventing itself, borrowing ideas informed by an ever-increasing assembly of scientific domains, from biology, chemistry, (statistical) physics, mathematics, computer science, statistics, bioinformatics, and more recently data sciences. We review the recent progress arising from this integration of knowledge, from the development of specific computer architecture to allow for longer timescales in physics-based simulations of protein folding to the recent advances in predicting contacts in proteins based on detection of coevolution using very large data sets of aligned protein sequences.

International audience; We propose a novel stochastic method to generate Brownian paths conditioned to start at an initial point and end at a given final point during a fixed time t f under a given potential U (x). These paths are sampled with a probability given by the overdamped Langevin dynamics. We show that these paths can be exactly generated by a local Stochastic Partial Differential Equation (SPDE). This equation cannot be solved in general but we present several approximations that are valid either in the low temperature regime or in the presence of barrier crossing. We show that this method warrants the generation of statistically independent transition paths. It is computationally very efficient. We illustrate the method first on two simple potentials, the two dimensional Mueller potential as well as on the Mexican hat potential, and then on the multi-dimensional problem of conformational transitions in proteins using the "Mixed Elastic Network Model" as a benchmark.

International audience; Barbiturates induce anesthesia by modulating the activity of anionic and cationic pentameric ligand-gated ion channels (pLGICs). Despite more than a century of use in clinical practice, the prototypic binding site for this class of drugs within pLGICs is yet to be described. In this study, we present the first X-ray structures of barbiturates bound to GLIC, a cationic prokaryotic pLGIC with excellent structural homology to other relevant channels sensitive to general anesthetics and, as shown here, to barbiturates, at clinically relevant concentrations. Several derivatives of barbiturates containing anomalous scatterers were synthesized, and these derivatives helped us unambiguously identify a unique barbiturate binding site within the central ion channel pore in a closed conformation. In addition, docking calculations around the observed binding site for all three states of the receptor, including a model of the desensitized state, showed that barbiturates preferentially stabilize the closed state. The identification of this pore binding site sheds light on the mechanism of barbiturate inhibition of cationic pLGICs and allows the rationalization of several structural and functional features previously observed for barbiturates.

International audience; We present a method for establishing correspondences be- tween human cortical surfaces that exactly matches the positions of given point landmarks, while attaining the global minimum of an ob- jective function that quantifies how far the mapping deviates from con- formality. On each surface, a conformal transformation is applied to the Euclidean distance metric, resulting in a hyperbolic metric with isolated cone point singularities at the landmarks. Equivalently, each surface is mapped to a hyperbolic orbifold: a pillow-like surface with each point landmark corresponding to a pillow corner. An initial surface-to-surface mapping exactly aligns the landmarks, and gradient descent is used to find the single, global minimum of the Dirichlet energy of the remainder of the mapping. Using a population of real MRI-based cortical surfaces with manually labeled sulcus endpoints as landmarks, we evaluate the approach by how much it distorts surfaces and by its biological plau- sibility: how well it aligns previously-unseen anatomical landmarks and by how well it promotes expected associations between cortical thickness and age. We show that, compared to a painstakingly-tuned approach that balances a tradeoff between minimizing landmark mismatch and Dirichlet energy, our method has similar biological plausibility, superior surface distortion, a better theoretical foundation, and fewer arbitrary parameters to tune. We also compare to conformal mapper in the spher- ical domain to show that sacrificing exact conformality of the mapping does not cause noticeable reductions in biological plausibility.

International audience; 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.

International audience; The Poisson-Boltzmann (PB) formalism is among the most popular approaches to modeling the solvation of molecules. It assumes a continuum model for water, leading to a dielectric permittivity that only depends on position in space. In contrast, the dipolar Poisson-Boltzmann-Langevin (DPBL) formalism represents the solvent as a collection of orientable dipoles with nonuniform concentration; this leads to a nonlinear permittivity function that depends both on the position and on the local electric field at that position. The differences in the assumptions underlying these two models lead to significant differences in the equations they generate. The PB equation is a second order, elliptic, nonlinear partial differential equation (PDE). Its response coefficients correspond to the dielectric permittivity and are therefore constant within each subdomain of the system considered (i.e., inside and outside of the molecules considered). While the DPBL equation is also a second order, elliptic, nonlinear PDE, its response coefficients are nonlinear functions of the electrostatic potential. Many solvers have been developed for the PB equation; to our knowledge, none of these can be directly applied to the DPBL equation. The methods they use may adapt to the difference; their implementations however are PBE specific. We adapted the PBE solver originally developed by Holst and Saied [J. Comput. Chem. 16, 337 (1995)] to the problem of solving the DPBL equation. This solver uses a truncated Newton method with a multigrid preconditioner. Numerical evidences suggest that it converges for the DPBL equation and that the convergence is superlinear. It is found however to be slow and greedy in memory requirement for problems commonly encountered in computational biology and computational chemistry. To circumvent these problems, we propose two variants, a quasi-Newton solver based on a simplified, inexact Jacobian and an iterative self-consistent solver that is based directly on the PBE solver. While both methods are not guaranteed to converge, numerical evidences suggest that they do and that their convergence is also superlinear. Both variants are significantly faster than the solver based on the exact Jacobian, with a much smaller memory footprint. All three methods have been implemented in a new code named AQUASOL, which is freely available.

International audience; A new continuum model is presented for computing the solvation free energies of cations in water. It combines in a single formalism based on statistical thermodynamics the Poisson model for electrostatics with the Langevin dipole model to account for nonuniform water dipole distribution around the ions. An excellent match between experimental and computed solvation free energies is obtained for 10 monovalent and divalent ions.

Two new crystal structures of Bacillus stearothermophilus tryptophanyl-tRNA synthetase (TrpRS) afford evidence that a closed interdomain hinge angle requires a covalent bond between AMP and an occupant of either pyrophosphate or tryptophan subsite. They also are within experimental error of a cluster of structures observed in a nonequilibrium molecular dynamics simulation showing partial active-site assembly. Further, the highest energy structure in a minimum action pathway computed by using elastic network models for Open and Pretransition state (PreTS) conformations for the fully liganded TrpRS monomer is intermediate between that simulated structure and a partially disassembled structure from a nonequilibrium molecular dynamics trajectory for the unliganded PreTS. These mutual consistencies provide unexpected validation of inferences drawn from molecular simulations.

International audience; 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.

We describe a new way to calculate the electrostatic properties of macromolecules that goes beyond the classical Poisson-Boltzmann treatment with only a small extra CPU cost. The solvent region is no longer modeled as a homogeneous dielectric media but rather as an assembly of self-orienting interacting dipoles of variable density. The method effectively unifies both the Poisson-centric view and the Langevin Dipole model. The model results in a variable dielectric constant \documentclass[10pt]{article} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{pmc} \usepackage[Euler]{upgreek} \pagestyle{empty} \oddsidemargin -1.0in \begin{document} \begin{equation*}{\epsilon}({\vec{r}})\end{equation*}\end{document} in the solvent region and also in a variable solvent density \documentclass[10pt]{article} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{pmc} \usepackage[Euler]{upgreek} \pagestyle{empty} \oddsidemargin -1.0in \begin{document} \begin{equation*}{\rho}({\vec{r}})\end{equation*}\end{document} that depends on the nature of the closest exposed solute atoms. The model was calibrated using small molecules and ions solvation data with only two adjustable parameters, namely the size and dipolar moment of the solvent. Hydrophobicity scales derived from the solvent density profiles agree very well with independently derived hydrophobicity scales, both at the atomic or residue level. Dimerization interfaces in homodimeric proteins or lipid-binding regions in membrane proteins clearly appear as poorly solvated patches on the solute accessible surface. Comparison of the thermally averaged solvent density of this model with the one derived from molecular dynamics simulations shows qualitative agreement on a coarse-grained level. Because this calculation is much more rapid than that from molecular dynamics, applications of a density-profile-based solvation energy to the identification of the true structure among a set of decoys become computationally feasible. Various possible improvements of the model are discussed, as well as extensions of the formalism to treat mixtures of dipolar solvents of different sizes.

Background Cone degeneration is the hallmark of the inherited retinal disease retinitis pigmentosa. We have previously identified a trophic factor "Rod-derived Cone Viability Factor (RdCVF) that is secreted by rods and promote cone viability in a mouse model of the disease. Results Here we report the bioinformatic identification and the experimental analysis of RdCVF2, a second trophic factor belonging to the Rod-derived Cone Viability Factor family. The mouse RdCVF gene is known to be bifunctional, encoding both a long thioredoxin-like isoform (RdCVF-L) and a short isoform with trophic cone photoreceptor viability activity (RdCVF-S). RdCVF2 shares many similarities with RdCVF in terms of gene structure, expression in a rod-dependent manner and protein 3D structure. Furthermore, like RdCVF, the RdCVF2 short isoform exhibits cone rescue activity that is independent of its putative thiol-oxydoreductase activity. Conclusion Taken together, these findings define a new family of bifunctional genes which are: expressed in vertebrate retina, encode trophic cone viability factors, and have major therapeutic potential for human retinal neurodegenerative diseases such as retinitis pigmentosa.

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.

International audience; Normal mode analysis (NMA) is an efficient way to study collective motions in biomolecules that bypasses the computational costs and many limitations associated with full dynamics simulations. The NOMAD-Ref web server presented here provides tools for online calculation of the normal modes of large molecules (up to 100,000 atoms) maintaining a full all-atom representation of their structures, as well as access to a number of programs that utilize these collective motions for deformation and refinement of biomolecular structures. Applications include the generation of sets of decoys with correct stereochemistry but arbitrary large amplitude movements, the quantification of the overlap between alternative conformations of a molecule, refinement of structures against experimental data, such as X-ray diffraction structure factors or Cryo-EM maps and optimization of docked complexes by modeling receptor/ligand flexibility through normal mode motions. The server can be accessed at the URL http://lorentz.immstr.pasteur.fr/nomad-ref.php.

International audience; Multiple sequence alignment is one of the cornerstones of modern molecular biology. It is used to identify conserved motifs, to determine protein domains, in 2D/3D structure prediction by homology and in evolutionary studies. Recently, high-throughput technologies such as genome sequencing and structural proteomics have lead to an explosion in the amount of sequence and structure information available. In response, several new multiple alignment methods have been developed that improve both the efficiency and the quality of protein alignments. Consequently, the benchmarks used to evaluate and compare these methods must also evolve. We present here the latest release of the most widely used multiple alignment benchmark, BAliBASE, which provides high quality, manually refined, reference alignments based on 3D structural superpositions. Version 3.0 of BAliBASE includes new, more challenging test cases, representing the real problems encountered when aligning large sets of complex sequences. Using a novel, semiautomatic update protocol, the number of protein families in the benchmark has been increased and representative test cases are now available that cover most of the protein fold space. The total number of proteins in BAliBASE has also been significantly increased from 1444 to 6255 sequences. In addition, full-length sequences are now provided for all test cases, which represent difficult cases for both global and local alignment programs. Finally, the BAliBASE Web site (http://www-bio3d-igbmc.u-strasbg.fr/balibase) has been completely redesigned to provide a more user-friendly, interactive interface for the visualization of the BAliBASE reference alignments and the associated annotations.

A newly defined family of fungal lectins displays no significant sequence similarity to any protein in the databases. These proteins, made of about 140 amino acid residues, have sequence identities ranging from 38% to 65% and share binding specificity to N-acetyl galactosamine. One member of this family, the lectin XCL from Xerocomus chrysenteron, induces drastic changes in the actin cytoskeleton after sugar binding at the cell surface and internalization, and has potent insecticidal activity. The crystal structure of XCL to 1.4 A resolution reveals the architecture of this new lectin family. The fold of the protein is not related to any of the several lectin folds documented so far. Unexpectedly, the structure similarity is significant with actinoporins, a family of pore-forming toxins. The specific structural features and sequence signatures in each protein family suggest a potential sugar binding site in XCL and a possible evolutionary relationship between these proteins. Finally, the tetrameric assembly of XCL reveals a complex network of protomer-protomer interfaces and generates a large, hydrated cavity of 1000 A3, which may become accessible to larger solutes after a small conformational change of the protein.

The biological function of macromolecules, such as proteins and nucleic acids, relies heavily on their interactions with their partners. The prediction of how molecules interact and how they can create large assemblies acting as nanomachines is essential for our understanding of biology but also for therapeutics and nanotechnology design.Blind challenges in biology, such as the CAPRI worldwide experiment for docking, have shown that in silico studies and simulations, mainly using physics-based potentials and techniques, could give structural insights in atomic detail. They might however be of very limited accuracy, particularly in predicting the native molecular structure of proteins, RNAs and complexes.Resorting to simple geometric coarse-grained modelling and machine learning strategies, such as genetic algorithms and support vector machines, we have shown that scoring the putative complex structures can be very much improved to reach the accuracy needed for experiment design and analysis, at least in a semi-rigid body context. From that proof of concept studies, most of the prediction strategies for docking now use machine learning for scoring optimization.Being able to predict the structure and the way molecular partners deform upon binding is also key to obtain better predictions, in particular for non-coding RNAs that are essential to target oncogenes. Our efforts in RNA structure prediction techniques have shown that data based parameterization of energy functions and statistical techniques largely improve the accuracy of structure prediction.Reaching the large assemblies stage also requires to be able to assess the dynamics of molecules from partial experimental data. We developed an efficient sampling technique based on inverse kinematics that does not rely on constraint counting, and implicitly calculates the rigidity of the molecule. Paired with experimental data, it offers an integrative view of the dynamics of non-coding RNAs for biological processes. Combined with clustering techniques, it allows for efficient and flexible cross-docking analysis for protein-RNA complexes.

The biological function of macromolecules, such as proteins and nucleic acids, relies heavily on their interactions with their partners. The prediction of how molecules interact and how they can create large assemblies acting as nanomachines is essential for our understanding of biology but also for therapeutics and nanotechnology design.Blind challenges in biology, such as the CAPRI worldwide experiment for docking, have shown that in silico studies and simulations, mainly using physics-based potentials and techniques, could give structural insights in atomic detail. They might however be of very limited accuracy, particularly in predicting the native molecular structure of proteins, RNAs and complexes.Resorting to simple geometric coarse-grained modelling and machine learning strategies, such as genetic algorithms and support vector machines, we have shown that scoring the putative complex structures can be very much improved to reach the accuracy needed for experiment design and analysis, at least in a semi-rigid body context. From that proof of concept studies, most of the prediction strategies for docking now use machine learning for scoring optimization.Being able to predict the structure and the way molecular partners deform upon binding is also key to obtain better predictions, in particular for non-coding RNAs that are essential to target oncogenes. Our efforts in RNA structure prediction techniques have shown that data based parameterization of energy functions and statistical techniques largely improve the accuracy of structure prediction.Reaching the large assemblies stage also requires to be able to assess the dynamics of molecules from partial experimental data. We developed an efficient sampling technique based on inverse kinematics that does not rely on constraint counting, and implicitly calculates the rigidity of the molecule. Paired with experimental data, it offers an integrative view of the dynamics of non-coding RNAs for biological processes. Combined with clustering techniques, it allows for efficient and flexible cross-docking analysis for protein-RNA complexes.