Potentials of mean force for protein structure prediction vindicated, formalized and generalized

Unlocking the specificity of antimicrobial peptide interactions for membrane-targeted therapies

Antimicrobial peptides (AMPs) are increasingly recognized as potent therapeutic agents, with their selective affinity for pathological membranes, low toxicity profile, and minimal resistance development making them particularly attractive in the pharmaceutical landscape. This study offers a comprehensive analysis of the interaction between specific AMPs, including magainin-2, pleurocidin, CM15, LL37, and clavanin, with lipid bilayer models of very different compositions that have been ordinarily used as biological membrane models of healthy mammal, cancerous, and bacterial cells. Employing unbiased molecular dynamics simulations and metadynamics techniques, we have deciphered the intricate mechanisms by which these peptides recognize pathogenic and pathologic lipid patterns and integrate into lipid assemblies. Our findings reveal that the transverse component of the peptide's hydrophobic dipole moment is critical for membrane interaction, decisively influencing the molecule's orientation and expected therapeutic efficacy. Our approach also provides insight on the kinetic and dynamic dependence on the peptide orientation in the axial and azimuthal angles when coming close to the membrane. The aim is to establish a robust framework for the rational design of peptide-based, membrane-targeted therapies, as well as effective quantitative descriptors that can facilitate the automated design of novel AMPs for these therapies using machine learning methods.

Hierarchical Framework for Predicting Entropies in Bottom-Up Coarse-Grained Models

The thermodynamic entropy of coarse-grained (CG) models stands as one of the most important properties for quantifying the missing information during the CG process and for establishing transferable (or extendible) CG interactions. However, performing additional CG simulations on top of model construction often leads to significant additional computational overhead. In this work, we propose a simple hierarchical framework for predicting the thermodynamic entropies of various molecular CG systems. Our approach employs a decomposition of the CG interactions, enabling the estimation of the CG partition function and thermodynamic properties a priori. Starting from the ideal gas description, we leverage classical perturbation theory to systematically incorporate simple yet essential interactions, ranging from the hard sphere model to the generalized van der Waals model. Additionally, we propose an alternative approach based on multiparticle correlation functions, allowing for systematic improvements through higher-order correlations. Numerical applications to molecular liquids validate the high fidelity of our approach, and our computational protocols demonstrate that a reduced model with simple energetics can reasonably estimate the thermodynamic entropy of CG models without performing any CG simulations. Overall, our findings present a systematic framework for estimating not only the entropy but also other thermodynamic properties of CG models, relying solely on information from the reference system.

Multi-eGO: Model Improvements toward the Study of Complex Self-Assembly Processes

Structure-based models have been instrumental in simulating protein folding and suggesting hypotheses about the mechanisms involved. Nowadays, at least for fast-folding proteins, folding can be simulated in explicit solvent using classical molecular dynamics. However, other self-assembly processes, such as protein aggregation, are still far from being accessible. Recently, we proposed that a hybrid multistate structure-based model, multi-eGO, could help to bridge the gap toward the simulation of out-of-equilibrium, concentration-dependent self-assembly processes. Here, we further improve the model and show how multi-eGO can effectively and accurately learn the conformational ensemble of the amyloid β42 intrinsically disordered peptide, reproduce the well-established folding mechanism of the B1 immunoglobulin-binding domain of streptococcal protein G, and reproduce the aggregation as a function of the concentration of the transthyretin 105-115 amyloidogenic peptide. We envision that by learning from the dynamics of a few minima, multi-eGO can become a platform for simulating processes inaccessible to other simulation techniques.

Gaussian representation of coarse-grained interactions of liquids: Theory, parametrization, and transferability

Coarse-grained (CG) interactions determined via bottom-up methodologies can faithfully reproduce the structural correlations observed in fine-grained (atomistic resolution) systems, yet they can suffer from limited extensibility due to complex many-body correlations. As part of an ongoing effort to understand and improve the applicability of bottom-up CG models, we propose an alternative approach to address both accuracy and transferability. Our main idea draws from classical perturbation theory to partition the hard sphere repulsive term from effective CG interactions. We then introduce Gaussian basis functions corresponding to the system's characteristic length by linking these Gaussian sub-interactions to the local particle densities at each coordination shell. The remaining perturbative long-range interaction can be treated as a collective solvation interaction, which we show exhibits a Gaussian form derived from integral equation theories. By applying this numerical parametrization protocol to CG liquid systems, our microscopic theory elucidates the emergence of Gaussian interactions in common phenomenological CG models. To facilitate transferability for these reduced descriptions, we further infer equations of state to determine the sub-interaction parameter as a function of the system variables. The reduced models exhibit excellent transferability across the thermodynamic state points. Furthermore, we propose a new strategy to design the cross-interactions between distinct CG sites in liquid mixtures. This involves combining each Gaussian in the proper radial domain, yielding accurate CG potentials of mean force and structural correlations for multi-component systems. Overall, our findings establish a solid foundation for constructing transferable bottom-up CG models of liquids with enhanced extensibility.

Conformational Stability and Denaturation Processes of Proteins Investigated by Electrophoresis under Extreme Conditions

The functional structure of proteins results from marginally stable folded conformations. Reversible unfolding, irreversible denaturation, and deterioration can be caused by chemical and physical agents due to changes in the physicochemical conditions of pH, ionic strength, temperature, pressure, and electric field or due to the presence of a cosolvent that perturbs the delicate balance between stabilizing and destabilizing interactions and eventually induces chemical modifications. For most proteins, denaturation is a complex process involving transient intermediates in several reversible and eventually irreversible steps. Knowledge of protein stability and denaturation processes is mandatory for the development of enzymes as industrial catalysts, biopharmaceuticals, analytical and medical bioreagents, and safe industrial food. Electrophoresis techniques operating under extreme conditions are convenient tools for analyzing unfolding transitions, trapping transient intermediates, and gaining insight into the mechanisms of denaturation processes. Moreover, quantitative analysis of electrophoretic mobility transition curves allows the estimation of the conformational stability of proteins. These approaches include polyacrylamide gel electrophoresis and capillary zone electrophoresis under cold, heat, and hydrostatic pressure and in the presence of non-ionic denaturing agents or stabilizers such as polyols and heavy water. Lastly, after exposure to extremes of physical conditions, electrophoresis under standard conditions provides information on irreversible processes, slow conformational drifts, and slow renaturation processes. The impressive developments of enzyme technology with multiple applications in fine chemistry, biopharmaceutics, and nanomedicine prompted us to revisit the potentialities of these electrophoretic approaches. This feature review is illustrated with published and unpublished results obtained by the authors on cholinesterases and paraoxonase, two physiologically and toxicologically important enzymes.

Statistical potentials from the Gaussian scaling behaviour of chain fragments buried within protein globules

Knowledge-based approaches use the statistics collected from protein data-bank structures to estimate effective interaction potentials between amino acid pairs. Empirical relations are typically employed that are based on the crucial choice of a reference state associated to the null interaction case. Despite their significant effectiveness, the physical interpretation of knowledge-based potentials has been repeatedly questioned, with no consensus on the choice of the reference state. Here we use the fact that the Flory theorem, originally derived for chains in a dense polymer melt, holds also for chain fragments within the core of globular proteins, if the average over buried fragments collected from different non-redundant native structures is considered. After verifying that the ensuing Gaussian statistics, a hallmark of effectively non-interacting polymer chains, holds for a wide range of fragment lengths, although with significant deviations at short spatial scales, we use it to define a ‘bona fide’ reference state. Notably, despite the latter does depend on fragment length, deviations from it do not. This allows to estimate an effective interaction potential which is not biased by the presence of correlations due to the connectivity of the protein chain. We show how different sequence-independent effective statistical potentials can be derived using this approach by coarse-graining the protein representation at varying levels. The possibility of defining sequence-dependent potentials is explored.

Structure‐conditioned amino‐acid couplings: how contact geometry affects pairwise sequence preferences

Relating a protein's sequence to its conformation is a central challenge for both structure prediction and sequence design. Statistical contact potentials, as well as their more descriptive versions that account for side‐chain orientation and other geometric descriptors, have served as simplistic but useful means of representing second‐order contributions in sequence–structure relationships. Here we ask what happens when a pairwise potential is conditioned on the fully defined geometry of interacting backbones fragments. We show that the resulting structure‐conditioned coupling energies more accurately reflect pair preferences as a function of structural contexts. These structure‐conditioned energies more reliably encode native sequence information and more highly correlate with experimentally determined coupling energies. Clustering a database of interaction motifs by structure results in ensembles of similar energies and clustering them by energy results in ensembles of similar structures. By comparing many pairs of interaction motifs and showing that structural similarity and energetic similarity go hand‐in‐hand, we provide a tangible link between modular sequence and structure elements. This link is applicable to structural modeling, and we show that scoring CASP models with structured‐conditioned energies results in substantially higher correlation with structural quality than scoring the same models with a contact potential. We conclude that structure‐conditioned coupling energies are a good way to model the impact of interaction geometry on second‐order sequence preferences.

Reconciling Simulations and Experiments With BICePs: A Review

Bayesian Inference of Conformational Populations (BICePs) is an algorithm developed to reconcile simulated ensembles with sparse experimental measurements. The Bayesian framework of BICePs enables population reweighting as a post-simulation processing step, with several advantages over existing methods, including the proper use of reference potentials, and the estimation of a Bayes factor-like quantity called the BICePs score for model selection. Here, we summarize the theory underlying this method in context with related algorithms, review the history of BICePs applications to date, and discuss current shortcomings along with future plans for improvement.

Representations of protein structure for exploring the conformational space: A speed-accuracy trade-off

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We compare ten structural representations, either atomistic or coarse-grained.

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Thus, ten distance-dependent statistical potentials of mean force (PMF) were built.

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The Cβ-only and Cα + Cβ representations provide the best speed–accuracy trade-off.

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Including glycines through Cα, in a Cβ-only representation, yields a higher accuracy.

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We generalize the conclusions to the total information gain (TIG) scoring function.

The recent breakthrough in the field of protein structure prediction shows the relevance of using knowledge-based based scoring functions in combination with a low-resolution 3D representation of protein macromolecules. The choice of not using all atoms is barely supported by any data in the literature, and is mostly motivated by empirical and practical reasons, such as the computational cost of assessing the numerous folds of the protein conformational space. Here, we present a comprehensive study, carried on a large and balanced benchmark of predicted protein structures, to see how different types of structural representations rank in either accuracy or calculation speed, and which ones offer the best compromise between these two criteria. We tested ten representations, including low-resolution, high-resolution, and coarse-grained approaches. We also investigated the generalization of the findings to other formalisms than the widely-used “potential of mean force” (PMF) method. Thus, we observed that representing protein structures by their β carbons—combined or not with Cα—provides the best speed–accuracy trade-off, when using a “total information gain” scoring function. For statistical PMFs, using MARTINI backbone and side-chains beads is the best option. Finally, we also demonstrated the necessity of training the reference state on all atom types, and of including the Cα atoms of glycine residues, in a Cβ-based representation.

MHCII3D-Robust Structure Based Prediction of MHC II Binding Peptides

Knowledge of MHC II binding peptides is highly desired in immunological research, particularly in the context of cancer, autoimmune diseases, or allergies. The most successful prediction methods are based on machine learning methods trained on sequences of experimentally characterized binding peptides. Here, we describe a complementary approach called MHCII3D, which is based on structural scaffolds of MHC II-peptide complexes and statistical scoring functions (SSFs). The MHC II alleles reported in the Immuno Polymorphism Database are processed in a dedicated 3D-modeling pipeline providing a set of scaffold complexes for each distinct allotype sequence. Antigen protein sequences are threaded through the scaffolds and evaluated by optimized SSFs. We compared the predictive power of MHCII3D with different sequence-based machine learning methods. The Pearson correlation to experimentally determine IC₅₀ values for MHC II Automated Server Benchmarks data sets from IEDB (Immune Epitope Database) is 0.42, which is in the competitor methods range. We show that MHCII3D is quite robust in leaving one molecule out tests and is therefore not prone to overfitting. Finally, we provide evidence that MHCII3D can complement the current sequence-based methods and help to identify problematic entries in IEDB. Scaffolds and MHCII3D executables can be freely downloaded from our web pages.

An information gain-based approach for evaluating protein structure models

For three decades now, knowledge-based scoring functions that operate through the "potential of mean force" (PMF) approach have continuously proven useful for studying protein structures. Although these statistical potentials are not to be confused with their physics-based counterparts of the same name-i.e. PMFs obtained by molecular dynamics simulations-their particular success in assessing the native-like character of protein structure predictions has lead authors to consider the computed scores as approximations of the free energy. However, this physical justification is a matter of controversy since the beginning. Alternative interpretations based on Bayes' theorem have been proposed, but the misleading formalism that invokes the inverse Boltzmann law remains recurrent in the literature. In this article, we present a conceptually new method for ranking protein structure models by quality, which is (i) independent of any physics-based explanation and (ii) relevant to statistics and to a general definition of information gain. The theoretical development described in this study provides new insights into how statistical PMFs work, in comparison with our approach. To prove the concept, we have built interatomic distance-dependent scoring functions, based on the former and new equations, and compared their performance on an independent benchmark of 60,000 protein structures. The results demonstrate that our new formalism outperforms statistical PMFs in evaluating the quality of protein structural decoys. Therefore, this original type of score offers a possibility to improve the success of statistical PMFs in the various fields of structural biology where they are applied. The open-source code is available for download at https://gitlab.rpbs.univ-paris-diderot.fr/src/ig-score.

Intergenerational Transmission of Education and ADHD: Effects of Parental Genotypes

It remains a challenge to determine whether children resemble their parents due to nature, nurture, or a mixture of both. Here we used a design that exploits the distinction between transmitted and non-transmitted alleles in genetic transmission from parent to offspring. Two separate polygenic scores (PGS) were calculated on the basis of the transmitted and non-transmitted alleles. The effect of the non-transmitted PGS is necessarily mediated by parental phenotypes, insofar as they contribute to the rearing environment of the offspring (genetic nurturing). We calculated transmitted and non-transmitted PGSs associated with adult educational attainment (EA) and PGSs associated with childhood ADHD in a general population sample of trios, i.e. child or adult offspring and their parents (N = 1120-2518). We tested if the EA and ADHD (non-)transmitted PGSs were associated with childhood academic achievement and ADHD in offspring. Based on the earlier findings for shared environment, we hypothesized to find genetic nurturing for academic achievement, but not for ADHD. In adults, both transmitted (R2 = 7.6%) and non-transmitted (R2 = 1.7%) EA PGSs were associated with offspring EA, evidencing genetic nurturing. In children around age 12, academic achievement was associated with the transmitted EA PGSs (R2 = 5.7%), but we found no support for genetic nurturing (R2 ~ 0.1%). The ADHD PGSs were not significantly associated with academic achievement (R2 ~ 0.6%). ADHD symptoms in children were only associated with transmitted EA PGSs and ADHD PGSs (R2 = 1-2%). Based on these results, we conclude that the associations between parent characteristics and offspring outcomes in childhood are mainly to be attributable to the effects of genes that are shared by parents and children.

Discrimination power of knowledge-based potential dictated by the dominant energies in native protein structures

Extracting a well-designed energy function is important for protein structure evaluation. Knowledge-based potential functions are one type of the energy functions which can be obtained from known protein structures. The pairwise potential between atom types is approximated using Boltzmann's law which relates the frequency of atom types to its potential. The total energy is approximated as a summation of pairwise potential between the atomic pairs. In the present study, the performance of knowledge-based potential function was assessed based on the strength of interaction between groups of amino acids. The dominant energies involved in the pairwise potentials were revealed by eigenvalue analysis of the matrix, the elements of which represent the energy between amino acids. For this purpose, the matrix including the mean of the energies of residue-residue interaction types was constructed using 500 native protein structures. The matrix has a dominant eigenvalue and amino acids, with LEU, VAL, ILE, PHE, TYR, ALA and TRP having high values along the dominant eigenvector. The results show that the ranking of amino acids is consistent with the power of amino acids in discriminating native structures using K-alphabet reduced model. In the reduced interactions, only amino acids from a subset of all 20 amino acids, along with their interactions are considered to assess the energy. In the K-alphabet reduced model, the reduced structures are constructed based on only the K-amino acid types. The dominant K-alphabet reduced model derived for the k-first amino acids in the list [LEU, VAL, PHE, ILE, TYR, ALA, TRP] of amino acids has the best discrimination of native structure among all possible K-alphabet reduced models. Knowledge-based potentials might be improved with a new strategy.

Accurate calculation of side chain packing and free energy with applications to protein molecular dynamics

To address the large gap between time scales that can be easily reached by molecular simulations and those required to understand protein dynamics, we present a rapid self-consistent approximation of the side chain free energy at every integration step. In analogy with the adiabatic Born-Oppenheimer approximation for electronic structure, the protein backbone dynamics are simulated as preceding according to the dictates of the free energy of an instantaneously-equilibrated side chain potential. The side chain free energy is computed on the fly, allowing the protein backbone dynamics to traverse a greatly smoothed energetic landscape. This computation results in extremely rapid equilibration and sampling of the Boltzmann distribution. Our method, termed Upside, employs a reduced model involving the three backbone atoms, along with the carbonyl oxygen and amide proton, and a single (oriented) side chain bead having multiple locations reflecting the conformational diversity of the side chain's rotameric states. We also introduce a novel, maximum-likelihood method to parameterize the side chain interactions using protein structures. We demonstrate state-of-the-art accuracy for predicting χ1 rotamer states while consuming only milliseconds of CPU time. Our method enables rapidly equilibrating coarse-grained simulations that can nonetheless contain significant molecular detail. We also show that the resulting free energies of the side chains are sufficiently accurate for de novo folding of some proteins.

Contact Potential for Structure Prediction of Proteins and Protein Complexes from Potts Model

The energy function is the key component of protein modeling methodology. This work presents a semianalytical approach to the development of contact potentials for protein structure modeling. Residue-residue and atom-atom contact energies were derived by maximizing the probability of observing native sequences in a nonredundant set of protein structures. The optimization task was formulated as an inverse statistical mechanics problem applied to the Potts model. Its solution by pseudolikelihood maximization provides consistent estimates of coupling constants at atomic and residue levels. The best performance was achieved when interacting atoms were grouped according to their physicochemical properties. For individual protein structures, the performance of the contact potentials in distinguishing near-native structures from the decoys is similar to the top-performing scoring functions. The potentials also yielded significant improvement in the protein docking success rates. The potentials recapitulated experimentally determined protein stability changes upon point mutations and protein-protein binding affinities. The approach offers a different perspective on knowledge-based potentials and may serve as the basis for their further development.

MyPMFs: a simple tool for creating statistical potentials to assess protein structural models

Evaluating the model quality of protein structures that evolve in environments with particular physicochemical properties requires scoring functions that are adapted to their specific residue compositions and/or structural characteristics. Thus, computational methods developed for structures from the cytosol cannot work properly on membrane or secreted proteins. Here, we present MyPMFs, an easy-to-use tool that allows users to train statistical potentials of mean force (PMFs) on the protein structures of their choice, with all parameters being adjustable. We demonstrate its use by creating an accurate statistical potential for transmembrane protein domains. We also show its usefulness to study the influence of the physical environment on residue interactions within protein structures. Our open-source software is freely available for download at https://github.com/bibip-impmc/mypmfs.

Modeling transcriptional activation changes to Gal4 variants via structure-based computational mutagenesis

As a DNA binding transcriptional activator, Gal4 promotes the expression of genes responsible for galactose metabolism. The Gal4 protein from Saccharomyces cerevisiae (baker’s yeast) has become a model for studying eukaryotic transcriptional activation in general because its regulatory properties mirror those of several eukaryotic organisms, including mammals. Given the availability of a crystallographic structure for Gal4, here we implement an in silico mutagenesis technique that makes use of a four-body knowledge-based energy function, in order to empirically quantify the structural impacts associated with single residue substitutions on the Gal4 protein. These results were used to examine the structure-function relationship in Gal4 based on a recently published experimental mutagenesis study, whereby functional changes to a uniformly distributed set of 1,068 single residue Gal4 variants were obtained by measuring their transcriptional activation levels relative to wild-type. A significant correlation was observed between computed (scalar) structural effect data and measured activity values for this collection of single residue Gal4 variants. Additionally, attribute vectors quantifying position-specific environmental impacts were generated for each of the Gal4 variants via computational mutagenesis, and we implemented supervised classification and regression statistical machine learning algorithms to train predictive models of variant Gal4 activity based on these structural changes. All models performed well under cross-validation testing, with balanced accuracy reaching 91% among the classification models, and with the actual and predicted activity values displaying a correlation as high as r = 0.80 for the regression models. Reliable predictions of transcriptional activation levels for Gal4 variants that have yet to be studied can be instantly generated by submitting their respective structure-based feature vectors to the trained models for testing. Such a computational pre-screening of Gal4 variants may potentially reduce costs associated with running large-scale mutagenesis experiments.

Model Selection Using BICePs: A Bayesian Approach for Force Field Validation and Parameterization

The Bayesian Inference of Conformational Populations (BICePs) algorithm reconciles theoretical predictions of conformational state populations with sparse and/or noisy experimental measurements. Among its key advantages is its ability to perform objective model selection through a quantity we call the BICePs score, which reflects the integrated posterior evidence in favor of a given model, computed through free energy estimation methods. Here, we explore how the BICePs score can be used for force field validation and parametrization. Using a 2D lattice protein as a toy model, we demonstrate that BICePs is able to select the correct value of an interaction energy parameter given ensemble-averaged experimental distance measurements. We show that if conformational states are sufficiently fine-grained, the results are robust to experimental noise and measurement sparsity. Using these insights, we apply BICePs to perform force field evaluations for all-atom simulations of designed β-hairpin peptides against experimental NMR chemical shift measurements. These tests suggest that BICePs scores can be used for model selection in the context of all-atom simulations. We expect this approach to be particularly useful for the computational foldamer design as a tool for improving general-purpose force fields given sparse experimental measurements.

Finding the needle in the haystack: towards solving the protein-folding problem computationally

Prediction of protein tertiary structures from amino acid sequence and understanding the mechanisms of how proteins fold, collectively known as "the protein folding problem," has been a grand challenge in molecular biology for over half a century. Theories have been developed that provide us with an unprecedented understanding of protein folding mechanisms. However, computational simulation of protein folding is still difficult, and prediction of protein tertiary structure from amino acid sequence is an unsolved problem. Progress toward a satisfying solution has been slow due to challenges in sampling the vast conformational space and deriving sufficiently accurate energy functions. Nevertheless, several techniques and algorithms have been adopted to overcome these challenges, and the last two decades have seen exciting advances in enhanced sampling algorithms, computational power and tertiary structure prediction methodologies. This review aims at summarizing these computational techniques, specifically conformational sampling algorithms and energy approximations that have been frequently used to study protein-folding mechanisms or to de novo predict protein tertiary structures. We hope that this review can serve as an overview on how the protein-folding problem can be studied computationally and, in cases where experimental approaches are prohibitive, help the researcher choose the most relevant computational approach for the problem at hand. We conclude with a summary of current challenges faced and an outlook on potential future directions.

A Generative Angular Model of Protein Structure Evolution

Recently described stochastic models of protein evolution have demonstrated that the inclusion of structural information in addition to amino acid sequences leads to a more reliable estimation of evolutionary parameters. We present a generative, evolutionary model of protein structure and sequence that is valid on a local length scale. The model concerns the local dependencies between sequence and structure evolution in a pair of homologous proteins. The evolutionary trajectory between the two structures in the protein pair is treated as a random walk in dihedral angle space, which is modeled using a novel angular diffusion process on the two-dimensional torus. Coupling sequence and structure evolution in our model allows for modeling both “smooth” conformational changes and “catastrophic” conformational jumps, conditioned on the amino acid changes. The model has interpretable parameters and is comparatively more realistic than previous stochastic models, providing new insights into the relationship between sequence and structure evolution. For example, using the trained model we were able to identify an apparent sequence–structure evolutionary motif present in a large number of homologous protein pairs. The generative nature of our model enables us to evaluate its validity and its ability to simulate aspects of protein evolution conditioned on an amino acid sequence, a related amino acid sequence, a related structure or any combination thereof.

Establishing knowledge on the sequence arrangement pattern of nucleated protein folding

The heat-tolerance mechanisms of (hyper)thermophilic proteins provide a unique opportunity to investigate the unsolved protein folding problem. In an attempt to determine whether the interval between residues in sequence might play a role in determining thermostability, we constructed a sequence interval-dependent value function to calculate the residue pair frequency. Additionally, we identified a new sequence arrangement pattern, where like-charged residues tend to be adjacently assembled, while unlike-charged residues are distributed over longer intervals, using statistical analysis of a large sequence database. This finding indicated that increasing the intervals between unlike-charged residues can increase protein thermostability, with the arrangement patterns of these charged residues serving as thermodynamically favorable nucleation points for protein folding. Additionally, we identified that the residue pairs K-E, R-E, L-V and V-V involving long sequence intervals play important roles involving increased protein thermostability. This work demonstrated a novel approach for considering sequence intervals as keys to understanding protein folding. Our findings of novel relationships between residue arrangement and protein thermostability can be used in industry and academia to aid the design of thermostable proteins.

Bayesian inference of protein ensembles from SAXS data

The inherent flexibility of intrinsically disordered proteins (IDPs) and multi-domain proteins with intrinsically disordered regions (IDRs) presents challenges to structural analysis. These macromolecules need to be represented by an ensemble of conformations, rather than a single structure. Small-angle X-ray scattering (SAXS) experiments capture ensemble-averaged data for the set of conformations. We present a Bayesian approach to ensemble inference from SAXS data, called Bayesian ensemble SAXS (BE-SAXS). We address two issues with existing methods: the use of a finite ensemble of structures to represent the underlying distribution, and the selection of that ensemble as a subset of an initial pool of structures. This is achieved through the formulation of a Bayesian posterior of the conformational space. BE-SAXS modifies a structural prior distribution in accordance with the experimental data. It uses multi-step expectation maximization, with alternating rounds of Markov-chain Monte Carlo simulation and empirical Bayes optimization. We demonstrate the method by employing it to obtain a conformational ensemble of the antitoxin PaaA2 and comparing the results to a published ensemble.

SARAH Domain-Mediated MST2-RASSF Dimeric Interactions

RASSF enzymes act as key apoptosis activators and tumor suppressors, being downregulated in many human cancers, although their exact regulatory roles remain unknown. A key downstream event in the RASSF pathway is the regulation of MST kinases, which are main effectors of RASSF-induced apoptosis. The regulation of MST1/2 includes both homo- and heterodimerization, mediated by helical SARAH domains, though the underlying molecular interaction mechanism is unclear. Here, we study the interactions between RASSF1A, RASSF5, and MST2 SARAH domains by using both atomistic molecular simulation techniques and experiments. We construct and study models of MST2 homodimers and MST2-RASSF SARAH heterodimers, and we identify the factors that control their high molecular stability. In addition, we also analyze both computationally and experimentally the interactions of MST2 SARAH domains with a series of synthetic peptides particularly designed to bind to it, and hope that our approach can be used to address some of the challenging problems in designing new anti-cancer drugs.

We model the conformational changes and protein-protein interactions of enzymes involved in signaling along the Hippo pathway—a key molecular mechanism that controls the process of programmed cell death in eukaryotic cells, including cells affected by cancer. Combining modern computational modeling techniques with experimental information from X-ray crystallography and systems biology studies, can unveil detailed molecular interactions and lead to novel drugs. Here, we study the atomistic mechanisms and interactions between MST2 and RASSF-type kinases, through their respective SARAH domains—highly conserved, long, terminal α-helices, which play essential roles in the activation of MST kinases and, therefore, in modulating apoptosis. In spite of their key roles in mediating cell signaling pathways, there is little structural information available for the RASSF SARAH domains and their dimerization with the MST2 SARAH domains. In particular, the RASSF1A crystal structure is not available yet. Here, we model, refine and validate atomistic structural models of dimers of the RASSF1A and MST2 SARAH domains, studying the interaction and the dynamic behavior of these molecular complexes using homology modeling, docking and full atomistic molecular dynamics simulations. Experimentally, we validate our approach by designing a novel peptide that can disrupt effectively MST2 homo and hetero SARAH dimers.

An Atomistic Statistically Effective Energy Function for Computational Protein Design

Shortcomings in the definition of effective free-energy surfaces of proteins are recognized to be a major contributory factor responsible for the low success rates of existing automated methods for computational protein design (CPD). The formulation of an atomistic statistically effective energy function (SEEF) suitable for a wide range of CPD applications and its derivation from structural data extracted from protein domains and protein-ligand complexes are described here. The proposed energy function comprises nonlocal atom-based and local residue-based SEEFs, which are coupled using a novel atom connectivity number factor to scale short-range, pairwise, nonbonded atomic interaction energies and a surface-area-dependent cavity energy term. This energy function was used to derive additional SEEFs describing the unfolded-state ensemble of any given residue sequence based on computed average energies for partially or fully solvent-exposed fragments in regions of irregular structure in native proteins. Relative thermal stabilities of 97 T4 bacteriophage lysozyme mutants were predicted from calculated energy differences for folded and unfolded states with an average unsigned error (AUE) of 0.84 kcal mol(-1) when compared to experiment. To demonstrate the utility of the energy function for CPD, further validation was carried out in tests of its capacity to recover cognate protein sequences and to discriminate native and near-native protein folds, loop conformers, and small-molecule ligand binding poses from non-native benchmark decoys. Experimental ligand binding free energies for a diverse set of 80 protein complexes could be predicted with an AUE of 2.4 kcal mol(-1) using an additional energy term to account for the loss in ligand configurational entropy upon binding. The atomistic SEEF is expected to improve the accuracy of residue-based coarse-grained SEEFs currently used in CPD and to extend the range of applications of extant atom-based protein statistical potentials.

Coarse-grained modeling of RNA 3D structure

Functional RNA molecules depend on three-dimensional (3D) structures to carry out their tasks within the cell. Understanding how these molecules interact to carry out their biological roles requires a detailed knowledge of RNA 3D structure and dynamics as well as thermodynamics, which strongly governs the folding of RNA and RNA-RNA interactions as well as a host of other interactions within the cellular environment. Experimental determination of these properties is difficult, and various computational methods have been developed to model the folding of RNA 3D structures and their interactions with other molecules. However, computational methods also have their limitations, especially when the biological effects demand computation of the dynamics beyond a few hundred nanoseconds. For the researcher confronted with such challenges, a more amenable approach is to resort to coarse-grained modeling to reduce the number of data points and computational demand to a more tractable size, while sacrificing as little critical information as possible. This review presents an introduction to the topic of coarse-grained modeling of RNA 3D structures and dynamics, covering both high- and low-resolution strategies. We discuss how physics-based approaches compare with knowledge based methods that rely on databases of information. In the course of this review, we discuss important aspects in the reasoning process behind building different models and the goals and pitfalls that can result.

Towards an accurate prediction of the thermal stability of homologous proteins

Protein thermostability has been the focus of growing research interests in the last decades since its understanding and control play important roles in the optimization of a wide series of bioprocesses of academic and industrial importance. The complexity of this issue is rooted in the fact that the mechanisms ensuring thermal resistance are not unique and specific, but rather family- or even protein-dependent. Therefore, and despite the amount of research already accomplished, obtaining fast and precise thermal stability predictions is still a challenge, especially on a large scale. This article deepens the study of protein thermal stability and is focused on the prediction of its best descriptor, the melting temperature Tm. The relations between Tm and a series of factors that are expected to influence the protein stability are analyzed and discussed. Different Tm-prediction methods that utilize these factors, sometimes with additional information about homologous proteins, are introduced, and their individual performances are evaluated. The best methods are based on temperature-dependent statistical potentials, on the environmental temperature of the host organism, on the fraction of charged residues, and on the number of residues. They are combined to build an improved prediction method with significantly increased score. The root mean square deviation between the computed and experimental Tm-values for 45 proteins of known structure from 11 families is about 7°C in cross-validation and decreases to 5°C when 10% outliers are removed. The associated linear correlation coefficients are equal to .91 and .95, respectively.

Predicting RNA 3D structure using a coarse-grain helix-centered model

A 3D model of RNA structure can provide information about its function and regulation that is not possible with just the sequence or secondary structure. Current models suffer from low accuracy and long running times and either neglect or presume knowledge of the long-range interactions which stabilize the tertiary structure. Our coarse-grained, helix-based, tertiary structure model operates with only a few degrees of freedom compared with all-atom models while preserving the ability to sample tertiary structures given a secondary structure. It strikes a balance between the precision of an all-atom tertiary structure model and the simplicity and effectiveness of a secondary structure representation. It provides a simplified tool for exploring global arrangements of helices and loops within RNA structures. We provide an example of a novel energy function relying only on the positions of stems and loops. We show that coupling our model to this energy function produces predictions as good as or better than the current state of the art tools. We propose that given the wide range of conformational space that needs to be explored, a coarse-grain approach can explore more conformations in less iterations than an all-atom model coupled to a fine-grain energy function. Finally, we emphasize the overarching theme of providing an ensemble of predicted structures, something which our tool excels at, rather than providing a handful of the lowest energy structures.

Feature extraction by statistical contact potentials and wavelet transform for predicting subcellular localizations in gram negative bacterial proteins

Predicting the localization of a protein has become a useful practice for inferring its function. Most of the reported methods to predict subcellular localizations in Gram-negative bacterial proteins make use of standard protein representations that generally do not take into account the distribution of the amino acids and the structural information of the proteins. Here, we propose a protein representation based on the structural information contained in the pairwise statistical contact potentials. The wavelet transform decodes the information contained in the primary structure of the proteins, allowing the identification of patterns along the proteins, which are used to characterize the subcellular localizations. Then, a support vector machine classifier is trained to categorize them. Cellular compartments like periplasm and extracellular medium are difficult to predict, having a high false negative rate. The wavelet-based method achieves an overall high performance while maintaining a low false negative rate, particularly, on "periplasm" and "extracellular medium". Our results suggest the proposed protein characterization is a useful alternative to representing and predicting protein sequences over the classical and cutting edge protein depictions.

Bayesian inference of conformational state populations from computational models and sparse experimental observables

We present a Bayesian inference approach to estimating conformational state populations from a combination of molecular modeling and sparse experimental data. Unlike alternative approaches, our method is designed for use with small molecules and emphasizes high-resolution structural models, using inferential structure determination with reference potentials, and Markov Chain Monte Carlo to sample the posterior distribution of conformational states. As an application of the method, we determine solution-state conformational populations of the 14-membered macrocycle cineromycin B, using a combination of previously published sparse Nuclear Magnetic Resonance (NMR) observables and replica-exchange molecular dynamic/Quantum Mechanical (QM)-refined conformational ensembles. Our results agree better with experimental data compared to previous modeling efforts. Bayes factors are calculated to quantify the consistency of computational modeling with experiment, and the relative importance of reference potentials and other model parameters.

ROTAS: a rotamer-dependent, atomic statistical potential for assessment and prediction of protein structures

Background

Multibody potentials accounting for cooperative effects of molecular interactions have shown better accuracy than typical pairwise potentials. The main challenge in the development of such potentials is to find relevant structural features that characterize the tightly folded proteins. Also, the side-chains of residues adopt several specific, staggered conformations, known as rotamers within protein structures. Different molecular conformations result in different dipole moments and induce charge reorientations. However, until now modeling of the rotameric state of residues had not been incorporated into the development of multibody potentials for modeling non-bonded interactions in protein structures.

Results

In this study, we develop a new multibody statistical potential which can account for the influence of rotameric states on the specificity of atomic interactions. In this potential, named “rotamer-dependent atomic statistical potential” (ROTAS), the interaction between two atoms is specified by not only the distance and relative orientation but also by two state parameters concerning the rotameric state of the residues to which the interacting atoms belong. It was clearly found that the rotameric state is correlated to the specificity of atomic interactions. Such rotamer-dependencies are not limited to specific type or certain range of interactions. The performance of ROTAS was tested using 13 sets of decoys and was compared to those of existing atomic-level statistical potentials which incorporate orientation-dependent energy terms. The results show that ROTAS performs better than other competing potentials not only in native structure recognition, but also in best model selection and correlation coefficients between energy and model quality.

Conclusions

A new multibody statistical potential, ROTAS accounting for the influence of rotameric states on the specificity of atomic interactions was developed and tested on decoy sets. The results show that ROTAS has improved ability to recognize native structure from decoy models compared to other potentials. The effectiveness of ROTAS may provide insightful information for the development of many applications which require accurate side-chain modeling such as protein design, mutation analysis, and docking simulation.

Electronic supplementary material

The online version of this article (doi:10.1186/1471-2105-15-307) contains supplementary material, which is available to authorized users.

Protein structure validation and refinement using amide proton chemical shifts derived from quantum mechanics

We present the ProCS method for the rapid and accurate prediction of protein backbone amide proton chemical shifts--sensitive probes of the geometry of key hydrogen bonds that determine protein structure. ProCS is parameterized against quantum mechanical (QM) calculations and reproduces high level QM results obtained for a small protein with an RMSD of 0.25 ppm (r = 0.94). ProCS is interfaced with the PHAISTOS protein simulation program and is used to infer statistical protein ensembles that reflect experimentally measured amide proton chemical shift values. Such chemical shift-based structural refinements, starting from high-resolution X-ray structures of Protein G, ubiquitin, and SMN Tudor Domain, result in average chemical shifts, hydrogen bond geometries, and trans-hydrogen bond ((h3)J(NC')) spin-spin coupling constants that are in excellent agreement with experiment. We show that the structural sensitivity of the QM-based amide proton chemical shift predictions is needed to obtain this agreement. The ProCS method thus offers a powerful new tool for refining the structures of hydrogen bonding networks to high accuracy with many potential applications such as protein flexibility in ligand binding.

Efficient Parameter Estimation of Generalizable Coarse-Grained Protein Force Fields Using Contrastive Divergence: A Maximum Likelihood Approach

Maximum Likelihood (ML) optimization schemes are widely used for parameter inference. They maximize the likelihood of some experimentally observed data, with respect to the model parameters iteratively, following the gradient of the logarithm of the likelihood. Here, we employ a ML inference scheme to infer a generalizable, physics-based coarse-grained protein model (which includes Go̅-like biasing terms to stabilize secondary structure elements in room-temperature simulations), using native conformations of a training set of proteins as the observed data. Contrastive divergence, a novel statistical machine learning technique, is used to efficiently approximate the direction of the gradient ascent, which enables the use of a large training set of proteins. Unlike previous work, the generalizability of the protein model allows the folding of peptides and a protein (protein G) which are not part of the training set. We compare the same force field with different van der Waals (vdW) potential forms: a hard cutoff model, and a Lennard-Jones (LJ) potential with vdW parameters inferred or adopted from the CHARMM or AMBER force fields. Simulations of peptides and protein G show that the LJ model with inferred parameters outperforms the hard cutoff potential, which is consistent with previous observations. Simulations using the LJ potential with inferred vdW parameters also outperforms the protein models with adopted vdW parameter values, demonstrating that model parameters generally cannot be used with force fields with different energy functions. The software is available at https://sites.google.com/site/crankite/.

The electrostatic profile of consecutive Cβ atoms applied to protein structure quality assessment

The structure of a protein provides insight into its physiological interactions with other components of the cellular soup. Methods that predict putative structures from sequences typically yield multiple, closely-ranked possibilities. A critical component in the process is the model quality assessing program (MQAP), which selects the best candidate from this pool of structures. Here, we present a novel MQAP based on the physical properties of sidechain atoms. We propose a method for assessing the quality of protein structures based on the electrostatic potential difference (EPD) of Cβ atoms in consecutive residues. We demonstrate that the EPDs of Cβ atoms on consecutive residues provide unique signatures of the amino acid types. The EPD of Cβ atoms are learnt from a set of 1000 non-homologous protein structures with a resolution cuto of 1.6 Å obtained from the PISCES database. Based on the Boltzmann hypothesis that lower energy conformations are proportionately sampled more, and on Annsen's thermodynamic hypothesis that the native structure of a protein is the minimum free energy state, we hypothesize that the deviation of observed EPD values from the mean values obtained in the learning phase is minimized in the native structure. We achieved an average specificity of 0.91, 0.94 and 0.93 on hg_structal, 4state_reduced and ig_structal decoy sets, respectively, taken from the Decoys `R' Us database. The source code and manual is made available at https://github.com/sanchak/mqap and permanently available on 10.5281/zenodo.7134.

Inference of structure ensembles of flexible biomolecules from sparse, averaged data

We present the theoretical foundations of a general principle to infer structure ensembles of flexible biomolecules from spatially and temporally averaged data obtained in biophysical experiments. The central idea is to compute the Kullback-Leibler optimal modification of a given prior distribution with respect to the experimental data and its uncertainty. This principle generalizes the successful inferential structure determination method and recently proposed maximum entropy methods. Tractability of the protocol is demonstrated through the analysis of simulated nuclear magnetic resonance spectroscopy data of a small peptide.

Formulation of probabilistic models of protein structure in atomic detail using the reference ratio method

We propose a method to formulate probabilistic models of protein structure in atomic detail, for a given amino acid sequence, based on Bayesian principles, while retaining a close link to physics. We start from two previously developed probabilistic models of protein structure on a local length scale, which concern the dihedral angles in main chain and side chains, respectively. Conceptually, this constitutes a probabilistic and continuous alternative to the use of discrete fragment and rotamer libraries. The local model is combined with a nonlocal model that involves a small number of energy terms according to a physical force field, and some information on the overall secondary structure content. In this initial study we focus on the formulation of the joint model and the evaluation of the use of an energy vector as a descriptor of a protein's nonlocal structure; hence, we derive the parameters of the nonlocal model from the native structure without loss of generality. The local and nonlocal models are combined using the reference ratio method, which is a well-justified probabilistic construction. For evaluation, we use the resulting joint models to predict the structure of four proteins. The results indicate that the proposed method and the probabilistic models show considerable promise for probabilistic protein structure prediction and related applications.

Protein structure quality assessment based on the distance profiles of consecutive backbone Cα atoms

Predicting the three dimensional native state structure of a protein from its primary sequence is an unsolved grand challenge in molecular biology. Two main computational approaches have evolved to obtain the structure from the protein sequence - ab initio/de novo methods and template-based modeling - both of which typically generate multiple possible native state structures. Model quality assessment programs (MQAP) validate these predicted structures in order to identify the correct native state structure. Here, we propose a MQAP for assessing the quality of protein structures based on the distances of consecutive Cα atoms. We hypothesize that the root-mean-square deviation of the distance of consecutive Cα (RDCC) atoms from the ideal value of 3.8 Å, derived from a statistical analysis of high quality protein structures (top100H database), is minimized in native structures. Based on tests with the top100H set, we propose a RDCC cutoff value of 0.012 Å, above which a structure can be filtered out as a non-native structure. We applied the RDCC discriminator on decoy sets from the Decoys 'R' Us database to show that the native structures in all decoy sets tested have RDCC below the 0.012 Å cutoff. While most decoy sets were either indistinguishable using this discriminator or had very few violations, all the decoy structures in the fisa decoy set were discriminated by applying the RDCC criterion. This highlights the physical non-viability of the fisa decoy set, and possible issues in benchmarking other methods using this set. The source code and manual is made available at https://github.com/sanchak/mqap and permanently available on 10.5281/zenodo.7134.

PHAISTOS: a framework for Markov chain Monte Carlo simulation and inference of protein structure

We present a new software framework for Markov chain Monte Carlo sampling for simulation, prediction, and inference of protein structure. The software package contains implementations of recent advances in Monte Carlo methodology, such as efficient local updates and sampling from probabilistic models of local protein structure. These models form a probabilistic alternative to the widely used fragment and rotamer libraries. Combined with an easily extendible software architecture, this makes PHAISTOS well suited for Bayesian inference of protein structure from sequence and/or experimental data. Currently, two force-fields are available within the framework: PROFASI and OPLS-AA/L, the latter including the generalized Born surface area solvent model. A flexible command-line and configuration-file interface allows users quickly to set up simulations with the desired configuration. PHAISTOS is released under the GNU General Public License v3.0. Source code and documentation are freely available from http://phaistos.sourceforge.net. The software is implemented in C++ and has been tested on Linux and OSX platforms.

Extracting knowledge from protein structure geometry

Protein structure prediction techniques proceed in two steps, namely the generation of many structural models for the protein of interest, followed by an evaluation of all these models to identify those that are native-like. In theory, the second step is easy, as native structures correspond to minima of their free energy surfaces. It is well known however that the situation is more complicated as the current force fields used for molecular simulations fail to recognize native states from misfolded structures. In an attempt to solve this problem, we follow an alternate approach and derive a new potential from geometric knowledge extracted from native and misfolded conformers of protein structures. This new potential, Metric Protein Potential (MPP), has two main features that are key to its success. Firstly, it is composite in that it includes local and nonlocal geometric information on proteins. At the short range level, it captures and quantifies the mapping between the sequences and structures of short (7-mer) fragments of protein backbones through the introduction of a new local energy term. The local energy term is then augmented with a nonlocal residue-based pairwise potential, and a solvent potential. Secondly, it is optimized to yield a maximized correlation between the energy of a structural model and its root mean square (RMS) to the native structure of the corresponding protein. We have shown that MPP yields high correlation values between RMS and energy and that it is able to retrieve the native structure of a protein from a set of high-resolution decoys.

The electrostatic profile of consecutive Cβ atoms applied to protein structure quality assessment

The structure of a protein provides insight into its physiological interactions with other components of the cellular soup. Methods that predict putative structures from sequences typically yield multiple, closely-ranked possibilities. A critical component in the process is the model quality assessing program (MQAP), which selects the best candidate from this pool of structures. Here, we present a novel MQAP based on the physical properties of sidechain atoms. We propose a method for assessing the quality of protein structures based on the electrostatic potential difference (EPD) of Cβ atoms in consecutive residues. We demonstrate that the EPDs of Cβ atoms on consecutive residues provide unique signatures of the amino acid types. The EPD of Cβ atoms are learnt from a set of 1000 non-homologous protein structures with a resolution cuto of 1.6 Å obtained from the PISCES database. Based on the Boltzmann hypothesis that lower energy conformations are proportionately sampled more, and on Annsen's thermodynamic hypothesis that the native structure of a protein is the minimum free energy state, we hypothesize that the deviation of observed EPD values from the mean values obtained in the learning phase is minimized in the native structure. We achieved an average specificity of 0.91, 0.94 and 0.93 on hg_structal, 4state_reduced and ig_structal decoy sets, respectively, taken from the Decoys `R' Us database. The source code and manual is made available at https://github.com/sanchak/mqap and permanently available on 10.5281/zenodo.7134.

Scientific benchmarks for guiding macromolecular energy function improvement

Accurate energy functions are critical to macromolecular modeling and design. We describe new tools for identifying inaccuracies in energy functions and guiding their improvement, and illustrate the application of these tools to the improvement of the Rosetta energy function. The feature analysis tool identifies discrepancies between structures deposited in the PDB and low-energy structures generated by Rosetta; these likely arise from inaccuracies in the energy function. The optE tool optimizes the weights on the different components of the energy function by maximizing the recapitulation of a wide range of experimental observations. We use the tools to examine three proposed modifications to the Rosetta energy function: improving the unfolded state energy model (reference energies), using bicubic spline interpolation to generate knowledge-based torisonal potentials, and incorporating the recently developed Dunbrack 2010 rotamer library (Shapovalov & Dunbrack, 2011).

BCL::Score--knowledge based energy potentials for ranking protein models represented by idealized secondary structure elements

The topology of most experimentally determined protein domains is defined by the relative arrangement of secondary structure elements, i.e. α-helices and β-strands, which make up 50–70% of the sequence. Pairing of β-strands defines the topology of β-sheets. The packing of side chains between α-helices and β-sheets defines the majority of the protein core. Often, limited experimental datasets restrain the position of secondary structure elements while lacking detail with respect to loop or side chain conformation. At the same time the regular structure and reduced flexibility of secondary structure elements make these interactions more predictable when compared to flexible loops and side chains. To determine the topology of the protein in such settings, we introduce a tailored knowledge-based energy function that evaluates arrangement of secondary structure elements only. Based on the amino acid C_β atom coordinates within secondary structure elements, potentials for amino acid pair distance, amino acid environment, secondary structure element packing, β-strand pairing, loop length, radius of gyration, contact order and secondary structure prediction agreement are defined. Separate penalty functions exclude conformations with clashes between amino acids or secondary structure elements and loops that cannot be closed. Each individual term discriminates for native-like protein structures. The composite potential significantly enriches for native-like models in three different databases of 10,000–12,000 protein models in 80–94% of the cases. The corresponding application, “BCL::ScoreProtein,” is available at www.meilerlab.org.