Testing similarity measures with continuous and discrete protein models

ELViM: Exploring Biomolecular Energy Landscapes through Multidimensional Visualization

Molecular dynamics (MD) simulations provide a powerful means of exploring the dynamic behavior of biomolecular systems at the atomic level. However, analyzing the vast data sets generated by MD simulations poses significant challenges. This article discusses the energy landscape visualization method (ELViM), a multidimensional reduction technique inspired by the energy landscape theory. ELViM transcends one-dimensional representations, offering a comprehensive analysis of the effective conformational phase space without the need for predefined reaction coordinates. We apply the ELViM to study the folding landscape of the antimicrobial peptide Polybia-MP1, showcasing its versatility in capturing complex biomolecular dynamics. Using dissimilarity matrices and a force-scheme approach, the ELViM provides intuitive visualizations, revealing structural correlations and local conformational signatures. The method is demonstrated to be adaptable, robust, and applicable to various biomolecular systems.

Probing Mastoparan-like Antimicrobial Peptides Interaction with Model Membrane Through Energy Landscape Analysis

Antimicrobial Peptides (AMPs) have emerged as promising alternatives to conventional antibiotics due to their capacity to disrupt the lipid packing of bacterial cell membranes. This mechanism of action may prevent the development of resistance by bacteria. Understanding their role in lipid packing disruption and their structural properties upon interaction with bacterial membranes is highly desirable. In this study, we employed Molecular Dynamics simulations and the Energy Landscape Visualization Method (ELViM) to characterize and compare the conformational ensembles of mastoparan-like Polybia-MP1 and its analogous H-MP1, in which histidines replace lysine residues. Two situations were analyzed: (i) the peptides in their free state in an aqueous solution containing water and ions and (ii) the peptides spontaneously adsorbing onto an anionic lipid bilayer, used as a bacteria membrane mimetic. ELViM was used to project a single effective conformational phase space for both peptides, providing a comparative analysis. This projection enabled us to map the conformational ensembles of each peptide in an aqueous solution and assess the structural effects of substituting lysines with histidines in H-MP1. Furthermore, a single conformational phase space analysis was employed to describe structural changes during the adsorption process using the same framework. We show that ELViM provides a comprehensive analysis, able to identify discrepancies in the conformational ensembles of these peptides that may affect their affinity to the membrane and adsorption kinetics.

Characterizing the Folding Transition-State Ensembles in the Energy Landscape of an RNA Tetraloop

Molecular dynamics (MD) simulations have become increasingly powerful and can now describe the folding/unfolding of small biomolecules in atomic detail. However, a major challenge in MD simulations is to represent the complex energy landscape of biomolecules using a small number of reaction coordinates. In this study, we investigate the folding pathways of an RNA tetraloop, gcGCAAgc, using five classical MD simulations with a combined simulation time of approximately 120 μs. Our approach involves analyzing the tetraloop dynamics, including the folding transition state ensembles, using the energy landscape visualization method (ELViM). The ELViM is an approach that uses internal distances to compare any two conformations, allowing for a detailed description of the folding process without requiring root mean square alignment of structures. This method has previously been applied to describe the energy landscape of disordered β-amyloid peptides and other proteins. The ELViM results in a non-linear projection of the multidimensional space, providing a comprehensive representation of the tetraloop's energy landscape. Our results reveal four distinct transition-state regions and establish the paths that lead to the folded tetraloop structure. This detailed analysis of the tetraloop's folding process has important implications for understanding RNA folding, and the ELViM approach can be used to study other biomolecules.

PROTHON: A Local Order Parameter-Based Method for Efficient Comparison of Protein Ensembles

The comparison of protein conformational ensembles is of central importance in structural biology. However, there are few computational methods for ensemble comparison, and those that are readily available, such as ENCORE, utilize methods that are sufficiently computationally expensive to be prohibitive for large ensembles. Here, a new method is presented for efficient representation and comparison of protein conformational ensembles. The method is based on the representation of a protein ensemble as a vector of probability distribution functions (pdfs), with each pdf representing the distribution of a local structural property such as the number of contacts between C_β atoms. Dissimilarity between two conformational ensembles is quantified by the Jensen-Shannon distance between the corresponding set of probability distribution functions. The method is validated for conformational ensembles generated by molecular dynamics simulations of ubiquitin, as well as experimentally derived conformational ensembles of a 130 amino acid truncated form of human tau protein. In the ubiquitin ensemble data set, the method was up to 88 times faster than the existing ENCORE software, while simultaneously utilizing 48 times fewer computing cores. We make the method available as a Python package, called PROTHON, and provide a GitHub page with the Python source code at https://github.com/PlotkinLab/Prothon.

Refining evERdock: Improved selection of good protein-protein complex models achieved by MD optimization and use of multiple conformations

A method for evaluating binding free energy differences of protein-protein complex structures generated by protein docking was recently developed by some of us. The method, termed evERdock, combined short (2 ns) molecular dynamics (MD) simulations in explicit water and solution theory in the energy representation (ER) and succeeded in selecting the near-native complex structures from a set of decoys. In the current work, we performed longer (up to 100 ns) MD simulations before employing ER analysis in order to further refine the structures of the decoy set with improved binding free energies. Moreover, we estimated the binding free energies for each complex structure based on an average value from five individual MD snapshots. After MD simulations, all decoys exhibit a decrease in binding free energy, suggesting that proper equilibration in explicit solvent resulted in more favourably bound complexes. During the MD simulations, non-native structures tend to become unstable and in some cases dissociate, while near-native structures maintain a stable interface. The energies after the MD simulations show an improved correlation between similarity criteria (such as interface root-mean-square distance) to the native (crystal) structure and the binding free energy. In addition, calculated binding free energies show sensitivity to the number of contacts, which was demonstrated to reflect the relative stability of structures at earlier stages of the MD simulation. We therefore conclude that the additional equilibration step along with the use of multiple conformations can make the evERdock scheme more versatile under low computational cost.

The optical signature of 2,6-bis((E)-2-(benzoxazol-2-yl)vinyl)naphthalene (BBVN) laser dye: a TDDFT study

The NTO hole-particle representation of excitation demonstrates that terminal benzoxazole nuclei in BBVN promote charge displacement in absorption/emission.

The computational prediction of Raman and ROA spectra of charged histidine tautomers in aqueous solution

Histidine is a key component of a number of enzymatic mechanisms, and undertakes many functionalities in biochemical systems.

Equilibration of Protein States: A Time Dependent Free-Energy Disconnectivity Graph

The process of equilibration of protein states in a three-stranded antiparallel β-sheet miniprotein is studied using a time-dependent free energy disconnectivity graph. To determine the rates of transitions, the molecular dynamics simulation results of a recent work (Kalgin, I. V.; J. Phys. Chem. B 2013, 117, 6092) are employed. The vertices of the graph are the free energies of characteristic states of the protein, and the edges are the transition state free energies. To determine the latter, the "complete" partition function (Eyring, 1935) is used, which includes the translational partition function corresponding to the ballistic motion of the system along the reaction coordinate. The distance along the reaction coordinate that enters the translational partition function is taken to be proportional to the observation time and thus measures the number of representative points that cross the transition state surface during given time. As the time increases, the free energy barriers between the clusters of characteristic conformations (native-like, helical, and β-sheet conformations of different degree of organization) decrease and (local) equilibrium between the clusters is established. With time, these clusters are grouped into larger clusters, extending the equilibrium to a larger portion of protein states.

Rapid determination of RMSDs corresponding to macromolecular rigid body motions

Finding the root mean sum of squared deviations (RMSDs) between two coordinate vectors that correspond to the rigid body motion of a macromolecule is an important problem in structural bioinformatics, computational chemistry, and molecular modeling. Standard algorithms compute the RMSD with time proportional to the number of atoms in the molecule. Here, we present RigidRMSD, a new algorithm that determines a set of RMSDs corresponding to a set of rigid body motions of a macromolecule in constant time with respect to the number of atoms in the molecule. Our algorithm is particularly useful for rigid body modeling applications, such as rigid body docking, and also for high-throughput analysis of rigid body modeling and simulation results. We also introduce a constant-time rotation RMSD as a similarity measure for rigid molecules. A C++ implementation of our algorithm is available at http://nano-d.inrialpes.fr/software/RigidRMSD.

Distribution of Reciprocal of Interatomic Distances: A Fast Structural Metric

We present a structural metric based on the Distribution of Reciprocal of Interatomic Distances (DRID) for evaluating geometrical similarity between two conformations of a molecule. A molecular conformation is described by a vector of 3N orientation-independent DRID descriptors where N is the number of molecular centroids, for example, the non-hydrogen atoms in all nonsymmetric groups of a peptide. For two real-world applications (pairwise comparison of snapshots from an explicit solvent simulation of a protease/peptide substrate complex and implicit solvent simulations of reversible folding of a 20-residue β-sheet peptide), the DRID-based metric is shown to be about 5 and 15 times faster than coordinate root-mean-square deviation (cRMSD) and distance root-mean-square deviation (dRMSD), respectively. A single core of a mainstream processor can perform about 10(8) DRID comparisons in one-half a minute. Importantly, the DRID metric has closer similarity to kinetic distance than does either cRMSD or dRMSD. Therefore, DRID is suitable for clustering molecular dynamics trajectories for kinetic analysis, for example, by Markov state models. Moreover, conformational space networks and free energy profiles derived by DRID-based clustering preserve the kinetic information.

Clustering biomolecular complexes by residue contacts similarity

Inaccuracies in computational molecular modeling methods are often counterweighed by brute-force generation of a plethora of putative solutions. These are then typically sieved via structural clustering based on similarity measures such as the root mean square deviation (RMSD) of atomic positions. Albeit widely used, these measures suffer from several theoretical and technical limitations (e.g., choice of regions for fitting) that impair their application in multicomponent systems (N > 2), large-scale studies (e.g., interactomes), and other time-critical scenarios. We present here a simple similarity measure for structural clustering based on atomic contacts--the fraction of common contacts--and compare it with the most used similarity measure of the protein docking community--interface backbone RMSD. We show that this method produces very compact clusters in remarkably short time when applied to a collection of binary and multicomponent protein-protein and protein-DNA complexes. Furthermore, it allows easy clustering of similar conformations of multicomponent symmetrical assemblies in which chain permutations can occur. Simple contact-based metrics should be applicable to other structural biology clustering problems, in particular for time-critical or large-scale endeavors.

Coupled folding-binding in a hydrophobic/polar protein model: impact of synergistic folding and disordered flanks

Coupled folding-binding is central to the function of many intrinsically disordered proteins, yet not fully understood. With a continuous three-letter protein model, we explore the free-energy landscape of pairs of interacting sequences and how it is impacted by 1), variations in the binding mechanism; and 2), the addition of disordered flanks to the binding region. In particular, we focus on two sequences, one with 16 and one with 35 amino acids, which make a stable dimeric three-helix bundle at low temperatures. Three distinct binding mechanisms are realized by altering the stabilities of the individual monomers: docking, coupled folding-binding of a single α-helix, and synergistic folding and binding. Compared to docking, the free-energy barrier for binding is reduced when the single α-helix is allowed to fold upon binding, but only marginally. A greater reduction is found for synergistic folding, which in addition results in a binding transition state characterized by very few interchain contacts. Disordered flanking chain segments attached to the α-helix sequence can, despite a negligible impact on the dimer stability, lead to a downhill free-energy surface in which the barrier for binding is eliminated.

Efficient identification of near-native conformations in ab initio protein structure prediction using structural profiles

One of the major bottlenecks in many ab initio protein structure prediction methods is currently the selection of a small number of candidate structures for high-resolution refinement from large sets of low-resolution decoys. This step often includes a scoring by low-resolution energy functions and a clustering of conformations by their pairwise root mean square deviations (RMSDs). As an efficient selection is crucial to reduce the overall computational cost of the predictions, any improvement in this direction can increase the overall performance of the predictions and the range of protein structures that can be predicted. We show here that the use of structural profiles, which can be predicted with good accuracy from the amino acid sequences of proteins, provides an efficient means to identify good candidate structures.

Using least median of squares for structural superposition of flexible proteins

BACKGROUND

The conventional superposition methods use an ordinary least squares (LS) fit for structural comparison of two different conformations of the same protein. The main problem of the LS fit that it is sensitive to outliers, i.e. large displacements of the original structures superimposed.

RESULTS

To overcome this problem, we present a new algorithm to overlap two protein conformations by their atomic coordinates using a robust statistics technique: least median of squares (LMS). In order to effectively approximate the LMS optimization, the forward search technique is utilized. Our algorithm can automatically detect and superimpose the rigid core regions of two conformations with small or large displacements. In contrast, most existing superposition techniques strongly depend on the initial LS estimating for the entire atom sets of proteins. They may fail on structural superposition of two conformations with large displacements. The presented LMS fit can be considered as an alternative and complementary tool for structural superposition.

CONCLUSION

The proposed algorithm is robust and does not require any prior knowledge of the flexible regions. Furthermore, we show that the LMS fit can be extended to multiple level superposition between two conformations with several rigid domains. Our fit tool has produced successful superpositions when applied to proteins for which two conformations are known. The binary executable program for Windows platform, tested examples, and database are available from https://engineering.purdue.edu/PRECISE/LMSfit.

Amyloid beta-protein monomer folding: free-energy surfaces reveal alloform-specific differences

Alloform-specific differences in structural dynamics between amyloid beta-protein (Abeta) 40 and Abeta42 appear to underlie the pathogenesis of Alzheimer's disease. To elucidate these differences, we performed microsecond timescale replica-exchange molecular dynamics simulations to sample the conformational space of the Abeta monomer and constructed its free-energy surface. We find that neither peptide monomer is unstructured, but rather that each may be described as a unique statistical coil in which five relatively independent folding units exist, comprising residues 1-5, 10-13, 17-22, 28-37, and 39-42, which are connected by four turn structures. The free-energy surfaces of both peptides are characterized by two large basins, comprising conformers with either substantial alpha-helix or beta-sheet content. Conformational transitions within and between these basins are rapid. The two additional hydrophobic residues at the Abeta42 C-terminus, Ile41 and Ala42, significantly increase contacts within the C-terminus, and between the C-terminus and the central hydrophobic cluster (Leu17-Ala21). As a result, the beta-structure of Abeta42 is more stable than that of Abeta40, and the conformational equilibrium in Abeta42 shifts towards beta-structure. These results suggest that drugs stabilizing alpha-helical Abeta conformers (or destabilizing the beta-sheet state) would block formation of neurotoxic oligomers. The atomic-resolution conformer structures determined in our simulations may serve as useful targets for this purpose. The conformers also provide starting points for simulations of Abeta oligomerization-a process postulated to be the key pathogenetic event in Alzheimer's disease.

Energy landscapes of the monomer and dimer of the Alzheimer's peptide Abeta(1-28)

The cytotoxicity of Alzheimer's disease has been linked to the self-assembly of the 4042 amino acid of the amyloid-beta (Abeta) peptide into oligomers. To understand the assembly process, it is important to characterize the very first steps of aggregation at an atomic level of detail. Here, we focus on the N-terminal fragment 1-28, known to form fibrils in vitro. Circular dichroism and NMR experiments indicate that the monomer of Abeta(1-28) is alpha-helical in a membranelike environment and random coil in aqueous solution. Using the activation-relaxation technique coupled with the OPEP coarse grained force field, we determine the structures of the monomer and of the dimer of Abeta(1-28). In agreement with experiments, we find that the monomer is predominantly random coil in character, but displays a non-negligible beta-strand probability in the N-terminal region. Dimerization impacts the structure of each chain and leads to an ensemble of intertwined conformations with little beta-strand content in the region Leu17-Ala21. All these structural characteristics are inconsistent with the amyloid fibril structure and indicate that the dimer has to undergo significant rearrangement en route to fibril formation.

Local quality assessment in homology models using statistical potentials and support vector machines

In this study, we address the problem of local quality assessment in homology models. As a prerequisite for the evaluation of methods for predicting local model quality, we first examine the problem of measuring local structural similarities between a model and the corresponding native structure. Several local geometric similarity measures are evaluated. Two methods based on structural superposition are found to best reproduce local model quality assessments by human experts. We then examine the performance of state-of-the-art statistical potentials in predicting local model quality on three qualitatively distinct data sets. The best statistical potential, DFIRE, is shown to perform on par with the best current structure-based method in the literature, ProQres. A combination of different statistical potentials and structural features using support vector machines is shown to provide somewhat improved performance over published methods.

Hidden symmetries in the primary sequences of beta-barrel family

In this paper, we analyze the symmetries of beta-barrel proteins at both structure and sequence levels by using a modified recurrent quantification analysis. It shows that the structures and sequences have the same two-fold symmetry, although the later diverged considerably. This result may be helpful to understand the mechanism of protein evolution.

Automatic procedure for using models of proteins in molecular replacement

In a crystallography experiment, a crystal is irradiated with X-rays whose diffracted waves are collected and measured. The reconstruction of the structure of the molecule in the crystal requires knowledge of the phase of the diffracted waves, information that is lost in the passage from the three-dimensional structure of the molecule to its diffraction pattern. It can be recovered using experimental methods such as heavy-atom isomorphous replacement and anomalous scattering or by molecular replacement, which relies on the availability of an atomic model of the target structure. This can be the structure of the target protein itself, if a previous structure determination is available, or a computational model or, in some cases, the structure of a homologous protein. It is not straightforward to predict beforehand whether or not a computational model will work in a molecular replacement experiment, although some rules of thumb exist. The consensus is that even minor differences in the quality of the model, which are rather difficult to estimate a priori, can have a significant effect on the outcome of the procedure. We describe here a method for quickly assessing whether a protein structure can be solved by molecular replacement. The procedure consists in submitting the sequence of the target protein to a selected list of freely available structure prediction servers, cluster the resulting models, select the representative structures of each cluster and use them as search models in an automatic phasing procedure. We tested the procedure using the structure factors of newly released proteins of known structure downloaded from the Protein Data Bank as soon as they were made available. Using our automatic procedure we were able to obtain an interpretable electron density map in more than half the cases.

Monte Carlo sampling of near-native structures of proteins with applications

Since a protein's dynamic fluctuation inside cells affects the protein's biological properties, we present a novel method to study the ensemble of near-native structures (NNS) of proteins, namely, the conformations that are very similar to the experimentally determined native structure. We show that this method enables us to (i) quantify the difficulty of predicting a protein's structure, (ii) choose appropriate simplified representations of protein structures, and (iii) assess the effectiveness of knowledge-based potential functions. We found that well-designed simple representations of protein structures are likely as accurate as those more complex ones for certain potential functions. We also found that the widely used contact potential functions stabilize NNS poorly, whereas potential functions incorporating local structure information significantly increase the stability of NNS.

Principal eigenvector of contact matrices and hydrophobicity profiles in proteins

With the aim of studying the relationship between protein sequences and their native structures, we adopted vectorial representations for both sequence and structure. The structural representation was based on the principal eigenvector of the fold's contact matrix (PE). As has been recently shown, the latter encodes sufficient information for reconstructing the whole contact matrix. The sequence was represented through a hydrophobicity profile (HP), using a generalized hydrophobicity scale that we obtained from the principal eigenvector of a residue-residue interaction matrix, and denoted as interactivity scale. Using this novel scale, we defined the optimal HP of a protein fold, and, by means of stability arguments, predicted to be strongly correlated with the PE of the fold's contact matrix. This prediction was confirmed through an evolutionary analysis, which showed that the PE correlates with the HP of each individual sequence adopting the same fold and, even more strongly, with the average HP of this set of sequences. Thus, protein sequences evolve in such a way that their average HP is close to the optimal one, implying that neutral evolution can be viewed as a kind of motion in sequence space around the optimal HP. Our results indicate that the correlation coefficient between N-dimensional vectors constitutes a natural metric in the vectorial space in which we represent both protein sequences and protein structures, which we call vectorial protein space. In this way, we define a unified framework for sequence-to-sequence, sequence-to-structure and structure-to-structure alignments. We show that the interactivity scale is nearly optimal both for the comparison of sequences to sequences and sequences to structures.

Exploring the energy landscape of model proteins: a metric criterion for the determination of dynamical connectivity

A method to reconstruct the energy landscape of small peptides is presented with reference to a two-dimensional off-lattice model. The starting point is a statistical analysis of the configurational distances between generic minima and directly connected pairs (DCP). As the mutual distance of DCP is typically much smaller than that of generic pairs, a metric criterion can be established to identify the great majority of DCP. Advantages and limits of this approach are thoroughly analyzed for three different heteropolymeric chains. A funnel-like structure of the energy landscape is found in all of the three cases, but the escape rates clearly reveal that the native configuration is more easily accessible (and is significantly more stable) for the sequence that is expected to behave as a real protein.

Improving sequence-based fold recognition by using 3D model quality assessment

MOTIVATION

The ability of a simple method (MODCHECK) to determine the sequence-structure compatibility of a set of structural models generated by fold recognition is tested in a thorough benchmark analysis. Four Model Quality Assessment Programs (MQAPs) were tested on 188 targets from the latest LiveBench-9 automated structure evaluation experiment. We systematically test and evaluate whether the MQAP methods can successfully detect native-like models.

RESULTS

We show that compared with the other three methods tested MODCHECK is the most reliable method for consistently performing the best top model selection and for ranking the models. In addition, we show that the choice of model similarity score used to assess a model's similarity to the experimental structure can influence the overall performance of these tools. Although these MQAP methods fail to improve the model selection performance for methods that already incorporate protein three dimension (3D) structural information, an improvement is observed for methods that are purely sequence-based, including the best profile-profile methods. This suggests that even the best sequence-based fold recognition methods can still be improved by taking into account the 3D structural information.

CONTACT

d.jones@cs.ucl.ac.uk

Two-state folding over a weak free-energy barrier

We present a Monte Carlo study of a model protein with 54 amino acids that folds directly to its native three-helix-bundle state without forming any well-defined intermediate state. The free-energy barrier separating the native and unfolded states of this protein is found to be weak, even at the folding temperature. Nevertheless, we find that melting curves to a good approximation can be described in terms of a simple two-state system, and that the relaxation behavior is close to single exponential. The motion along individual reaction coordinates is roughly diffusive on timescales beyond the reconfiguration time for a single helix. A simple estimate based on diffusion in a square-well potential predicts the relaxation time within a factor of two.