A coarse-grained normal mode approach for macromolecules: an efficient implementation and application to Ca(2+)-ATPase

Scalable computation of anisotropic vibrations for large macromolecular assemblies

The Normal Mode Analysis (NMA) is a standard approach to elucidate the anisotropic vibrations of macromolecules at their folded states, where low-frequency collective motions can reveal rearrangements of domains and changes in the exposed surface of macromolecules. Recent advances in structural biology have enabled the resolution of megascale macromolecules with millions of atoms. However, the calculation of their vibrational modes remains elusive due to the prohibitive cost associated with constructing and diagonalizing the underlying eigenproblem and the current approaches to NMA are not readily adaptable for efficient parallel computing on graphic processing unit (GPU). Here, we present eigenproblem construction and diagonalization approach that implements level-structure bandwidth-reducing algorithms to transform the sparse computation in NMA to a globally-sparse-yet-locally-dense computation, allowing batched tensor products to be most efficiently executed on GPU. We map, optimize, and compare several low-complexity Krylov-subspace eigensolvers, supplemented by techniques such as Chebyshev filtering, sum decomposition, external explicit deflation and shift-and-inverse, to allow fast GPU-resident calculations. The method allows accurate calculation of the first 1000 vibrational modes of some largest structures in PDB ( > 2.4 million atoms) at least 250 times faster than existing methods.

Coarse-Graining Waters: Unveiling The Effective Hydrophilicity/Hydrophobicity of Individual Protein Atoms and The Roles of Waters' Hydrogens

There have been many coarse-graining methods developed that aim to reduce the sizes of simulated systems and their computational costs. In this work, we develop a new coarse-graining method, called coarse-graining-delta (or δ-CG in short), that reduces the degrees of freedom of the potential energy surface by coarse-graining relative locations of atoms from their unit centers. Our method extends and generalizes the methods used in the coarse-grained normal mode analysis and enables us to study the roles of the individual removed atoms in a system, which have been difficult to study in molecular dynamics simulations. By applying δ-CG to coarse-grain three-point water molecules into single-point solvent particles, we successfully identify the effective hydrophilicity and hydrophobicity of all the individual protein atom types, which collectively correlate well with the known hydrophilic, hydrophobic, and amphipathic characteristics of amino acids. Moreover, our investigation shows that water's hydrogens have two roles in interacting with protein atoms. First, water molecules adjust their poses around different amino acids and their atoms, and the statistical preferences of the hydrogen poses near the atoms determine the effective hydrophilicity and hydrophobicity of the atoms, which have not been successfully addressed before. Second, the collective dynamics of the hydrogens assist the water molecules in escaping from the potential energy wells of the hydrophilic atoms. Our method also shows that coarse-graining a system mathematically leads to breaking antisymmetry of the nonbonded interactions; as a result, two interacting coarse-grained units exert different forces on each other. Our study indicates that the accuracy of coarse-grained force fields, such as the MARTINI force field and the UNRES force field, can be improved in two ways: (i) refining their potential energy functions and coefficients by analyzing the coarse-grained potential energy surface using δ-CG, and (ii) introducing non-antisymmetric interactions.

Internal Normal Mode Analysis Applied to RNA Flexibility and Conformational Changes

We investigated the capability of internal normal modes to reproduce RNA flexibility and predict observed RNA conformational changes and, notably, those induced by the formation of RNA-protein and RNA-ligand complexes. Here, we extended our iNMA approach developed for proteins to study RNA molecules using a simplified representation of the RNA structure and its potential energy. Three data sets were also created to investigate different aspects. Despite all the approximations, our study shows that iNMA is a suitable method to take into account RNA flexibility and describe its conformational changes opening the route to its applicability in any integrative approach where these properties are crucial.

Coarse-Grained Models for Vault Normal Model Analysis

Recent experiments have shown that the molecular complex of vault has large conformational changes at its shoulder and cap regions in solution. From the comparison of two configuration structures, it has been found that the shoulder region can twist and move outward, while the cap region will rotate and push upward correspondingly. To further understand these experimental results, in this paper, we study the vault dynamics for the first time. Since vault has an extremely large-sized structure with around 63,336 Cα atoms, traditional normal mode method with the Cα coarse-grained representation will fall short. We employ a newly invented multiscale virtual particle-based anisotropic network model (MVP-ANM). To reduce the complexity, the 39-folder vault structure is coarse-grained to about 6000 virtual particles, which significantly reduces the computational cost while still maintaining the basic structure information. Among the 14 low frequency eigenmodes from Mode 7 to Mode 20, two eigenmodes, i.e., Mode 9 and Mode 20, are found to be directly associated with the experimental observations. In Mode 9, shoulder region undergoes a significant expansion while the cap part is lifted upward. In Mode 20, a clear rotation of both shoulder and cap regions is well observed. Our results are consistent with the experimental observations. More importantly, these low frequency eigenmodes indicate that the vault waist, shoulder and lower cap regions are the most likely regions for the opening of the vault particle. And the opening mechanism is highly likely to be rotation and expansion at these regions. As far as we know, this is the first work to provide the normal mode analysis for the vault complex.

Biomolecular Topology: Modelling and Analysis

With the great advancement of experimental tools, a tremendous amount of biomolecular data has been generated and accumulated in various databases. The high dimensionality, structural complexity, the nonlinearity, and entanglements of biomolecular data, ranging from DNA knots, RNA secondary structures, protein folding configurations, chromosomes, DNA origami, molecular assembly, to others at the macromolecular level, pose a severe challenge in their analysis and characterization. In the past few decades, mathematical concepts, models, algorithms, and tools from algebraic topology, combinatorial topology, computational topology, and topological data analysis, have demonstrated great power and begun to play an essential role in tackling the biomolecular data challenge. In this work, we introduce biomolecular topology, which concerns the topological problems and models originated from the biomolecular systems. More specifically, the biomolecular topology encompasses topological structures, properties and relations that are emerged from biomolecular structures, dynamics, interactions, and functions. We discuss the various types of biomolecular topology from structures (of proteins, DNAs, and RNAs), protein folding, and protein assembly. A brief discussion of databanks (and databases), theoretical models, and computational algorithms, is presented. Further, we systematically review related topological models, including graphs, simplicial complexes, persistent homology, persistent Laplacians, de Rham-Hodge theory, Yau-Hausdorff distance, and the topology-based machine learning models.

Response of Terahertz Protein Vibrations to Ligand Binding: Calmodulin-Peptide Complexes as a Case Study

Terahertz vibrations are sensitive reporters of the structure and interactions of proteins. Ligand binding alters the nature and distribution of these collective vibrations. The ligand-induced changes in the terahertz protein vibrations contribute to the binding entropy and to the overall thermodynamic stability of the resultant protein-ligand complexes. Here, we have examined the response of the low-frequency (below 6 terahertz) collective vibrations of the calcium-loaded calmodulin (CaM) to binding to five different ligands, both in the presence and absence of water, using normal-mode analysis and molecular dynamics simulations. A comparison of the vibrational spectra of hydrated and dry systems reveals that protein-solvent interactions stiffen the terahertz protein vibrations and that these solvent-coupled collective vibrations contribute significantly to the hydration-sensitive variation in the vibrational entropy of CaM. In the absence of water, the low-frequency vibrations of CaM are stiffened by ligand binding. On the contrary, the number and the cumulative vibrational entropy of low-frequency vibrational modes (ω < 200 cm^-1) of the hydrated CaM are increased noticeably after binding to the peptides, indicating binding-induced softening of collective vibrations of the protein. Although the calculated and experimental binding affinities of the chosen complexes correlated reasonably well, no systematic correlation was observed between the protein vibrational entropy and the binding affinity. The results underscored the importance of the interplay of protein-ligand and solvent interactions in modulating the low-frequency vibrations of proteins.

Extracting the Dynamic Motion of Proteins Using Normal Mode Analysis

Normal mode analysis (NMA) is a technique for describing the conformational states accessible to a protein in a minimum energy conformation. NMA gives results similar to those produced by principal components analysis of a molecular dynamics simulation, but with only a fraction of the computational effort. Here, we provide a brief overview of the theory and describe three methods for carrying out NMA, including the use of one of the on-line services, the use of off-line software for calculating the projection of the modes calculated from one conformation onto another, and an all-atom NMA calculated using GROMACS. For all three methods, we will use the E1·2Ca²⁺ form of the Ca²⁺-ATPase as a concrete example.

Normal-mode driven exploration of protein domain motions

Domain motions involved in the function of proteins can often be well described as a combination of motions along a handfull of low-frequency modes, that is, with the values of a few normal coordinates. This means that, when the functional motion of a protein is unknown, it should prove possible to predict it, since it amounts to guess a few values. However, without the help of additional experimental data, using normal coordinates for generating accurate conformers far away from the initial one is not so straightforward. To do so, a new approach is proposed: instead of building conformers directly with the values of a subset of normal coordinates, they are built in two steps, the conformer built with normal coordinates being just used for defining a set of distance constraints, the final conformer being built so as to match them. Note that this approach amounts to transform the problem of generating accurate protein conformers using normal coordinates into a better known one: the distance-geometry problem, which is herein solved with the help of the ROSETTA software. In the present study, this approach allowed to rebuild accurately six large amplitude conformational changes, using at most six low-frequency normal coordinates. As a consequence of the low-dimensionality of the corresponding subspace, random exploration also proved enough for generating low-energy conformers close to the known end-point of the conformational change of the LAO binding protein, lysozyme T4 and adenylate kinase.

HOPMA: Boosting Protein Functional Dynamics with Colored Contact Maps

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

Deciphering the Mechanism of Inhibition of SERCA1a by Sarcolipin Using Molecular Simulations

SERCA1a is an ATPase calcium pump that transports Ca²⁺ from the cytoplasm to the sarco/endoplasmic reticulum lumen. Sarcolipin (SLN), a transmembrane peptide, regulates the activity of SERCA1a by decreasing its Ca²⁺ transport rate, but its mechanism of action is still not well-understood. To decipher this mechanism, we have performed normal mode analysis in the all-atom model, with the SERCA1a-SLN complex, or the isolated SERCA1a, embedded in an explicit membrane. The comparison of the results allowed us to provide an explanation at the atomic level for the action of SLN that is in good agreement with experimental observations. In our analyses, the presence of SLN locally perturbs the TM6 transmembrane helix and as a consequence modifies the position of D800, one of the key metal-chelating residues. Additionally, it reduces the flexibility of the gating residues, V304, and E309 in TM4, at the entrance of the Ca²⁺ binding sites, which would decrease the affinity for Ca²⁺. Unexpectedly, SLN has also an effect on the ATP binding site more than 35 Å away, due to the straightening of TM5, a long helix considered as the spine of the protein. The straightening of TM5 modifies the structure of the P-N linker that sits above it, and which comprises the ³⁵¹DKTG³⁵⁴ conserved motif, resulting in an increase of the distance between ATP and the phosphorylation site. As a consequence, the turn-over rate could be affected. All this gives SERCA1a the propensity to go toward a Ca²⁺ low-affinity E2-like state in the presence of SLN and toward a Ca²⁺ high-affinity E1-like state in the absence of SLN. In addition to a general mechanism of inhibition of SERCA1a regulatory peptides, this study also provides an insight into the conformational transition between the E2 and E1 states.

Weighted-persistent-homology-based machine learning for RNA flexibility analysis

With the great significance of biomolecular flexibility in biomolecular dynamics and functional analysis, various experimental and theoretical models are developed. Experimentally, Debye-Waller factor, also known as B-factor, measures atomic mean-square displacement and is usually considered as an important measurement for flexibility. Theoretically, elastic network models, Gaussian network model, flexibility-rigidity model, and other computational models have been proposed for flexibility analysis by shedding light on the biomolecular inner topological structures. Recently, a topology-based machine learning model has been proposed. By using the features from persistent homology, this model achieves a remarkable high Pearson correlation coefficient (PCC) in protein B-factor prediction. Motivated by its success, we propose weighted-persistent-homology (WPH)-based machine learning (WPHML) models for RNA flexibility analysis. Our WPH is a newly-proposed model, which incorporate physical, chemical and biological information into topological measurements using a weight function. In particular, we use local persistent homology (LPH) to focus on the topological information of local regions. Our WPHML model is validated on a well-established RNA dataset, and numerical experiments show that our model can achieve a PCC of up to 0.5822. The comparison with the previous sequence-information-based learning models shows that a consistent improvement in performance by at least 10% is achieved in our current model.

When Stiffness Matters: Mechanosensing in Heart Development and Disease

During embryonic morphogenesis, the heart undergoes a complex series of cellular phenotypic maturations (e.g., transition of myocytes from proliferative to quiescent or maturation of the contractile apparatus), and this involves stiffening of the extracellular matrix (ECM) acting in concert with morphogenetic signals. The maladaptive remodeling of the myocardium, one of the processes involved in determination of heart failure, also involves mechanical cues, with a progressive stiffening of the tissue that produces cellular mechanical damage, inflammation, and ultimately myocardial fibrosis. The assessment of the biomechanical dependence of the molecular machinery (in myocardial and non-myocardial cells) is therefore essential to contextualize the maturation of the cardiac tissue at early stages and understand its pathologic evolution in aging. Because systems to perform multiscale modeling of cellular and tissue mechanics have been developed, it appears particularly novel to design integrated mechano-molecular models of heart development and disease to be tested in ex vivo reconstituted cells/tissue-mimicking conditions. In the present contribution, we will discuss the latest implication of mechanosensing in heart development and pathology, describe the most recent models of cell/tissue mechanics, and delineate novel strategies to target the consequences of heart failure with personalized approaches based on tissue engineering and induced pluripotent stem cell (iPSC) technologies.

Predicting Protein Functional Motions: an Old Recipe with a New Twist

Large macromolecules, including proteins and their complexes, very often adopt multiple conformations. Some of them can be seen experimentally, for example with x-ray crystallography or cryo-electron microscopy. This structural heterogeneity is not occasional and is frequently linked with specific biological function. Thus, the accurate description of macromolecular conformational transitions is crucial for understanding fundamental mechanisms of life's machinery. We report on a real-time method to predict such transitions by extrapolating from instantaneous eigen motions, computed using the normal mode analysis, to a series of twists. We demonstrate the applicability of our approach to the prediction of a wide range of motions, including large collective opening-closing transitions and conformational changes induced by partner binding. We also highlight particularly difficult cases of very small transitions between crystal and solution structures. Our method guarantees preservation of the protein structure during the transition and allows accessing conformations that are unreachable with classical normal mode analysis. We provide practical solutions to describe localized motions with a few low-frequency modes and to relax some geometrical constraints along the predicted transitions. This work opens the way to the systematic description of protein motions, whatever their degree of collectivity. Our method is freely available as a part of the NOn-Linear rigid Block (NOLB) package.

Analysis of the vibrational and sound spectrum of over 100,000 protein structures and application in sonification

We report a high-throughput method that enables us to automatically compute the vibrational spectra of more than 100,000 proteins available in the Protein Data Bank to date, in a consistent manner. Using this new algorithm we report a comprehensive database of the normal mode frequencies of all known protein structures, which has not been available before. We then use the resulting frequency spectra of the proteins to generate audible sound by overlaying the molecular vibrations and translating them to the audible frequency range using the music theoretic concept of transpositional equivalence. The method, implemented as a Max audio device for use in a digital audio workstation (DAW), provides unparalleled insights into the rich vibrational signatures of protein structures, and offers a new way for creative expression by using it as a new type of musical instrument. This musical instrument is fully defined by the vibrational feature of almost all known protein structures, making it fundamentally different from all the traditional instruments that are limited by the material properties of a few types of conventional engineering materials, such as wood, metals or polymers.

A complex multiscale virtual particle model based elastic network model (CMVP-ENM) for the normal mode analysis of biomolecular complexes

The CMVP-ENM for virus normal mode analysis. With a special ratio parameter, CMVP-ENM can characterize the multi-material properties of biomolecular complexes and systematically enhance or suppress the modes for different components.

All-atom normal mode dynamics of HIV-1 capsid

Dynamics of biomolecular assemblies offer invaluable insights into their functional mechanisms. For extremely large biomolecular systems, such as HIV-1 capsid that has nearly 5 millions atoms, obtaining its normal mode dynamics using even coarse-grained models can be a challenging task. In this work, we have successfully carried out a normal mode analysis of an entire HIV-1 capsid in full all-atom details. This is made possible through our newly developed BOSE (Block of Selected Elasticity) model that is founded on the principle of resonance discovered in our recent work. The resonance principle makes it possible to most efficiently compute the vibrations of a whole capsid at any given frequency by projecting the motions of component capsomeres into a narrow subspace. We have conducted also assessments of the quality of the BOSE modes by comparing them with benchmark modes obtained directly from the original Hessian matrix. Our all-atom normal mode dynamics study of the HIV-1 capsid reveals the dynamic role of the pentamers in stabilizing the capsid structure and is in agreement with experimental findings that suggest capsid disassembly and uncoating start when the pentamers become destabilized. Our results on the dynamics of hexamer pores suggest that nucleotide transport should take place mostly at hexamers near pentamers, especially at the larger hemispherical end.

Supramolecular assemblies are large biomolecular complexes composed of hundreds or even thousands of protein chains. They function as molecular machines or as large containers that store or facilitate the chemical reactions of other molecules. Whatever they do, their functional mechanisms are tightly linked to their structures and intrinsic dynamics. Recently, due to breakthroughs in experimental techniques, many supramolecular assemblies have been determined, such as the capsid of human immunodeficiency virus (HIV) that is composed of nearly 5 millions of atoms. Computational studies of these systems are challenging due to their extremely large sizes. In this work, we have successfully carried out a dynamics study of an entire HIV capsid in full all-atom details. Our study reveals new insights into the dynamics of the N-terminal loops, the stabilizing role of the pentamers, and where the nucleotide transport may take place.

Fast Normal Mode Computations of Capsid Dynamics Inspired by Resonance

Increasingly more and larger structural complexes are being determined experimentally. The sizes of these systems pose a formidable computational challenge to the study of their vibrational dynamics by normal mode analysis. To overcome this challenge, this work presents a novel resonance-inspired approach. Tests on large shell structures of protein capsids demonstrate there is a strong resonance between the vibrations of a whole capsid and those of individual capsomeres. We then show how this resonance can be taken advantage of to significantly speed up normal mode computations.

Multiscale virtual particle based elastic network model (MVP-ENM) for normal mode analysis of large-sized biomolecules

In this paper, a multiscale virtual particle based elastic network model (MVP-ENM) is proposed for the normal mode analysis of large-sized biomolecules. The multiscale virtual particle (MVP) model is proposed for the discretization of biomolecular density data. With this model, large-sized biomolecular structures can be coarse-grained into virtual particles such that a balance between model accuracy and computational cost can be achieved. An elastic network is constructed by assuming "connections" between virtual particles. The connection is described by a special harmonic potential function, which considers the influence from both the mass distributions and distance relations of the virtual particles. Two independent models, i.e., the multiscale virtual particle based Gaussian network model (MVP-GNM) and the multiscale virtual particle based anisotropic network model (MVP-ANM), are proposed. It has been found that in the Debye-Waller factor (B-factor) prediction, the results from our MVP-GNM with a high resolution are as good as the ones from GNM. Even with low resolutions, our MVP-GNM can still capture the global behavior of the B-factor very well with mismatches predominantly from the regions with large B-factor values. Further, it has been demonstrated that the low-frequency eigenmodes from our MVP-ANM are highly consistent with the ones from ANM even with very low resolutions and a coarse grid. Finally, the great advantage of MVP-ANM model for large-sized biomolecules has been demonstrated by using two poliovirus virus structures. The paper ends with a conclusion.

NOLB: Nonlinear Rigid Block Normal-Mode Analysis Method

We present a new conceptually simple and computationally efficient method for nonlinear normal-mode analysis called NOLB. It relies on the rotations-translations of blocks (RTB) theoretical basis developed by Y.-H. Sanejouand and colleagues [ Durand et al. Biopolymers 1994 , 34 , 759 - 771 . Tama et al. Proteins: Struct., Funct., Bioinf . 2000 , 41 , 1 - 7 ]. We demonstrate how to physically interpret the eigenvalues computed in the RTB basis in terms of angular and linear velocities applied to the rigid blocks and how to construct a nonlinear extrapolation of motion out of these velocities. The key observation of our method is that the angular velocity of a rigid block can be interpreted as the result of an implicit force, such that the motion of the rigid block can be considered as a pure rotation about a certain center. We demonstrate the motions produced with the NOLB method on three different molecular systems and show that some of the lowest frequency normal modes correspond to the biologically relevant motions. For example, NOLB detects the spiral sliding motion of the TALE protein, which is capable of rapid diffusion along its target DNA. Overall, our method produces better structures compared to the standard approach, especially at large deformation amplitudes, as we demonstrate by visual inspection, energy, and topology analyses and also by the MolProbity service validation. Finally, our method is scalable and can be applied to very large molecular systems, such as ribosomes. Standalone executables of the NOLB normal-mode analysis method are available at https://team.inria.fr/nano-d/software/nolb-normal-modes/ . A graphical user interface created for the SAMSON software platform will be made available at https://www.samson-connect.net .

Generalized flexibility-rigidity index

Flexibility-rigidity index (FRI) has been developed as a robust, accurate, and efficient method for macromolecular thermal fluctuation analysis and B-factor prediction. The performance of FRI depends on its formulations of rigidity index and flexibility index. In this work, we introduce alternative rigidity and flexibility formulations. The structure of the classic Gaussian surface is utilized to construct a new type of rigidity index, which leads to a new class of rigidity densities with the classic Gaussian surface as a special case. Additionally, we introduce a new type of flexibility index based on the domain indicator property of normalized rigidity density. These generalized FRI (gFRI) methods have been extensively validated by the B-factor predictions of 364 proteins. Significantly outperforming the classic Gaussian network model, gFRI is a new generation of methodologies for accurate, robust, and efficient analysis of protein flexibility and fluctuation. Finally, gFRI based molecular surface generation and flexibility visualization are demonstrated.

Modeling global changes induced by local perturbations to the HIV-1 capsid

The HIV-1 capsid is a conical protein shell made up of hexamers and pentamers of the capsid protein. The capsid houses the viral genome and replication machinery, and its opening, or uncoating, within the host cell marks a critical step in the HIV-1 lifecycle. Binding of host factors such as TRIM5α and cyclophilin A (CypA) can alter the capsid's stability, accelerating or delaying the onset of uncoating and disrupting infectivity. We employ coarse-grained computational modeling to investigate the effects of point mutations and host factor binding on HIV-1 capsid stability. We find that the largest fluctuations occur in the low-curvature regions of the capsid, and that its structural dynamics are affected by perturbations at the inter-hexamer interfaces and near the CypA binding loop, suggesting roles for these features in capsid stability. Our models show that linking capsid proteins across hexamers attenuates vibration in the low-curvature regions of the capsid, but that linking within hexamers does not. These results indicate a possible mechanism through which CypA binding alters capsid stability and highlight the utility of coarse-grained network modeling for understanding capsid mechanics.

Multiscale Gaussian network model (mGNM) and multiscale anisotropic network model (mANM)

Gaussian network model (GNM) and anisotropic network model (ANM) are some of the most popular methods for the study of protein flexibility and related functions. In this work, we propose generalized GNM (gGNM) and ANM methods and show that the GNM Kirchhoff matrix can be built from the ideal low-pass filter, which is a special case of a wide class of correlation functions underpinning the linear scaling flexibility-rigidity index (FRI) method. Based on the mathematical structure of correlation functions, we propose a unified framework to construct generalized Kirchhoff matrices whose matrix inverse leads to gGNMs, whereas, the direct inverse of its diagonal elements gives rise to FRI method. With this connection, we further introduce two multiscale elastic network models, namely, multiscale GNM (mGNM) and multiscale ANM (mANM), which are able to incorporate different scales into the generalized Kirchhoff matrices or generalized Hessian matrices. We validate our new multiscale methods with extensive numerical experiments. We illustrate that gGNMs outperform the original GNM method in the B-factor prediction of a set of 364 proteins. We demonstrate that for a given correlation function, FRI and gGNM methods provide essentially identical B-factor predictions when the scale value in the correlation function is sufficiently large. More importantly, we reveal intrinsic multiscale behavior in protein structures. The proposed mGNM and mANM are able to capture this multiscale behavior and thus give rise to a significant improvement of more than 11% in B-factor predictions over the original GNM and ANM methods. We further demonstrate the benefits of our mGNM through the B-factor predictions of many proteins that fail the original GNM method. We show that the proposed mGNM can also be used to analyze protein domain separations. Finally, we showcase the ability of our mANM for the analysis of protein collective motions.

Flexibility-rigidity index for protein-nucleic acid flexibility and fluctuation analysis

Protein-nucleic acid complexes are important for many cellular processes including the most essential functions such as transcription and translation. For many protein-nucleic acid complexes, flexibility of both macromolecules has been shown to be critical for specificity and/or function. The flexibility-rigidity index (FRI) has been proposed as an accurate and efficient approach for protein flexibility analysis. In this article, we introduce FRI for the flexibility analysis of protein-nucleic acid complexes. We demonstrate that a multiscale strategy, which incorporates multiple kernels to capture various length scales in biomolecular collective motions, is able to significantly improve the state of art in the flexibility analysis of protein-nucleic acid complexes. We take the advantage of the high accuracy and O(N) computational complexity of our multiscale FRI method to investigate the flexibility of ribosomal subunits, which are difficult to analyze by alternative approaches. An anisotropic FRI approach, which involves localized Hessian matrices, is utilized to study the translocation dynamics in an RNA polymerase.

Simulation and Theory of Antibody Binding to Crowded Antigen-Covered Surfaces

In this paper we introduce a fully flexible coarse-grained model of immunoglobulin G (IgG) antibodies parametrized directly on cryo-EM data and simulate the binding dynamics of many IgGs to antigens adsorbed on a surface at increasing densities. Moreover, we work out a theoretical model that allows to explain all the features observed in the simulations. Our combined computational and theoretical framework is in excellent agreement with surface-plasmon resonance data and allows us to establish a number of important results. (i) Internal flexibility is key to maximize bivalent binding, flexible IgGs being able to explore the surface with their second arm in search for an available hapten. This is made clear by the strongly reduced ability to bind with both arms displayed by artificial IgGs designed to rigidly keep a prescribed shape. (ii) The large size of IgGs is instrumental to keep neighboring molecules at a certain distance (surface repulsion), which essentially makes antigens within reach of the second Fab always unoccupied on average. (iii) One needs to account independently for the thermodynamic and geometric factors that regulate the binding equilibrium. The key geometrical parameters, besides excluded-volume repulsion, describe the screening of free haptens by neighboring bound antibodies. We prove that the thermodynamic parameters govern the low-antigen-concentration regime, while the surface screening and repulsion only affect the binding at high hapten densities. Importantly, we prove that screening effects are concealed in relative measures, such as the fraction of bivalently bound antibodies. Overall, our model provides a valuable, accurate theoretical paradigm beyond existing frameworks to interpret experimental profiles of antibodies binding to multi-valent surfaces of different sorts in many contexts.

Antibodies are the main working horses of the human immune system. Remarkably, no matter the size or the shape of the pathological intruders, these extremely flexible three-lobe molecules are able to form a complex, thus eliciting an immune response. What makes antibodies so effective? To answer this and other questions, we have developed a simplified computational scheme to simulate the dynamics of many antibodies interacting with each other and with antigens. Coarse-grained models are a great opportunity, as they give access to a true multi-scale approach to biologically relevant problems. In this work, our innovative method allowed us to simulate the binding process of many antibodies to surface-adsorbed antigens. This led us to elucidate and quantify many important physical aspects of their biological function in agreement with experiments, such as the role of their flexibility and crowding effects at the hapten-covered surface, which were shown to finely regulate the avidity.

Communication: Capturing protein multiscale thermal fluctuations

Existing elastic network models are typically parametrized at a given cutoff distance and often fail to properly predict the thermal fluctuation of many macromolecules that involve multiple characteristic length scales. We introduce a multiscale flexibility-rigidity index (mFRI) method to resolve this problem. The proposed mFRI utilizes two or three correlation kernels parametrized at different length scales to capture protein interactions at corresponding scales. It is about 20% more accurate than the Gaussian network model (GNM) in the B-factor prediction of a set of 364 proteins. Additionally, the present method is able to deliver accurate predictions for some large macromolecules on which GNM fails to produce accurate predictions. Finally, for a protein of N residues, mFRI is of linear scaling (O(N)) in computational complexity, in contrast to the order of O(N(3)) for GNM.

Computational approaches to detect allosteric pathways in transmembrane molecular machines

Many of the functions of transmembrane proteins involved in signal processing and transduction across the cell membrane are determined by allosteric couplings that propagate the functional effects well beyond the original site of activation. Data gathered from breakthroughs in biochemistry, crystallography, and single molecule fluorescence have established a rich basis of information for the study of molecular mechanisms in the allosteric couplings of such transmembrane proteins. The mechanistic details of these couplings, many of which have therapeutic implications, however, have only become accessible in synergy with molecular modeling and simulations. Here, we review some recent computational approaches that analyze allosteric coupling networks (ACNs) in transmembrane proteins, and in particular the recently developed Protein Interaction Analyzer (PIA) designed to study ACNs in the structural ensembles sampled by molecular dynamics simulations. The power of these computational approaches in interrogating the functional mechanisms of transmembrane proteins is illustrated with selected examples of recent experimental and computational studies pursued synergistically in the investigation of secondary active transporters and GPCRs. This article is part of a Special Issue entitled: Membrane Proteins edited by J.C. Gumbart and Sergei Noskov.

CHARMM Force-Fields with Modified Polyphosphate Parameters Allow Stable Simulation of the ATP-Bound Structure of Ca(2+)-ATPase

Adenosine triphosphate (ATP) is an indispensable energy source in cells. In a wide variety of biological phenomena like glycolysis, muscle contraction/relaxation, and active ion transport, chemical energy released from ATP hydrolysis is converted to mechanical forces to bring about large-scale conformational changes in proteins. Investigation of structure-function relationships in these proteins by molecular dynamics (MD) simulations requires modeling of ATP in solution and ATP bound to proteins with accurate force-field parameters. In this study, we derived new force-field parameters for the triphosphate moiety of ATP based on the high-precision quantum calculations of methyl triphosphate. We tested our new parameters on membrane-embedded sarcoplasmic reticulum Ca(2+)-ATPase and four soluble proteins. The ATP-bound structure of Ca(2+)-ATPase remains stable during MD simulations, contrary to the outcome in shorter simulations using original parameters. Similar results were obtained with the four ATP-bound soluble proteins. The new force-field parameters were also tested by investigating the range of conformations sampled during replica-exchange MD simulations of ATP in explicit water. Modified parameters allowed a much wider range of conformational sampling compared with the bias toward extended forms with original parameters. A diverse range of structures agrees with the broad distribution of ATP conformations in proteins deposited in the Protein Data Bank. These simulations suggest that the modified parameters will be useful in studies of ATP in solution and of the many ATP-utilizing proteins.

A new algorithm for construction of coarse-grained sites of large biomolecules

The development of coarse-grained (CG) models for large biomolecules remains a challenge in multiscale simulations, including a rigorous definition of CG representations for them. In this work, we proposed a new stepwise optimization imposed with the boundary-constraint (SOBC) algorithm to construct the CG sites of large biomolecules, based on the s cheme of essential dynamics CG. By means of SOBC, we can rigorously derive the CG representations of biomolecules with less computational cost. The SOBC is particularly efficient for the CG definition of large systems with thousands of residues. The resulted CG sites can be parameterized as a CG model using the normal mode analysis based fluctuation matching method. Through normal mode analysis, the obtained modes of CG model can accurately reflect the functionally related slow motions of biomolecules. The SOBC algorithm can be used for the construction of CG sites of large biomolecules such as F-actin and for the study of mechanical properties of biomaterials.

Bridging between NMA and Elastic Network Models: Preserving All-Atom Accuracy in Coarse-Grained Models

Dynamics can provide deep insights into the functional mechanisms of proteins and protein complexes. For large protein complexes such as GroEL/GroES with more than 8,000 residues, obtaining a fine-grained all-atom description of its normal mode motions can be computationally prohibitive and is often unnecessary. For this reason, coarse-grained models have been used successfully. However, most existing coarse-grained models use extremely simple potentials to represent the interactions within the coarse-grained structures and as a result, the dynamics obtained for the coarse-grained structures may not always be fully realistic. There is a gap between the quality of the dynamics of the coarse-grained structures given by all-atom models and that by coarse-grained models. In this work, we resolve an important question in protein dynamics computations—how can we efficiently construct coarse-grained models whose description of the dynamics of the coarse-grained structures remains as accurate as that given by all-atom models? Our method takes advantage of the sparseness of the Hessian matrix and achieves a high efficiency with a novel iterative matrix projection approach. The result is highly significant since it can provide descriptions of normal mode motions at an all-atom level of accuracy even for the largest biomolecular complexes. The application of our method to GroEL/GroES offers new insights into the mechanism of this biologically important chaperonin, such as that the conformational transitions of this protein complex in its functional cycle are even more strongly connected to the first few lowest frequency modes than with other coarse-grained models.

Proteins and other biomolecules are not static but are constantly in motion. Moreover, they possess intrinsic collective motion patterns that are tightly linked to their functions. Thus, an accurate and detailed description of their motions can provide deep insights into their functional mechanisms. For large protein complexes with hundreds of thousands of atoms or more, an atomic level description of the motions can be computationally prohibitive, and so coarse-grained models with fewer structural details are often used instead. However, there can be a big gap between the quality of motions derived from atomic models and those from coarse-grained models. In this work, we solve an important problem in protein dynamics studies: how to preserve the atomic-level accuracy in describing molecular motions while using coarse-grained models? We accomplish this by developing a novel iterative matrix projection method that dramatically speeds up the computations. This method is significant since it promises accurate descriptions of protein motions approaching an all-atom level even for the largest biomolecular complexes. Results shown here for a large molecular chaperonin demonstrate how this can provide new insights into its functional process.

Computation of the memory functions in the generalized Langevin models for collective dynamics of macromolecules

We present a numerical method to approximate the memory functions in the generalized Langevin models for the collective dynamics of macromolecules. We first derive the exact expressions of the memory functions, obtained from projection to subspaces that correspond to the selection of coarse-grain variables. In particular, the memory functions are expressed in the forms of matrix functions, which will then be approximated by Krylov-subspace methods. It will also be demonstrated that the random noise can be approximated under the same framework, and the second fluctuation-dissipation theorem is automatically satisfied. The accuracy of the method is examined through several numerical examples.

Fast and anisotropic flexibility-rigidity index for protein flexibility and fluctuation analysis

Protein structural fluctuation, typically measured by Debye-Waller factors, or B-factors, is a manifestation of protein flexibility, which strongly correlates to protein function. The flexibility-rigidity index (FRI) is a newly proposed method for the construction of atomic rigidity functions required in the theory of continuum elasticity with atomic rigidity, which is a new multiscale formalism for describing excessively large biomolecular systems. The FRI method analyzes protein rigidity and flexibility and is capable of predicting protein B-factors without resorting to matrix diagonalization. A fundamental assumption used in the FRI is that protein structures are uniquely determined by various internal and external interactions, while the protein functions, such as stability and flexibility, are solely determined by the structure. As such, one can predict protein flexibility without resorting to the protein interaction Hamiltonian. Consequently, bypassing the matrix diagonalization, the original FRI has a computational complexity of O(N(2)). This work introduces a fast FRI (fFRI) algorithm for the flexibility analysis of large macromolecules. The proposed fFRI further reduces the computational complexity to O(N). Additionally, we propose anisotropic FRI (aFRI) algorithms for the analysis of protein collective dynamics. The aFRI algorithms permit adaptive Hessian matrices, from a completely global 3N × 3N matrix to completely local 3 × 3 matrices. These 3 × 3 matrices, despite being calculated locally, also contain non-local correlation information. Eigenvectors obtained from the proposed aFRI algorithms are able to demonstrate collective motions. Moreover, we investigate the performance of FRI by employing four families of radial basis correlation functions. Both parameter optimized and parameter-free FRI methods are explored. Furthermore, we compare the accuracy and efficiency of FRI with some established approaches to flexibility analysis, namely, normal mode analysis and Gaussian network model (GNM). The accuracy of the FRI method is tested using four sets of proteins, three sets of relatively small-, medium-, and large-sized structures and an extended set of 365 proteins. A fifth set of proteins is used to compare the efficiency of the FRI, fFRI, aFRI, and GNM methods. Intensive validation and comparison indicate that the FRI, particularly the fFRI, is orders of magnitude more efficient and about 10% more accurate overall than some of the most popular methods in the field. The proposed fFRI is able to predict B-factors for α-carbons of the HIV virus capsid (313 236 residues) in less than 30 seconds on a single processor using only one core. Finally, we demonstrate the application of FRI and aFRI to protein domain analysis.

On the relationship between low-frequency normal modes and the large-scale conformational changes of proteins

Normal mode analysis is a computational technique that allows to study the dynamics of biological macromolecules. It was first applied to small protein cases, more than thirty years ago. The interest in this technique then raised when it was realized that it can provide insights about the large-scale conformational changes a protein can experience, for instance upon ligand binding. As it was also realized that studying highly simplified protein models can provide similar insights, meaning that this kind of analysis can be both quick and simple to handle, several applications were proposed, in the context of various structural biology techniques. This review focuses on these applications, as well as on how the functional relevance of the lowest-frequency modes of proteins was established.

iMODS: internal coordinates normal mode analysis server

Normal mode analysis (NMA) in internal (dihedral) coordinates naturally reproduces the collective functional motions of biological macromolecules. iMODS facilitates the exploration of such modes and generates feasible transition pathways between two homologous structures, even with large macromolecules. The distinctive internal coordinate formulation improves the efficiency of NMA and extends its applicability while implicitly maintaining stereochemistry. Vibrational analysis, motion animations and morphing trajectories can be easily carried out at different resolution scales almost interactively. The server is versatile; non-specialists can rapidly characterize potential conformational changes, whereas advanced users can customize the model resolution with multiple coarse-grained atomic representations and elastic network potentials. iMODS supports advanced visualization capabilities for illustrating collective motions, including an improved affine-model-based arrow representation of domain dynamics. The generated all-heavy-atoms conformations can be used to introduce flexibility for more advanced modeling or sampling strategies. The server is free and open to all users with no login requirement at http://imods.chaconlab.org.

Aligning experimental and theoretical anisotropic B-factors: water models, normal-mode analysis methods, and metrics

The strength of X-ray crystallography in providing the information for protein dynamics has been under appreciated. The anisotropic B-factors (ADPs) from high-resolution structures are invaluable in studying the relationship among structure, dynamics, and function. Here, starting from an in-depth evaluation of the metrics used for comparing the overlap between two ellipsoids, we applied normal-mode analysis (NMA) to predict the theoretical ADPs and then align them with experimental results. Adding an extra layer of explicitly treated water on protein surface significantly improved the energy minimization results and better reproduced the anisotropy of experimental ADPs. In comparing experimental and theoretical ADPs, we focused on the overlap in shape, the alignment of dominant directions, and the similarity in magnitude. The choices of water molecules, NMA methods, and the metrics for evaluating the overlap of ADPs determined final results. This study provides useful information for exploring the physical basis and the application potential of experimental ADPs.

Molecular nonlinear dynamics and protein thermal uncertainty quantification

This work introduces molecular nonlinear dynamics (MND) as a new approach for describing protein folding and aggregation. By using a mode system, we show that the MND of disordered proteins is chaotic while that of folded proteins exhibits intrinsically low dimensional manifolds (ILDMs). The stability of ILDMs is found to strongly correlate with protein energies. We propose a novel method for protein thermal uncertainty quantification based on persistently invariant ILDMs. Extensive comparison with experimental data and the state-of-the-art methods in the field validate the proposed new method for protein B-factor prediction.

Efficient prediction of protein conformational pathways based on the hybrid elastic network model

Various computational models have gained immense attention by analyzing the dynamic characteristics of proteins. Several models have achieved recognition by fulfilling either theoretical or experimental predictions. Nonetheless, each method possesses limitations, mostly in computational outlay and physical reality. These limitations remind us that a new model or paradigm should advance theoretical principles to elucidate more precisely the biological functions of a protein and should increase computational efficiency. With these critical caveats, we have developed a new computational tool that satisfies both physical reality and computational efficiency. In the proposed hybrid elastic network model (HENM), a protein structure is represented as a mixture of rigid clusters and point masses that are connected with linear springs. Harmonic analyses based on the HENM have been performed to generate normal modes and conformational pathways. The results of the hybrid normal mode analyses give new physical insight to the 70S ribosome. The feasibility of the conformational pathways of hybrid elastic network interpolation (HENI) was quantitatively evaluated by comparing three different overlap values proposed in this paper. A remarkable observation is that the obtained mode shapes and conformational pathways are consistent with each other. Our timing results show that HENM has some advantage in computational efficiency over a coarse-grained model, especially for large proteins, even though it takes longer to construct the HENM. Consequently, the proposed HENM will be one of the best alternatives to the conventional coarse-grained ENMs and all-atom based methods (such as molecular dynamics) without loss of physical reality.