Coarse-graining of protein structures for the normal mode studies

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.

From System Modeling to System Analysis: The Impact of Resolution Level and Resolution Distribution in the Computer-Aided Investigation of Biomolecules

The ever increasing computer power, together with the improved accuracy of atomistic force fields, enables researchers to investigate biological systems at the molecular level with remarkable detail. However, the relevant length and time scales of many processes of interest are still hardly within reach even for state-of-the-art hardware, thus leaving important questions often unanswered. The computer-aided investigation of many biological physics problems thus largely benefits from the usage of coarse-grained models, that is, simplified representations of a molecule at a level of resolution that is lower than atomistic. A plethora of coarse-grained models have been developed, which differ most notably in their granularity; this latter aspect determines one of the crucial open issues in the field, i.e. the identification of an optimal degree of coarsening, which enables the greatest simplification at the expenses of the smallest information loss. In this review, we present the problem of coarse-grained modeling in biophysics from the viewpoint of system representation and information content. In particular, we discuss two distinct yet complementary aspects of protein modeling: on the one hand, the relationship between the resolution of a model and its capacity of accurately reproducing the properties of interest; on the other hand, the possibility of employing a lower resolution description of a detailed model to extract simple, useful, and intelligible information from the latter.

Inferring functional units in ion channel pores via relative entropy

Coarse-grained protein models approximate the first-principle physical potentials. Among those modeling approaches, the relative entropy framework yields promising and physically sound results, in which a mapping from the target protein structure and dynamics to a model is defined and subsequently adjusted by an entropy minimization of the model parameters. Minimization of the relative entropy is equivalent to maximization of the likelihood of reproduction of (configurational ensemble) observations by the model. In this study, we extend the relative entropy minimization procedure beyond parameter fitting by a second optimization level, which identifies the optimal mapping to a (dimension-reduced) topology. We consider anisotropic network models of a diverse set of ion channels and assess our findings by comparison to experimental results.

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.

Cofilin reduces the mechanical properties of actin filaments: approach with coarse-grained methods

An actin filament is an essential cytoskeleton protein in a cell. Various proteins bind to actin for cell functions such as migration, division, and shape control. ADF/cofilin is a protein that severs actin filaments and is related to their dynamics. Actin is known to have excellent mechanical properties. Binding cofilin reduces its mechanical properties, and is related to the severing process. In this research, we applied a coarse-grained molecular dynamics simulation (CGMD) method to obtain actin filaments and cofilin-bound actin (cofilactin) filaments. Using these two obtained models, we constructed an elastic network model-based structure and conducted a normal mode analysis. Based on the low-frequency normal modes of the filament structure, we applied the continuum beam theory to calculate the mechanical properties of the actin and cofilactin filaments. The CGMD method provided structurally accurate actin and cofilactin filaments in relation to the mechanical properties, which showed good agreement with the established experimental results.

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.

Perspective: Coarse-grained models for biomolecular systems

By focusing on essential features, while averaging over less important details, coarse-grained (CG) models provide significant computational and conceptual advantages with respect to more detailed models. Consequently, despite dramatic advances in computational methodologies and resources, CG models enjoy surging popularity and are becoming increasingly equal partners to atomically detailed models. This perspective surveys the rapidly developing landscape of CG models for biomolecular systems. In particular, this review seeks to provide a balanced, coherent, and unified presentation of several distinct approaches for developing CG models, including top-down, network-based, native-centric, knowledge-based, and bottom-up modeling strategies. The review summarizes their basic philosophies, theoretical foundations, typical applications, and recent developments. Additionally, the review identifies fundamental inter-relationships among the diverse approaches and discusses outstanding challenges in the field. When carefully applied and assessed, current CG models provide highly efficient means for investigating the biological consequences of basic physicochemical principles. Moreover, rigorous bottom-up approaches hold great promise for further improving the accuracy and scope of CG models for biomolecular systems.

Role of sequence and structural polymorphism on the mechanical properties of amyloid fibrils

Amyloid fibrils playing a critical role in disease expression, have recently been found to exhibit the excellent mechanical properties such as elastic modulus in the order of 10 GPa, which is comparable to that of other mechanical proteins such as microtubule, actin filament, and spider silk. These remarkable mechanical properties of amyloid fibrils are correlated with their functional role in disease expression. This suggests the importance in understanding how these excellent mechanical properties are originated through self-assembly process that may depend on the amino acid sequence. However, the sequence-structure-property relationship of amyloid fibrils has not been fully understood yet. In this work, we characterize the mechanical properties of human islet amyloid polypeptide (hIAPP) fibrils with respect to their molecular structures as well as their amino acid sequence by using all-atom explicit water molecular dynamics (MD) simulation. The simulation result suggests that the remarkable bending rigidity of amyloid fibrils can be achieved through a specific self-aggregation pattern such as antiparallel stacking of β strands (peptide chain). Moreover, we have shown that a single point mutation of hIAPP chain constituting a hIAPP fibril significantly affects the thermodynamic stability of hIAPP fibril formed by parallel stacking of peptide chain, and that a single point mutation results in a significant change in the bending rigidity of hIAPP fibrils formed by antiparallel stacking of β strands. This clearly elucidates the role of amino acid sequence on not only the equilibrium conformations of amyloid fibrils but also their mechanical properties. Our study sheds light on sequence-structure-property relationships of amyloid fibrils, which suggests that the mechanical properties of amyloid fibrils are encoded in their sequence-dependent molecular architecture.

COARSE-GRAINED MODELING OF PROTEIN UNFOLDING DYNAMICS

We present a new dynamic elastic network model (DENM) that describes the unfolding process of a force-loaded protein. The protein interaction network and its potentials are constructed based on information of its native-state structure obtained from the Protein Data Bank, with network nodes positioned at the C_α coordinates of the protein backbone. Specifically, to mimic the unfolding process, i.e., to simulate the process of overcoming the local energy barrier on the free energy landscape with force loading, the noncovalent protein network bonds (i.e., hydrogen bonds, salt bridges, hydrophobic contacts, etc.) are broken one-by-one with a certain probability, while the strong covalent bonds along the backbone (i.e., peptide bonds, disulfide bonds, etc.) are kept intact. The jumping event from local energy minima (bonds breaking rate) are chosen according to Kramer's theory and the Bell model. Moreover, we exploit the self-similar structure of proteins at different scales to design an effective coarse-graining procedure for DENM with optimal parameter selection. The robustness of DENM is validated by coarse-grained molecular dynamics (MD) simulation against atomistic MD simulation of force-extension processes of the Fibrinogen and Titin Immunoglobulin proteins. We observe that the native structure of the proteins determines the total unfolding dynamics (including large deviations) and not just the fluctuations around the native state.

Mapping polymerization and allostery of hemoglobin S using point mutations

Hemoglobin is a complex system that undergoes conformational changes in response to oxygen, allosteric effectors, mutations, and environmental changes. Here, we study allostery and polymerization of hemoglobin and its variants by application of two previously described methods: (i) AllosMod for simulating allostery dynamics given two allosterically related input structures and (ii) a machine-learning method for dynamics- and structure-based prediction of the mutation impact on allostery (Weinkam et al. J. Mol. Biol. 2013, 425, 647-661), now applicable to systems with multiple coupled binding sites, such as hemoglobin. First, we predict the relative stabilities of substates and microstates of hemoglobin, which are determined primarily by entropy within our model. Next, we predict the impact of 866 annotated mutations on hemoglobin's oxygen binding equilibrium. We then discuss a subset of 30 mutations that occur in the presence of the sickle cell mutation and whose effects on polymerization have been measured. Seven of these HbS mutations occur in three predicted druggable binding pockets that might be exploited to directly inhibit polymerization; one of these binding pockets is not apparent in the crystal structure, but only in structures generated by AllosMod. For the 30 mutations, we predict that mutation-induced conformational changes within a single tetramer tend not to significantly impact polymerization; instead, these mutations more likely impact polymerization by directly perturbing a polymerization interface. Finally, our analysis of allostery allows us to hypothesize why hemoglobin evolved to have multiple subunits and a persistent low frequency sickle cell mutation.

PIM: phase integrated method for normal mode analysis of biomolecules in a crystalline environment

In this study, a normal mode analysis, named phase integrated method (PIM), is developed for computing modes of biomolecules in a crystalline environment. PIM can calculate low-frequency modes on one or a few asymmetric units (AUs) and generate exact modes of a whole unit cell according to space group symmetry, while the translational symmetry between unit cells is maintained via the periodic boundary condition. Therefore, the method can dramatically reduce computational cost in mode calculation in the presence of crystal symmetry. PIM also has an option to map modes onto a single AU to form an orthonormalized mode set, which can be directly applied to normal-mode-based thermal parameter refinement in X-ray crystallography. The performance of PIM was tested on all 65 space groups available in protein crystals (one protein for each space group) and on another set of 83 ultra-high-resolution X-ray structures. The results showed that considering space group symmetry in mode calculation is crucial for accurately describing vibrational motion in a crystalline environment. Moreover, the optimal inter-AU packing stiffness was found to be about 60% of that of intra-AU interactions (non-bonded interaction only).

On the applicability of elastic network normal modes in small-molecule docking

Incorporating backbone flexibility into protein-ligand docking is still a challenging problem. In protein-protein docking, normal mode analysis (NMA) has become increasingly popular as it can be used to describe the collective motions of a biological system, but the question of whether NMA can also be useful in predicting the conformational changes observed upon small-molecule binding has only been addressed in a few case studies. Here, we describe a large-scale study on the applicability of NMA for protein-ligand docking using 433 apo/holo pairs of the Astex data sets. On the basis of sets of the first normal modes from the apo structure, we first generated for each paired holo structure a set of conformations that optimally reproduce its C(α) trace with respect to the underlying normal mode subspace. Using AutoDock, GOLD, and FlexX we then docked the original ligands into these conformations to assess how the docking performance depends on the number of modes used to reproduce the holo structure. The results of our study indicate that, even for such a best-case scenario, the use of normal mode analysis in small-molecule docking is restricted and that a general rule on how many modes to use does not seem to exist or at least is not easy to find.

Virtual interface substructure synthesis method for normal mode analysis of super-large molecular complexes at atomic resolution

Normal mode analysis of large biomolecular complexes at atomic resolution remains challenging in computational structure biology due to the requirement of large amount of memory space and central processing unit time. In this paper, we present a method called virtual interface substructure synthesis method or VISSM to calculate approximate normal modes of large biomolecular complexes at atomic resolution. VISSM introduces the subunit interfaces as independent substructures that join contacting molecules so as to keep the integrity of the system. Compared with other approximate methods, VISSM delivers atomic modes with no need of a coarse-graining-then-projection procedure. The method was examined for 54 protein-complexes with the conventional all-atom normal mode analysis using CHARMM simulation program and the overlap of the first 100 low-frequency modes is greater than 0.7 for 49 complexes, indicating its accuracy and reliability. We then applied VISSM to the satellite panicum mosaic virus (SPMV, 78,300 atoms) and to F-actin filament structures of up to 39-mer, 228,813 atoms and found that VISSM calculations capture functionally important conformational changes accessible to these structures at atomic resolution. Our results support the idea that the dynamics of a large biomolecular complex might be understood based on the motions of its component subunits and the way in which subunits bind one another.

Normal mode analysis with molecular geometry restraints: bridging molecular mechanics and elastic models

A new method for normal mode analysis is reported for all-atom structures using molecular geometry restraints (MGR). Similar to common molecular mechanics force fields, the MGR potential contains short- and long-range terms. The short-range terms are defined by molecular geometry, i.e., bond lengths, angles and dihedrals; the long-range term is similar to that in elastic network models. Each interaction term uses a single force constant parameter, and is determined by fitting against a set of known structures. Tests on proteins/non-proteins show that MGR can produce low frequency eigenvectors closer to all-atom force-field-based methods than conventional elastic network models. Moreover, the "tip effect", found in low frequency eigenvectors in elastic network models, is reduced in MGR to the same level of the modes produced by force-field-based methods. The results suggest that molecular geometry plays an important role, in addition to molecular shape, in determining low frequency deformational motions. MGR does not require initial energy minimization, and is applicable to almost any structure, including the one with missing atoms, bad contacts, or bad geometries, frequently observed in low-resolution structure determination and refinement. The method bridges the two major representations in normal mode analyses, i.e., the molecular mechanics models and elastic network models.

Multiscale network model for large protein dynamics

Protein dynamics is essential for gaining insight into biological functions of proteins. Although protein dynamics is well delineated by molecular model, the molecular model is computationally prohibited for simulating large protein structures. In this work, we provide a multiscale network model (MNM) that allows the efficient computation on low-frequency normal modes related to structural deformation of proteins as well as dynamic behavior of functional sites. Specifically, MNM consists of two regions, one of which is described as a low-resolution structure, while the other is dictated by a high-resolution structure. The high-resolution regions using all alpha carbons of the protein are mainly binding site parts, which play a critical function in molecules, while the low-resolution parts are constructed from a further coarse-grained model (not using all alpha carbons). The feasibility of MNM to observe the cooperative motion of a protein structure was validated. It was shown that the MNM enables us to understand functional motion of proteins with computational efficiency.

Large Protein Dynamics Described by Hierarchical-Component Mode Synthesis

Protein dynamics has played a pivotal role in understanding the biological function of protein. For investigation of such dynamics, normal-mode analysis (NMA) has been broadly employed with atomistic model and/or coarse-grained models such as elastic network model (ENM). For large protein complexes, NMA with even ENM encounters the expensive computational process such as diagonalization of Hessian (stiffness) matrix. Here, we suggest the hierarchical-component mode synthesis (hCMS), which allows the fast computation of low-frequency normal modes related to conformational change. Specifically, a large protein structure is regarded as a combination of several structural units, for which the eigen-value problem is utilized for obtaining the frequencies and their normal modes for each structural unit, and consequently, such frequencies and normal modes are assembled with geometrical constraint for interface between structural units in order to find the low-frequency normal modes of a large protein complex. It is shown that hCMS is able to provide the normal modes with accuracy, quantitatively comparable to those of original NMA. This implies that hCMS may enable the computationally efficient analysis of large protein dynamics.

Statistical thermodynamics of residue fluctuations in native proteins

Statistical thermodynamics of residue fluctuations of native proteins in a temperature, pressure, and force reservoir is formulated. The general theory is discussed in terms of harmonic and anharmonic fluctuations of residues. The two elastic network models based on the harmonic approximation, the anisotropic network and the Gaussian network models are discussed as the limiting cases of the general theory. The heat capacity and the correlations between the energy fluctuations and residue fluctuations are obtained for the harmonic approximation. The formulation is extended to large fluctuations of residues in order to account for effects of anharmonicity. The fluctuation probability function is constructed for this purpose as a tensorial Hermite series expansion with higher order moments of fluctuations as coefficients. Evaluation of the higher order moments using the proposed statistical thermodynamics model is explained. The formulation is applied to a hexapeptide and the fluctuations of residues obtained by molecular dynamics simulations are characterized in the framework of the model developed. In particular, coupling of two different modes in the nonlinear model is discussed in detail.

The ribosome structure controls and directs mRNA entry, translocation and exit dynamics

The protein-synthesizing ribosome undergoes large motions to effect the translocation of tRNAs and mRNA; here, the domain motions of this system are explored with a coarse-grained elastic network model using normal mode analysis. Crystal structures are used to construct various model systems of the 70S complex with/without tRNA, elongation factor Tu and the ribosomal proteins. Computed motions reveal the well-known ratchet-like rotational motion of the large subunits, as well as the head rotation of the small subunit and the high flexibility of the L1 and L7/L12 stalks, even in the absence of ribosomal proteins. This result indicates that these experimentally observed motions during translocation are inherently controlled by the ribosomal shape and only partially dependent upon GTP hydrolysis. Normal mode analysis further reveals the mobility of A- and P-tRNAs to increase in the absence of the E-tRNA. In addition, the dynamics of the E-tRNA is affected by the absence of the ribosomal protein L1. The mRNA in the entrance tunnel interacts directly with helicase proteins S3 and S4, which constrain the mRNA in a clamp-like fashion, as well as with protein S5, which likely orients the mRNA to ensure correct translation. The ribosomal proteins S7, S11 and S18 may also be involved in assuring translation fidelity by constraining the mRNA at the exit site of the channel. The mRNA also interacts with the 16S 3' end forming the Shine-Dalgarno complex at the initiation step; the 3' end may act as a 'hook' to reel in the mRNA to facilitate its exit.

A minimalist network model for coarse-grained normal mode analysis and its application to biomolecular x-ray crystallography

In this article, we report a method for coarse-grained normal mode analysis called the minimalist network model. The main features of the method are that it can deliver accurate low-frequency modes on structures without undergoing initial energy minimization and that it also retains the details of molecular interactions. The method does not require any additional adjustable parameters after coarse graining and is computationally very fast. Tests on modeling the experimentally measured anisotropic displacement parameters in biomolecular x-ray crystallography demonstrate that the method can consistently perform better than other commonly used methods including our own one. We expect this method to be effective for applications such as structural refinement and conformational sampling.

Effects of surface water on protein dynamics studied by a novel coarse-grained normal mode approach

Normal mode analysis (NMA) has received much attention as a direct approach to extract the collective motions of macromolecules. However, the stringent requirement of computational resources by classical all-atom NMA limits the size of the macromolecules to which the method is normally applied. We implemented a novel coarse-grained normal mode approach based on partitioning the all-atom Hessian matrix into relevant and nonrelevant parts. It is interesting to note that, using classical all-atom NMA results as a reference, we found that this method generates more accurate results than do other coarse-grained approaches, including elastic network model and block normal mode approaches. Moreover, this new method is effective in incorporating the energetic contributions from the nonrelevant atoms, including surface water molecules, into the coarse-grained protein motions. The importance of such improvements is demonstrated by the effect of surface water to shift vibrational modes to higher frequencies and by an increase in overlap of the coarse-grained eigenvector space (the motion directions) with that obtained from molecular dynamics simulations of solvated protein in a water box. These results not only confirm the quality of our method but also point out the importance of incorporating surface structural water in studying protein dynamics.