Anharmonic Normal Mode Analysis of Elastic Network Model Improves the Modeling of Atomic Fluctuations in Protein Crystal Structures

Protein conformational dynamics, despite its significant anharmonicity, has been widely explored by normal mode analysis (NMA) based on atomic or coarse-grained potential functions. To account for the anharmonic aspects of protein dynamics, this study proposes, and has performed, an anharmonic NMA (ANMA) based on the C_α-only elastic network models, which assume elastic interactions between pairs of residues whose C_α atoms or heavy atoms are within a cutoff distance. The key step of ANMA is to sample an anharmonic potential function along the directions of eigenvectors of the lowest normal modes to determine the mean-squared fluctuations along these directions. ANMA was evaluated based on the modeling of anisotropic displacement parameters (ADPs) from a list of 83 high-resolution protein crystal structures. Significant improvement was found in the modeling of ADPs by ANMA compared with standard NMA. Further improvement in the modeling of ADPs is attained if the interactions between a protein and its crystalline environment are taken into account. In addition, this study has determined the optimal cutoff distances for ADP modeling based on elastic network models, and these agree well with the peaks of the statistical distributions of distances between C_α atoms or heavy atoms derived from a large set of protein crystal structures.     

Buckling of a growing tissue and the emergence of two-dimensional patterns

The process of biological growth and the associated generation of residual stress has previously been considered as a driving mechanism for tissue buckling and pattern selection in numerous areas of biology. Here, we develop a two-dimensional thin plate theory to simulate the growth of cultured intestinal epithelial cells on a deformable substrate, with the goal of elucidating how a tissue engineer might best recreate the regular array of invaginations (crypts of Lieberkühn) found in the wall of the mammalian intestine. We extend the standard von Kármán equations to incorporate inhomogeneity in the plate’s mechanical properties and surface stresses applied to the substrate by cell proliferation. We determine numerically the configurations of a homogeneous plate under uniform cell growth, and show how tethering to an underlying elastic foundation can be used to promote higher-order buckled configurations. We then examine the independent effects of localised softening of the substrate and spatial patterning of cellular growth, demonstrating that (within a two-dimensional framework, and contrary to the predictions of one-dimensional models) growth patterning constitutes a more viable mechanism for control of crypt distribution than does material inhomogeneity.     

Temperature‐induced unfolding behavior of proteins studied by tensorial elastic network model

Motivated by single molecule experiments and recent molecular dynamics (MD) studies, we propose a simple and computationally efficient method based on a tensorial elastic network model to investigate the unfolding pathways of proteins under temperature variation. The tensorial elastic network model, which relies on the native state topology of a protein, combines the anisotropic network model, the bond bending elasticity, and the backbone twist elasticity to successfully predicts both the isotropic and anisotropic fluctuations in a manner similar to the Gaussian network model and anisotropic network model. The unfolding process is modeled by breaking the native contacts between residues one by one, and by assuming a threshold value for strain fluctuation. Using this method, we simulated the unfolding processes of four well‐characterized proteins: chymotrypsin inhibitor, barnase, ubiquitein, and adenalyate kinase. We found that this step‐wise process leads to two or more cooperative, first‐order‐like transitions between partial denaturation states. The sequence of unfolding events obtained using this method is consistent with experimental and MD studies. The results also imply that the native topology of proteins “encrypts” information regarding their unfolding process. Proteins 2016; 84:1767–1775. © 2016 Wiley Periodicals, Inc.     

Protein structural variation in computational models and crystallographic data

Normal mode analysis offers an efficient way of modeling the conformational flexibility of protein structures. We use anisotropic displacement parameters from crystallography to test the quality of prediction of both the magnitude and directionality of conformational flexibility. Normal modes from four simple elastic network model potentials and from the CHARMM force field are calculated for a data set of 83 diverse, ultrahigh-resolution crystal structures. While all five potentials provide good predictions of the magnitude of flexibility, all-atom potentials have a clear edge at prediction of directionality, and the CHARMM potential has the highest prediction quality. The low-frequency modes from different potentials are similar, but those computed from the CHARMM potential show the greatest difference from the elastic network models. The comprehensive evaluation demonstrates the costs and benefits of using normal mode potentials of varying complexity.     

Stress-sensitive tissue regeneration in viscoelastic biomaterials subjected to modulated tensile strain

This research contribution addresses the mechanochemistry of intra-tissue mass transfer for nutrients, oxygen, growth factors, and other essential ingredients that anchorage-dependent cells require for successful proliferation on biocompatible surfaces. The unsteady state reaction-diffusion equation (i.e., modified diffusion equation) is solved according to the von Kármán-Pohlhausen integral method of boundary layer analysis when nutrient consumption and tissue regeneration are stimulated by harmonically imposed stress. The mass balance with diffusion and stress-sensitive kinetics represents a rare example where the Damköhler and Deborah numbers appear together in an effort to simulate the development of mass transfer boundary layers in porous viscoelastic biomaterials. The Boltzmann superposition integral is employed to calculate time-dependent strain in terms of the real and imaginary components of dynamic compliance for viscoelastic solids that transmit harmonic excitation to anchorage-dependent cells. Rates of nutrient consumption under stress-free conditions are described by third-order kinetics which include local mass densities of nutrients, oxygen, and attached cells that maintain dynamic equilibrium with active protein sites in the porous matrix. Thinner nutrient mass transfer boundary layers are stabilized at shorter dimensionless diffusion times when the stress-free intra-tissue Damköhler number increases above its initial-condition-sensitive critical value. The critical stress-sensitive intra-tissue Damköhler number, above which it is necessary to consider the effect of harmonic strain on nutrient consumption and tissue regeneration, is proportional to the Deborah number and corresponds to a larger fraction of the stress-free intra-tissue Damköhler number in rigid biomaterials.     

Predicting the order in which contacts are broken during single molecule protein stretching experiments

We combine two methods to enable the prediction of the order in which contacts are broken under external stretching forces in single molecule experiments. These two methods are Gō-like models and elastic network models. The Gō-like models have shown remarkable success in representing many aspects of protein behavior, including the reproduction of experimental data obtained from atomic force microscopy. The simple elastic network models are often used successfully to predict the fluctuations of residues around their mean positions, comparing favorably with the experimentally measured crystallographic B-factors. The behavior of biomolecules under external forces has been demonstrated to depend principally on their elastic properties and the overall shape of their structure. We have studied in detail the muscle protein titin and green fluorescent protein and tested for ten other proteins. First, we stretch the proteins computationally by performing stochastic dynamics simulations with the Gō-like model. We obtain the force-displacement curves and unfolding scenarios of possible mechanical unfolding. We then use the elastic network model to calculate temperature factors (B-factors) and compare the slowest modes of motion for the stretched proteins and compare them with the predicted order of breaking contacts between residues in the Gō-like model. Our results show that a simple Gaussian network model is able to predict contacts that break in the next time stage of stretching. Additionally, we have found that the contact disruption is strictly correlated with the highest force exerted by the backbone on these residues. Our prediction of bond-breaking agrees well with the unfolding scenario obtained with the Gō-like model. We anticipate that this method will be a useful new tool for interpreting stretching experiments.     

Molecular mechanism of domain swapping in proteins: an analysis of slower motions

Domain swapping is a structural phenomenon that plays an important role in the mechanism of oligomerization of some proteins. The monomer units in the oligomeric structure become entangled with each other. Here we investigate the mechanism of domain swapping in diphtheria toxin and the structural criteria required for it to occur by analyzing the slower modes of motion with elastic network models, Gaussian network model and anisotropic network model. We take diphtheria toxin as a representative of this class of domain-swapped proteins and show that the domain, which is being swapped in the dimeric state, rotates and twists, in going from the "open" to the "closed" state, about a hinge axis that passes through the middle of the loop extending between two domains. A combination of the intra- and intermolecular contacts of the dimer is almost equivalent to that of the monomer, which shows that the relative orientations of the residues in both forms are almost identical. This is also reflected in the calculated B-factors when compared with the experimentally determined B-factors in x-ray crystal structures. The slowest modes of both the monomer and dimer show a common hinge centered on residue 387. The differences in distances between the monomer and the dimer also shows the hinge at nearly the same location (residue 381). Finally, the first three dominant modes of anisotropic network model together shows a twisting motion about the hinge centered on residue 387. We further identify the location of hinges for a set of another 12 domain swapped proteins and give the quantitative measures of the motions of the swapped domains toward their "closed" state, i.e., the overlap and correlation between vectors.     

Tensorial elastic network model for protein dynamics: Integration of the anisotropic network model with bond‐bending and twist elasticities

We present a tensorial elastic network model (TNM) to describe the equilibrium fluctuations of proteins near their native fold structure. The model combines the anisotropic network model (ANM), bond bending elasticity, and backbone twist elasticity, and can predict both the isotropic fluctuations, similar to the Gaussian network model (GNM), and anisotropic fluctuations, similar to the ANM. TNM performs equally well for B‐factor predictions as GNM and predicts the anisotropy of B‐factors better than ANM. The model also outperforms the ANM in its predictability of the complete anisotropic displacement parameters. Proteins 2012; © 2012 Wiley Periodicals, Inc.     

Structural flexibility in proteins: impact of the crystal environment

MOTIVATION: In the study of the structural flexibility of proteins, crystallographic Debye-Waller factors are the most important experimental information used in the calibration and validation of computational models, such as the very successful elastic network models (ENMs). However, these models are applied to single protein molecules, whereas the experiments are performed on crystals. Moreover, the energy scale in standard ENMs is undefined and must be obtained by fitting to the same data that the ENM is trying to predict, reducing the predictive power of the model. RESULTS: We develop an elastic network model for the whole protein crystal in order to study the influence of crystal packing and lattice vibrations on the thermal fluctuations of the atom positions. We use experimental values for the compressibility of the crystal to establish the energy scale of our model. We predict the elastic constants of the crystal and compare with experimental data. Our main findings are (1) crystal packing modifies the atomic fluctuations considerably and (2) thermal fluctuations are not the dominant contribution to crystallographic Debye-Waller factors. AVAILABILITY: The programs developed for this work are available as supplementary material at Bioinformatics Online.     

Evaluating elastic network models of crystalline biological molecules with temperature factors, correlated motions, and diffuse x-ray scattering

In this study, the variance-covariance matrix of protein motions is used to compare several elastic network models within the theoretical framework of x-ray scattering from crystals. A set of 33 ultra-high resolution structures is used to characterize the average scaling behavior of the vibrational density of states and make comparisons between experimental and theoretical temperature factors. Detailed investigations of the vibrational density of states, correlations, and predicted diffuse x-ray scatter are carried out for crystalline Staphylococcal nuclease; correlations and diffuse x-ray scatter are also compared to predictions from the translation, libration, screw model and a liquid-like dynamics model. We show that elastic network models developed to best predict temperature factors without regard for the crystal environment have relatively strong long-range interactions that yield very short-ranged atom-atom correlations. Further, we find that the low-frequency modes dominate the variance-covariance matrix only for those models with a physically reasonable vibrational density of states, and the fraction of modes required to converge the correlations is higher than that typically used for elastic network model studies. The practical implications are explored using computed diffuse x-ray scatter, which can be measured experimentally.     

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.     

Homogenized stiffness matrices for mineralized collagen fibrils and lamellar bone using unit cell finite element models

Mineralized collagen fibrils have been usually analyzed like a two-phase composite material where crystals are considered as platelets that constitute the reinforcement phase. Different models have been used to describe the elastic behavior of the material. In this work, it is shown that when Halpin–Tsai equations are applied to estimate elastic constants from typical constituent properties, not all crystal dimensions yield a model that satisfy thermodynamic restrictions. We provide the ranges of platelet dimensions that lead to positive definite stiffness matrices. On the other hand, a finite element model of a mineralized collagen fibril unit cell under periodic boundary conditions is analyzed. By applying six canonical load cases, homogenized stiffness matrices are numerically calculated. Results show a monoclinic behavior of the mineralized collagen fibril. In addition, a 5-layer lamellar structure is also considered where crystals rotate in adjacent layers of a lamella. The stiffness matrix of each layer is calculated applying Lekhnitskii transformations, and a new finite element model under periodic boundary conditions is analyzed to calculate the homogenized 3D anisotropic stiffness matrix of a unit cell of lamellar bone. Results are compared with the rule-of-mixtures showing in general good agreement.     

Unsteady hydrodynamic forces acting on a robotic arm and its flow field: Application to the crawl stroke

This study aims to clarify the mechanisms by which unsteady hydrodynamic forces act on the hand of a swimmer during a crawl stroke. Measurements were performed for a hand attached to a robotic arm with five degrees of freedom independently controlled by a computer. The computer was programmed so the hand and arm mimicked a human performing the stroke. We directly measured forces on the hand and pressure distributions around it at 200 Hz; flow fields underwater near the hand were obtained via 2D particle image velocimetry (PIV). The data revealed two mechanisms that generate unsteady forces during a crawl stroke. One is the unsteady lift force generated when hand movement changes direction during the stroke, leading to vortex shedding and bound vortex created around it. This bound vortex circulation results in a lift that contributes to the thrust. The other occurs when the hand moves linearly with a large angle of attack, creating a Kármán vortex street. This street alternatively sheds clockwise and counterclockwise vortices, resulting in a quasi-steady drag contributing to the thrust. We presume that professional swimmers benefit from both mechanisms. Further studies are necessary in which 3D flow fields are measured using a 3D PIV system and a human swimmer.     

A finite element framework for computation of protein normal modes and mechanical response

A computational framework based on the Finite Element Method is presented to calculate the normal modes and mechanical response of proteins and their supramolecular assemblies. Motivated by elastic network models, proteins are treated as continuum elastic solids with molecular volume defined by their solvent-excluded surface. The discretized Finite Element representation is obtained using a surface simplification algorithm that facilitates the generation of models of arbitrary prescribed spatial resolution. The procedure is applied to a mutant of T4 phage lysozyme, G-actin, syntenin, cytochrome-c', beta-tubulin, and the supramolecular assembly filamentous actin (F-actin). Equilibrium thermal fluctuations of alpha-carbon atoms and their inter-residue correlations compare favorably with all-atom-based results, the Rotational-Translational Block procedure, and experiment. Additionally, the free vibration and compressive buckling responses of F-actin are in quantitative agreement with experiment. The proposed methodology is applicable to any protein or protein assembly and facilitates the incorporation of specific atomic-level interactions, including aqueous-electrolyte-mediated electrostatic effects and solvent damping. The procedure is equally applicable to proteins with known atomic coordinates as it is to electron density maps of proteins, protein complexes, and supramolecular assemblies of unknown atomic structure.     

Flexible fitting to cryo-electron microscopy maps with coarse-grained elastic network models

Cryo-electron microscopy has become an important tool for protein structure determination in recent decades. Since proteins may exist in multiple conformational states, combining high resolution X-ray or NMR structures with cryo-electron microscopy maps is a useful approach to obtain proteins in different functional states. Flexible fitting methods used in cryo-electron microscopy aim to obtain an unknown protein conformation from a high resolution structure and a cryo-electron microscopy map. Since all-atom flexible fitting is computationally expensive, many efficient flexible fitting algorithms that utilize coarse-grained elastic network models have been proposed. In this study, we investigated performance of three coarse-grained elastic network model-based flexible fitting methods (EMFF, iModFit, NMFF) using 25 protein pairs at four resolutions. This study shows that the application of coarse-grained elastic network models to flexible fitting of cryo-electron microscopy maps can provide fast and fruitful models of various conformational states of proteins.     

A comparative analysis of the equilibrium dynamics of a designed protein inferred from NMR,X‐ray,and computations

A detailed analysis of high‐resolution structural data and computationally predicted dynamics was carried out for a designed sugar‐binding protein. The mean‐square deviations in the positions of residues derived from nuclear magnetic resonance (NMR) models and those inferred from X‐ray crystallographic B‐factors for two different crystal forms were compared with the predictions based on the Gaussian Network Model (GNM) and the results from molecular dynamics (MD) simulations. GNM systematically yielded a higher correlation than MD, with experimental data, suggesting that the lack of atomistic details in the coarse‐grained GNM is more than compensated for by the mathematically exact evaluation of fluctuations using the native contacts topology. Evidence is provided that particular loop motions are curtailed by intermolecular contacts in the crystal environment causing a discrepancy between theory and experiments. Interestingly, the information conveyed by X‐ray crystallography becomes more consistent with NMR models and computational predictions when ensembles of X‐ray models are considered. Less precise (broadly distributed) ensembles indeed appear to describe the accessible conformational space under native state conditions better than B‐factors. Our results highlight the importance of using multiple conformations obtained by alternative experimental methods, and analyzing results from both coarse‐grained models and atomic simulations, for accurate assessment of motions accessible to proteins under native state conditions. Proteins 2009. © 2009 Wiley‐Liss, Inc.     

Sampling of the native conformational ensemble of myoglobin via structures in different crystalline environments

Proteins sample multiple conformational substates in their native environment, but the process of crystallization selects the conformers that allow for close packing. The population of conformers can be shifted by varying the environment through a range of crystallization conditions, often resulting in different space groups and changes in the packing arrangements. Three high resolution structures of myoglobin (Mb) in different crystal space groups are presented, including one in a new space group P6(1)22 and two structures in space groups P2(1)2(1)2(1) and P6. We compare coordinates and anisotropic displacement parameters (ADPs) from these three structures plus an existing structure in space group P2(1). While the overall changes are small, there is substantial variation in several external regions with varying patterns of crystal contacts across the space group packing arrangements. The structural ensemble containing four different crystal forms displays greater conformational variance (Calpha rmsd of 0.54-0.79 A) in comparison to a collection of four Mb structures with different ligands and mutations in the same crystal form (Calpha rmsd values of 0.28-0.37 A). The high resolution of the data enables comparison of both the magnitudes and directions of ADPs, which are found to be suppressed by crystal contacts. A composite dynamic profile of Mb structural variation from the four structures was compared with an independent structural ensemble developed from NMR refinement. Despite the limitations and biases of each method, the ADPs of the crystallographic ensemble closely match the positional variance from the solution NMR ensemble with linear correlation of 0.8. This suggests that crystal packing selects conformers representative of the solution ensemble, and several different crystal forms give a more complete view of the plasticity of a protein structure.