Deriving protein dynamical properties from weighted protein contact number

It has recently been shown that in proteins the atomic mean-square displacement (or B-factor) can be related to the number of the neighboring atoms (or protein contact number), and that this relationship allows one to compute the B-factor profiles directly from protein contact number. This method, referred to as the protein contact model, is appealing, since it requires neither trajectory integration nor matrix diagonalization. As a result, the protein contact model can be applied to very large proteins and can be implemented as a high-throughput computational tool to compute atomic fluctuations in proteins. Here, we show that this relationship can be further refined to that between the atomic mean-square displacement and the weighted protein contact-number, the weight being the square of the reciprocal distance between the contacting pair. In addition, we show that this relationship can be utilized to compute the cross-correlation of atomic motion (the B-factor is essentially the auto-correlation of atomic motion). For a nonhomologous dataset comprising 972 high-resolution X-ray protein structures (resolution <2.0 A and sequence identity <25%), the mean correlation coefficient between the X-ray and computed B-factors based on the weighted protein contact-number model is 0.61, which is better than those of the original contact-number model (0.51) and other methods. We also show that the computed correlation maps based on the weighted contact-number model are globally similar to those computed through normal model analysis for some selected cases. Our results underscore the relationship between protein dynamics and protein packing. We believe that our method will be useful in the study of the protein structure-dynamics relationship.     

vGNM: a better model for understanding the dynamics of proteins in crystals

The dynamics of proteins are important for understanding their functions. In recent years, the simple coarse-grained Gaussian Network Model (GNM) has been fairly successful in interpreting crystallographic B-factors. However, the model clearly ignores the contribution of the rigid body motions and the effect of crystal packing. The model cannot explain the fact that the same protein may have significantly different B-factors under different crystal packing conditions. In this work, we propose a new GNM, called vGNM, which takes into account both the contribution of the rigid body motions and the effect of crystal packing, by allowing the amplitude of the internal modes to be variables. It hypothesizes that the effect of crystal packing should cause some modes to be amplified and others to become less important. In doing so, vGNM is able to resolve the apparent discrepancy in experimental B-factors among structures of the same protein but with different crystal packing conditions, which GNM cannot explain. With a small number of parameters, vGNM is able to reproduce experimental B-factors for a large set of proteins with significantly better correlations (having a mean value of 0.81 as compared to 0.59 by GNM). The results of applying vGNM also show that the rigid body motions account for nearly 60% of the total fluctuations, in good agreement with previous findings.     

On the relationship between the protein structure and protein dynamics

Recently, we have developed a method (Shih et al., Proteins: Structure, Function, and Bioinformatics 2007;68: 34-38) to compute correlation of fluctuations of proteins. This method, referred to as the protein fixed-point (PFP) model, is based on the positional vectors of atoms issuing from the fixed point, which is the point of the least fluctuations in proteins. One corollary from this model is that atoms lying on the same shell centered at the fixed point will have the same thermal fluctuations. In practice, this model provides a convenient way to compute the average dynamical properties of proteins directly from the geometrical shapes of proteins without the need of any mechanical models, and hence no trajectory integration or sophisticated matrix operations are needed. As a result, it is more efficient than molecular dynamics simulation or normal mode analysis. Though in the previous study the PFP model has been successfully applied to a number of proteins of various folds, it is not clear to what extent this model will be applied. In this article, we have carried out the comprehensive analysis of the PFP model for a dataset comprising 972 high-resolution X-ray structures with pairwise sequence identity or=0.5. Our result shows that the fixed-point model is indeed quite general and will be a useful tool for high throughput analysis of dynamical properties of proteins.     

All-Atom Contact Model for Understanding Protein Dynamics from Crystallographic B-Factors

An all-atom local contact model is described that can be used to predict protein motions underlying isotropic crystallographic B-factors. It uses a mean-field approximation to represent the motion of an atom in a harmonic potential generated by the surrounding atoms resting at their equilibrium positions. Based on a 400-ns molecular dynamics simulation of ubiquitin in explicit water, it is found that each surrounding atom stiffens the spring constant by a term that on average scales exponentially with the interatomic distance. This model combines features of the local density model by Halle and the local contact model by Zhang and Brüschweiler. When applied to a nonredundant set of 98 ultra-high resolution protein structures, an average correlation coefficient of 0.75 is obtained for all atoms. The systematic inclusion of crystal contact contributions and fraying effects is found to enhance the performance substantially. Because the computational cost of the local contact model scales linearly with the number of protein atoms, it is applicable to proteins of any size for the prediction of B-factors of both backbone and side-chain atoms. The model performs as well as or better than several other models tested, such as rigid-body motional models, the local density model, and various forms of the elastic network model. It is concluded that at the currently achievable level of accuracy, collective intramolecular motions are not essential for the interpretation of B-factors.     

Structural compliance: A new metric for protein flexibility

Proteins are the active players in performing essential molecular activities throughout biology, and their dynamics has been broadly demonstrated to relate to their mechanisms. The intrinsic fluctuations have often been used to represent their dynamics and then compared to the experimental B-factors. However, proteins do not move in a vacuum and their motions are modulated by solvent that can impose forces on the structure. In this paper, we introduce a new structural concept, which has been called the structural compliance, for the evaluation of the global and local deformability of the protein structure in response to intramolecular and solvent forces. Based on the application of pairwise pulling forces to a protein elastic network, this structural quantity has been computed and sometimes is even found to yield an improved correlation with the experimental B-factors, meaning that it may serve as a better metric for protein flexibility. The inverse of structural compliance, namely the structural stiffness, has also been defined, which shows a clear anticorrelation with the experimental data. Although the present applications are made to proteins, this approach can also be applied to other biomolecular structures such as RNA. This present study considers only elastic network models, but the approach could be applied further to conventional atomic molecular dynamics. Compliance is found to have a slightly better agreement with the experimental B-factors, perhaps reflecting its bias toward the effects of local perturbations, in contrast to mean square fluctuations. The code for calculating protein compliance and stiffness is freely accessible at https://jerniganlab.github.io/Software/PACKMAN/Tutorials/compliance .     

Critical evaluation of simple network models of protein dynamics and their comparison with crystallographic B-factors

In recent years, elastic network models (ENM) have been widely used to describe low-frequency collective motions in proteins. These models are often validated and calibrated by fitting mean-square atomic displacements estimated from x-ray crystallography (B-factors). We show that a proper calibration procedure must account for the rigid-body motion and constraints imposed by the crystalline environment on the protein. These fundamental aspects of protein dynamics in crystals are often ignored in currently used ENMs, leading to potentially erroneous network parameters. Here we develop an ENM that properly takes the rigid-body motion and crystalline constraints into account. Its application to the crystallographic B-factors reveals that they are dominated by rigid-body motion and thus are poorly suited for the calibration of models for internal protein dynamics. Furthermore, the translation libration screw (TLS) model that treats proteins as rigid bodies is considerably more successful in interpreting the experimental B-factors than ENMs. This conclusion is reached on the basis of a comparative study of various models of protein dynamics. To evaluate their performance, we used a data set of 330 protein structures that combined the sets previously used in the literature to test and validate different models. We further propose an extended TLS model that treats the bulk of the protein as a rigid body while allowing for flexibility of chain ends. This model outperforms other simple models of protein dynamics in interpreting the crystallographic B-factors.     

Improved amino acid flexibility parameters

Protein molecules exhibit varying degrees of flexibility throughout their three-dimensional structures, with some segments showing little mobility while others may be so disordered as to be unresolvable by techniques such as X-ray crystallography. Atomic displacement parameters, or B-factors, from X-ray crystallographic studies give an experimentally determined indication of the degree of mobility in a protein structure. To provide better estimators of amino acid flexibility, we have examined B-factors from a large set of high-resolution crystal structures. Because of the differences among structures, it is necessary to normalize the B-factors. However, many proteins have segments of unusually high mobility, which must be accounted for before normalization can be performed. Accordingly, a median-based method from quality control studies was used to identify outliers. After removal of outliers from, and normalization of, each protein chain, the B-factors were collected for each amino acid in the set. It was found that the distribution of normalized B-factors followed a Gumbel, or extreme value distribution, and the location parameter, or mode, of this distribution was used as an estimator of flexibility for the amino acid. These new parameters have a higher correlation with experimentally determined B-factors than parameters from earlier methods.     

Molecular crowding inhibits intramolecular breathing motions in proteins

In aqueous solution some proteins undergo large-scale movements of secondary structures, subunits or domains, referred to as protein “breathing”, that define a native-state ensemble of structures. These fluctuations are sensitive to the nature and concentration of solutes and other proteins and are thereby expected to be different in the crowded interior of a cell than in dilute solution. Here we use a combination of wide angle X-ray scattering (WAXS) and computational modeling to derive a quantitative measure of the spatial scale of conformational fluctuations in a protein solution. Concentration-dependent changes in the observed scattering intensities are consistent with a model of structural fluctuations in which secondary structures undergo rigid-body motions relative to one another. This motion increases with decreasing protein concentration or increasing temperature. Analysis of a set of five structurally and functionally diverse proteins reveals a diversity of kinetic behaviors. Proteins with multiple disulfide bonds exhibit little or no increase in breathing in dilute solutions. The spatial extent of structural fluctuations appears highly dependent on both protein structure and concentration and is universally suppressed at very high protein concentrations.     

Dynamics may significantly influence the estimation of interatomic distances in biomolecular X-ray structures

Atomic positions obtained by X-ray crystallography are time and space averages over many molecules in the crystal. Importantly, interatomic distances, calculated between such average positions and frequently used in structural and mechanistic analyses, can be substantially different from the more appropriate time-average and ensemble-average interatomic distances. Using crystallographic B-factors, one can deduce corrections, which have so far been applied exclusively to small molecules, to obtain correct average distances as a function of the type of atomic motion. Here, using 4774 high-quality protein X-ray structures, we study the significance of such corrections for different types of atomic motion. Importantly, we show that for distances shorter than 5 Å, corrections greater than 0.5 Å may apply, especially for noncorrelated or anticorrelated motion. For example, 14% of the studied structures have at least one pair of atoms with a correction of ≥ 0.5 Å in the case of noncorrelated motion. Using molecular dynamics simulations of villin headpiece, ubiquitin, and SH3 domain unit cells, we demonstrate that the majority of average interatomic distances in these proteins agree with noncorrelated corrections, suggesting that such deviations may be truly relevant. Importantly, we demonstrate that the corrections do not significantly affect stereochemistry and the overall quality of final refined X-ray structures, but can provide marked improvements in starting unrefined models obtained from low-resolution X-ray data. Finally, we illustrate the potential mechanistic and biological significance of the calculated corrections for KcsA ion channel and show that they provide indirect evidence that motions in its selectivity filter are highly correlated.     

Concordance of residual dipolar couplings, backbone order parameters and crystallographic B-factors for a small alpha/beta protein: a unified picture of high probability, fast atomic motions in proteins

Using ensemble refinement of the third immunoglobulin binding domain (GB3) of streptococcal protein G (a small alpha/beta protein of 56 residues), we demonstrate that backbone (N-H, N-C', Calpha-Halpha, Calpha-C') residual dipolar coupling data in five independent alignment media, generalized order parameters from 15N relaxation data, and B-factors from a high-resolution (1.1A), room temperature crystal structure are entirely consistent with one another within experimental error. The optimal ensemble size representation is between four and eight, as assessed by complete cross-validation of the residual dipolar couplings. Thus, in the case of GB3, all three observables reflect the same low-amplitude anisotropic motions arising from fluctuations in backbone phi/psi torsion angles in the picosecond to nanosecond regime in both solution and crystalline environments, yielding a unified picture of fast, high-probability atomic motions in proteins. An understanding of these motions is crucial for understanding the impact of protein dynamics on protein function, since they provide part of the driving force for triggered conformational changes that occur, for example, upon ligand binding, signal transduction and enzyme catalysis.     

Multiscale modeling of macromolecular conformational changes combining concepts from rigidity and elastic network theory

The development of a two-step approach for multiscale modeling of macromolecular conformational changes is based on recent developments in rigidity and elastic network theory. In the first step, static properties of the macromolecule are determined by decomposing the molecule into rigid clusters by using the graph-theoretical approach FIRST and an all-atom representation of the protein. In this way, rigid clusters are not limited to consist of residues adjacent in sequence or secondary structure elements as in previous studies. Furthermore, flexible links between rigid clusters are identified and can be modeled as such subsequently. In the second step, dynamical properties of the molecule are revealed by the rotations-translations of blocks approach (RTB) using an elastic network model representation of the coarse-grained protein. In this step, only rigid body motions are allowed for rigid clusters, whereas links between them are treated as fully flexible. The approach was tested on a data set of 10 proteins that showed conformational changes on ligand binding. For efficiency, coarse-graining the protein results in a remarkable reduction of memory requirements and computational times by factors of 9 and 27 on average and up to 25 and 125, respectively. For accuracy, directions and magnitudes of motions predicted by our approach agree well with experimentally determined ones, despite embracing in extreme cases >50% of the protein into one rigid cluster. In fact, the results of our method are in general comparable with when no or a uniform coarse-graining is applied; and the results are superior if the movement is dominated by loop or fragment motions. This finding indicates that explicitly distinguishing between flexible and rigid regions is advantageous when using a simplified protein representation in the second step. Finally, motions of atoms in rigid clusters are also well predicted by our approach, which points to the need to consider mobile protein regions in addition to flexible ones when modeling correlated motions.     

Motions and structural variability within toxins: implication for their use as scaffolds for protein engineering

Animal toxins are small proteins built on the basis of a few disulfide bonded frameworks. Because of their high variability in sequence and biologic function, these proteins are now used as templates for protein engineering. Here we report the extensive characterization of the structure and dynamics of two toxin folds, the "three-finger" fold and the short alpha/beta scorpion fold found in snake and scorpion venoms, respectively. These two folds have a very different architecture; the short alpha/beta scorpion fold is highly compact, whereas the "three-finger" fold is a beta structure presenting large flexible loops. First, the crystal structure of the snake toxin alpha was solved at 1.8-A resolution. Then, long molecular dynamics simulations (10 ns) in water boxes of the snake toxin alpha and the scorpion charybdotoxin were performed, starting either from the crystal or the solution structure. For both proteins, the crystal structure is stabilized by more hydrogen bonds than the solution structure, and the trajectory starting from the X-ray structure is more stable than the trajectory started from the NMR structure. The trajectories started from the X-ray structure are in agreement with the experimental NMR and X-ray data about the protein dynamics. Both proteins exhibit fast motions with an amplitude correlated to their secondary structure. In contrast, slower motions are essentially only observed in toxin alpha. The regions submitted to rare motions during the simulations are those that exhibit millisecond time-scale motions. Lastly, the structural variations within each fold family are described. The localization and the amplitude of these variations suggest that the regions presenting large-scale motions should be those tolerant to large insertions or deletions.     

A simple model of backbone flexibility improves modeling of side-chain conformational variability

The considerable flexibility of side-chains in folded proteins is important for protein stability and function, and may have a role in mediating allosteric interactions. While sampling side-chain degrees of freedom has been an integral part of several successful computational protein design methods, the predictions of these approaches have not been directly compared to experimental measurements of side-chain motional amplitudes. In addition, protein design methods frequently keep the backbone fixed, an approximation that may substantially limit the ability to accurately model side-chain flexibility. Here, we describe a Monte Carlo approach to modeling side-chain conformational variability and validate our method against a large dataset of methyl relaxation order parameters derived from nuclear magnetic resonance (NMR) experiments (17 proteins and a total of 530 data points). We also evaluate a model of backbone flexibility based on Backrub motions, a type of conformational change frequently observed in ultra-high-resolution X-ray structures that accounts for correlated side-chain backbone movements. The fixed-backbone model performs reasonably well with an overall rmsd between computed and predicted side-chain order parameters of 0.26. Notably, including backbone flexibility leads to significant improvements in modeling side-chain order parameters for ten of the 17 proteins in the set. Greater accuracy of the flexible backbone model results from both increases and decreases in side-chain flexibility relative to the fixed-backbone model. This simple flexible-backbone model should be useful for a variety of protein design applications, including improved modeling of protein-protein interactions, design of proteins with desired flexibility or rigidity, and prediction of correlated motions within proteins.     

A simple way to compute protein dynamics without a mechanical model

We found that in proteins the average atomic fluctuation is linearly related to the square of the atomic distance from the center of mass of the protein. Using this simple relation, we can accurately compute the temperature factors of proteins of a wide range of sizes and folds, and the correlation of the fluctuations in proteins. This simple relation provides a direct link between protein dynamics and the static protein's geometrical shape and offers a simple way to compute protein dynamics without either long time trajectory integration or any matrix operations.     

Conformational dynamics of supramolecular protein assemblies

Supramolecular protein assemblies including molecular motors, cytoskeletal filaments, chaperones, and ribosomes play a central role in a broad array of cellular functions ranging from cell division and motility to RNA and protein synthesis and folding. Single-particle reconstructions of such assemblies have been growing rapidly in recent years, providing increasingly high resolution structural information under native conditions. While the static structure of these assemblies provides essential insight into their mechanism of biological function, their dynamical motions provide additional important information that cannot be inferred from structure alone. Here we present an unsupervised computational framework for the analysis of high molecular weight protein assemblies and use it to analyze the conformational dynamics of structures deposited in the Electron Microscopy Data Bank. Protein assemblies are modeled using a recently introduced coarse-grained modeling framework based on the finite element method, which is used to compute equilibrium thermal fluctuations, elastic strain energy distributions associated with specific conformational transitions, and dynamical correlations in distant molecular domains. Results are presented in detail for the ribosome-bound termination factor RF2 from Escherichia coli, the nuclear pore complex from Dictyostelium discoideum, and the chaperonin GroEL from E. coli. Elastic strain energy distributions reveal hinge-regions associated with specific conformational change pathways, and correlations in collective molecular motions reveal dynamical coupling between distant molecular domains that suggest new, as well as confirm existing, allosteric mechanisms. Results are publically available for use in further investigation and interpretation of biological function including cooperative transitions, allosteric communication, and molecular mechanics, as well as in further classification and refinement of electron microscopy based structures.     

Why protein R-factors are so large: a self-consistent analysis

The R-factor and R-free are commonly used to measure the quality of protein models obtained in X-ray crystallography. Well-refined protein structures usually have R-factors in the range of 20-25%, whereas intrinsic errors in the experimental data are usually around 5%. We use molecular dynamics simulations to perform a self-consistent analysis by which we determine the major factors contributing to large values of protein R-factors. The analysis shows that significant R-factor values can arise from the use of isotropic B-factors to model anisotropic protein motions and from coordinate errors. Even in the absence of coordinate errors, the use of isotropic B-factors can cause the R-factors to be around 10%; for coordinate errors smaller than 0.2 A, the two errors types make similar contributions. The inaccuracy of the energy function used and multistate protein dynamics are unlikely to make significant contributions to the large R-factors.     

Comparison of Full-atomic and Coarse-grained Models to Examine the Molecular Fluctuations of c-AMP Dependent Protein Kinase

Abstract

Molecular fluctuations of the native conformation of c-AMP dependent protein kinase (cAPK) have been investigated with three different approaches. The first approach is the full atomic normal mode analysis (NMA) with empirical force fields. The second and third approaches are based on a coarse-grained model with a single single-parameter- harmonic potential between close residues in the crystal structure of the molecule without any residue specificity. The second method calculates only the magnitude of fluctuations whereas the third method is developed to find the directionality of the fluctuations which are essential to understand the functional importance of biological molecules. The aim, in this study, is to determine whether using such coarse-grained models are appropriate for elucidating the global dynamic characteristics of large proteins which reduces the size of the system at least by a factor of ten. The mean-square fluctuations of C^α atoms and the residue cross-correlations are obtained by three approaches. These results are then compared to test the results of coarse grained models on the overall collective motions. AH three of the approaches show that highly flexible regions correspond to the activation and solvent exposed loops, whereas the conserved residues (especially in substrate binding regions) exhibit almost no flexibility, adding stability to the structure. The anti-correlated motions of the two lobes of the catalytic core provide flexibility to the molecule. High similarities among the results of these methods indicate that the slowest modes governing the most global motions are preserved in the coarse grained models for proteins. This finding may suggest that the general shapes of the structures are representative of their dynamic characteristics and the dominant motions of protein structures are robust at coarse-grained levels.     

Relation between sequence and structure of HIV-1 protease inhibitor complexes: a model system for the analysis of protein flexibility.

The flexibility of different regions of HIV-1 protease was examined by using a database consisting of 73 X-ray structures that differ in terms of sequence, ligands or both. The root-mean-square differences of the backbone for the set of structures were shown to have the same variation with residue number as those obtained from molecular dynamics simulations, normal mode analyses and X-ray B-factors. This supports the idea that observed structural changes provide a measure of the inherent flexibility of the protein, although specific interactions between the protease and the ligand play a secondary role. The results suggest that the potential energy surface of the HIV-1 protease is characterized by many local minima with small energetic differences, some of which are sampled by the different X-ray structures of the HIV-1 protease complexes. Interdomain correlated motions were calculated from the structural fluctuations and the results were also in agreement with molecular dynamics simulations and normal mode analyses. Implications of the results for the drug-resistance engendered by mutations are discussed briefly.