Reorientational contact-weighted elastic network model for the prediction of protein dynamics: comparison with NMR relaxation

Prediction of order parameters based on protein NMR structure ensemble and machine learning

The fast motions of proteins at the picosecond to nanosecond timescale, known as fast dynamics, are closely related to protein conformational entropy and rearrangement, which in turn affect catalysis, ligand binding and protein allosteric effects. The most used NMR approach to study fast protein dynamics is the model free method, which uses order parameter S2 to describe the amplitude of the internal motion of local group. However, to obtain order parameter through NMR experiments is quite complex and lengthy. In this paper, we present a machine learning approach for predicting backbone 1H-15N order parameters based on protein NMR structure ensemble. A random forest model is used to learn the relationship between order parameters and structural features. Our method achieves high accuracy in predicting backbone 1H-15N order parameters for a test dataset of 10 proteins, with a Pearson correlation coefficient of 0.817 and a root-mean-square error of 0.131.

Patterns in Protein Flexibility: A Comparison of NMR "Ensembles", MD Trajectories, and Crystallographic B-Factors

Proteins are molecular machines requiring flexibility to function. Crystallographic B-factors and Molecular Dynamics (MD) simulations both provide insights into protein flexibility on an atomic scale. Nuclear Magnetic Resonance (NMR) lacks a universally accepted analog of the B-factor. However, a lack of convergence in atomic coordinates in an NMR-based structure calculation also suggests atomic mobility. This paper describes a pattern in the coordinate uncertainties of backbone heavy atoms in NMR-derived structural “ensembles” first noted in the development of FindCore2 (previously called Expanded FindCore: DA Snyder, J Grullon, YJ Huang, R Tejero, GT Montelione, Proteins: Structure, Function, and Bioinformatics 82 (S2), 219–230) and demonstrates that this pattern exists in coordinate variances across MD trajectories but not in crystallographic B-factors. This either suggests that MD trajectories and NMR “ensembles” capture motional behavior of peptide bond units not captured by B-factors or indicates a deficiency common to force fields used in both NMR and MD calculations.

Identification of key sites controlling protein functional motions by using elastic network model combined with internal coordinates

The elastic network model (ENM) is an effective method to extract the intrinsic dynamical properties encoded in protein tertiary structures. We have proposed a new ENM-based analysis method to reveal the motion modes directly responsible for a specific protein function, in which an internal coordinate related to the specific function was introduced to construct the internal/Cartesian hybrid coordinate space. In the present work, the function-related internal coordinates combined with a linear perturbation method were applied to identify the key sites controlling specific protein functional motions. The change in the fluctuations of the internal coordinate in response to residue perturbation was calculated in the hybrid coordinate space by using the linear response theory. The residues with the large fluctuation changes were identified to be the key sites that allosterically control the specific protein function. Two proteins, i.e., human DNA polymerase β and the chaperonin from Methanococcus maripaludis, were investigated as case studies, in which several collective and local internal coordinates were applied to identify the functionally key residues of these two studied proteins. The calculation results are consistent with the experimental observations. It is found that different collective internal coordinates lead to similar results, where the predicted functionally key sites are located at similar positions in the protein structure. While for the local internal coordinates, the predicted key sites tend to be situated at the region near to the coordinate-involving residues. Our studies provide a starting point for further exploring other function-related internal coordinates for other interesting proteins.

Specific non-local interactions are not necessary for recovering native protein dynamics

The elastic network model (ENM) is a widely used method to study native protein dynamics by normal mode analysis (NMA). In ENM we need information about all pairwise distances, and the distance between contacting atoms is restrained to the native value. Therefore ENM requires O(N²) information to realize its dynamics for a protein consisting of N amino acid residues. To see if (or to what extent) such a large amount of specific structural information is required to realize native protein dynamics, here we introduce a novel model based on only O(N) restraints. This model, named the ‘contact number diffusion’ model (CND), includes specific distance restraints for only local (along the amino acid sequence) atom pairs, and semi-specific non-local restraints imposed on each atom, rather than atom pairs. The semi-specific non-local restraints are defined in terms of the non-local contact numbers of atoms. The CND model exhibits the dynamic characteristics comparable to ENM and more correlated with the explicit-solvent molecular dynamics simulation than ENM. Moreover, unrealistic surface fluctuations often observed in ENM were suppressed in CND. On the other hand, in some ligand-bound structures CND showed larger fluctuations of buried protein atoms interacting with the ligand compared to ENM. In addition, fluctuations from CND and ENM show comparable correlations with the experimental B-factor. Although there are some indications of the importance of some specific non-local interactions, the semi-specific non-local interactions are mostly sufficient for reproducing the native protein dynamics.

Predicting the Flexibility Profile of Ribosomal RNAs

Flexibility in biomolecules is an important determinant of biological functionality, which can be measured quantitatively by atomic Debye-Waller factor or B-factor. Although numerous works have been addressed on theoretical and computational studies of the B-factor profiles of proteins, the methods used for predicting B-factor values of nucleic acids, especially the complicated ribosomal RNAs (rRNAs), which are very functionally similar to proteins in providing matrix structures and in catalyzing biochemical reactions, still remain unexploited. In this article, we present a quantitative structure-flexibility relationship (QSFR) study with the aim at the quantitative prediction of rRNA B-factor based on primary sequences (sequence-based) and advanced structures (structure-based) by using both linear and nonlinear machine learning approaches, including partial least squares regression (PLS), least squares support vector machine (LSSVM), and Gaussian process (GP). By rigorously examining the performance and reliability of constructed statistical models and by comparing our models in detail to those developed previously for protein B-factors, we demonstrate that (i) rRNA B-factors could be predicted at a similar level of accuracy with that of protein, (ii) a structure-based approach performed much better as compared to sequence-based methods in modeling of rRNA B-factors, and (iii) rRNA flexibility is primarily governed by the local features of nonbonding potential landscapes, such as electrostatic and van der Waals forces.

Generalized spring tensor models for protein fluctuation dynamics and conformation changes

BACKGROUND

In the last decade, various coarse-grained elastic network models have been developed to study the large-scale motions of proteins and protein complexes where computer simulations using detailed all-atom models are not feasible. Among these models, the Gaussian Network Model (GNM) and Anisotropic Network Model (ANM) have been widely used. Both models have strengths and limitations. GNM can predict the relative magnitudes of protein fluctuations well, but due to its isotropy assumption, it can not be applied to predict the directions of the fluctuations. In contrast, ANM adds the ability to do the latter, but loses a significant amount of precision in the prediction of the magnitudes.

RESULTS

In this article, we develop a single model, called generalized spring tensor model (STeM), that is able to predict well both the magnitudes and the directions of the fluctuations. Specifically, STeM performs equally well in B-factor predictions as GNM and has the ability to predict the directions of fluctuations as ANM. This is achieved by employing a physically more realistic potential, the Go-like potential. The potential, which is more sophisticated than that of either GNM or ANM, though adds complexity to the derivation process of the Hessian matrix (which fortunately has been done once for all and the MATLAB code is freely available electronically at http://www.cs.iastate.edu/~gsong/STeM), causes virtually no performance slowdown.

CONCLUSIONS

Derived from a physically more realistic potential, STeM proves to be a natural solution in which advantages that used to exist in two separate models, namely GNM and ANM, are achieved in one single model. It thus lightens the burden to work with two separate models and to relate the modes of GNM with those of ANM at times. By examining the contributions of different interaction terms in the Gō potential to the fluctuation dynamics, STeM reveals, (i) a physical explanation for why the distance-dependent, inverse distance square (i.e., 1/(r)2) spring constants perform better than the uniform ones, and (ii), the importance of three-body and four-body interactions to properly modeling protein dynamics.

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.

Application of Multiplexed Replica Exchange Molecular Dynamics to the UNRES Force Field: Tests with alpha and alpha+beta Proteins

The replica exchange (RE) method is increasingly used to improve sampling in molecular dynamics (MD) simulations of biomolecular systems. Recently, we implemented the united-residue UNRES force field for mesoscopic MD. Initial results from UNRES MD simulations show that we are able to simulate folding events that take place in a microsecond or even a millisecond time scale. To speed up the search further, we applied the multiplexing replica exchange molecular dynamics (MREMD) method. The multiplexed variant (MREMD) of the RE method, developed by Rhee and Pande, differs from the original RE method in that several trajectories are run at a given temperature. Each set of trajectories run at a different temperature constitutes a layer. Exchanges are attempted not only within a single layer but also between layers. The code has been parallelized and scales up to 4000 processors. We present a comparison of canonical MD, REMD, and MREMD simulations of protein folding with the UNRES force-field. We demonstrate that the multiplexed procedure increases the power of replica exchange MD considerably and convergence of the thermodynamic quantities is achieved much faster.

Slow motions in chicken villin headpiece subdomain probed by cross-correlated NMR relaxation of amide NH bonds in successive residues

The villin headpiece subdomain (HP36) is a widely used system for protein-folding studies. Nuclear magnetic resonance cross-correlated relaxation rates arising from correlated fluctuations of two N-H(N) dipole-dipole interactions involving successive residues were measured at two temperatures at which HP36 is at least 99% folded. The experiment revealed the presence of motions slower than overall tumbling of the molecule. Based on the theoretical analysis of the spectral densities we show that the structural and dynamic contributions to the experimental cross-correlated relaxation rate can be separated under certain conditions. As a result, dynamic cross-correlated order parameters describing slow microsecond-to-millisecond motions of N-H bonds in neighboring residues can be introduced for any extent of correlations in the fluctuations of the two bond vectors. These dynamic cross-correlated order parameters have been extracted for HP36. The comparison of their values at two different temperatures indicates that when the temperature is raised, slow motions increase in amplitude. The increased amplitude of these fluctuations may reflect the presence of processes directly preceding the unfolding of the protein.

Evolutionary conservation of protein vibrational dynamics

The aim of the present work is to study the evolutionary divergence of vibrational protein dynamics. To this end, we used the Gaussian Network Model to perform a systematic analysis of normal mode conservation on a large dataset of proteins classified into homologous sets of family pairs and superfamily pairs. We found that the lowest most collective normal modes are the most conserved ones. More precisely, there is, on average, a linear correlation between normal mode conservation and mode collectivity. These results imply that the previously observed conservation of backbone flexibility (B-factor) profiles is due to the conservation of the most collective modes, which contribute the most to such profiles. We discuss the possible roles of normal mode robustness and natural selection in the determination of the observed behavior. Finally, we draw some practical implications for dynamics-based protein alignment and classification and discuss possible caveats of the present approach.

Protein conformational flexibility prediction using machine learning

Using a data set of 16 proteins, a neural network has been trained to predict backbone 15N generalized order parameters from the three-dimensional structures of proteins. The final network parameterization contains six input features. The average prediction accuracy, as measured by the Pearson's correlation coefficient between experimental and predicted values of the square of the generalized order parameter is >0.70. Predicted order parameters for non-terminal amino acid residues depends most strongly on the local packing density and the probability that the residue is located in regular secondary structure.

Coarse-grained models reveal functional dynamics--I. Elastic network models--theories, comparisons and perspectives

In this review, we summarize the progress on coarse-grained elastic network models (CG-ENMs) in the past decade. Theories were formulated to allow study of conformational dynamics in time/space frames of biological interest. Several highlighted models and their underlined hypotheses are introduced in physical depth. Important ENM offshoots, motivated to reproduce experimental data as well as to address the slow-mode-encoded configurational transitions, are also introduced. With the theoretical developments, computational cost is significantly reduced due to simplified potentials and coarse-grained schemes. Accumulating wealth of data suggest that ENMs agree equally well with experiment in describing equilibrium dynamics despite their distinct potentials and levels of coarse-graining. They however do differ in the slowest motional components that are essential to address large conformational changes of functional significance. The difference stems from the dissimilar curvatures of the harmonic energy wells described for each model. We also provide our views on the predictability of 'open to close' (open-->close) transitions of biomolecules on the basis of conformational selection theory. Lastly, we address the limitations of the ENM formalism which are partially alleviated by the complementary CG-MD approach, to be introduced in the second paper of this two-part series.

Coarse-grained biomolecular simulation with REACH: realistic extension algorithm via covariance Hessian

Coarse-graining of protein interactions provides a means of simulating large biological systems. Here, a coarse-graining method, REACH, is introduced, in which the force constants of a residue-scale elastic network model are calculated from the variance-covariance matrix obtained from atomistic molecular dynamics (MD) simulation. In test calculations, the C(alpha)-atoms variance-covariance matrices are calculated from the ensembles of 1-ns atomistic MD trajectories in monomeric and dimeric myoglobin, and used to derive coarse-grained force constants for the local and nonbonded interactions. Construction of analytical model functions of the distance-dependence of the interresidue force constants allows rapid calculation of the REACH normal modes. The model force constants from monomeric and dimeric myoglobin are found to be similar in magnitude to each other. The MD intra- and intermolecular mean-square fluctuations and the vibrational density of states are well reproduced by the residue-scale REACH normal modes without requiring rescaling of the force constant parameters. The temperature-dependence of the myoglobin REACH force constants reveals that the dynamical transition in protein internal fluctuations arises principally from softening of the elasticity in the nonlocal interactions. The REACH method is found to be a reliable way of determining spatiotemporal protein motion without the need for expensive computations of long atomistic MD simulations.

An overview of recent developments in the interpretation and prediction of fast internal protein dynamics

During the past decades, NMR spectroscopy has emerged as a unique tool for the study of protein dynamics. Indeed, relaxation studies on isotopically labeled proteins can provide information on the overall motions as well as the internal fast, sub-nanosecond, dynamics. Therefore, the interpretation and the prediction of spin relaxation rates in proteins are important issues that have motivated numerous theoretical and methodological developments, including the description of overall dynamics and its possible coupling to internal mobility, the introduction of models of internal dynamics, the determination of correlation functions from experimental data, and the relationship between relaxation and thermodynamical quantities. A brief account of recent developments that have proven useful in this domain are presented.

Insights into equilibrium dynamics of proteins from comparison of NMR and X-ray data with computational predictions

For a representative set of 64 nonhomologous proteins, each containing a structure solved by NMR and X-ray crystallography, we analyzed the variations in atomic coordinates between NMR models, the temperature (B) factors measured by X-ray crystallography, and the fluctuation dynamics predicted by the Gaussian network model (GNM). The NMR and X-ray data exhibited a correlation of 0.49. The GNM results, on the other hand, yielded a correlation of 0.59 with X-ray data and a distinctively better correlation (0.75) with NMR data. The higher correlation between GNM and NMR data, compared to that between GNM and X-ray B factors, is shown to arise from the differences in the spectrum of modes accessible in solution and in the crystal environment. Mainly, large-amplitude motions sampled in solution are restricted, if not inaccessible, in the crystalline environment of X-rays. Combined GNM and NMR analysis emerges as a useful tool for assessing protein dynamics.

Networks of coupled rotators: relationship between structures and internal dynamics in metal-binding proteins. Applications to apo- and holo-calbindin

This article presents an analysis of the internal dynamics of the Ca2+-binding protein calbindin, based on the Networks of Coupled Rotators (NCRs) introduced recently. Several fundamental and practical issues raised by this approach are investigated. The roles of various parameters of the model are examined. The NCR model is shown to account for the modifications of the internal dynamics upon Ca2+ binding by calbindin. Two alternative strategies to estimate local internal effective correlation times of the protein are proposed, which offer good agreement between predictions and experiment.