Conventional NMA as a better standard for evaluating elastic network models

The amounts of thermal vibrations and static disorder in protein X-ray crystallographic B-factors

Crystallographic B-factors provide direct dynamical information on the internal mobility of proteins that is closely linked to function, and are also widely used as a benchmark in assessing elastic network models. A significant question in the field is: what is the exact amount of thermal vibrations in protein crystallographic B-factors? This work sets out to answer this question. First, we carry out a thorough, statistically sound analysis of crystallographic B-factors of over 10 000 structures. Second, by employing a highly accurate all-atom model based on the well-known CHARMM force field, we obtain computationally the magnitudes of thermal vibrations of nearly 1000 structures. Our key findings are: (i) the magnitude of thermal vibrations, surprisingly, is nearly protein-independent, as a corollary to the universality for the vibrational spectra of globular proteins established earlier; (ii) the magnitude of thermal vibrations is small, less than 0.1 Å² at 100 K; (iii) the percentage of thermal vibrations in B-factors is the lowest at low resolution and low temperature (<10%) but increases to as high as 60% for structures determined at high resolution and at room temperature. The significance of this work is that it provides for the first time, using an extremely large dataset, a thorough analysis of B-factors and their thermal and static disorder components. The results clearly demonstrate that structures determined at high resolution and at room temperature have the richest dynamics information. Since such structures are relatively rare in the PDB database, the work naturally calls for more such structures to be determined experimentally.

Bridging between material properties of proteins and the underlying molecular interactions

In this work, we develop a novel method that bridges between material properties of proteins, particularly the modulus of elasticity, and the underlying molecular interactions. To this end, we employ both an all-atom normal mode analysis (NMA) model with the CHARMM force field and an elastic solid model for proteins and protein interfaces. And the "bridge" between the two models is a common physical property predictable by both models: the magnitude of thermal vibrations. This connection allows one to calibrate the Young's moduli of proteins and protein interface regions. We find that the Young's moduli of proteins are in the range of a few Gpa to 10 Gpa, while the Young's moduli of the interface regions are several times smaller. The work is significant as it represents the first attempt to systematically compute the elastic moduli of proteins from molecular interactions.

Shape-preserving elastic solid models of macromolecules

Mass-spring models have been a standard approach in molecular modeling for the last few decades, such as elastic network models (ENMs) that are widely used for normal mode analysis. In this work, we present a vastly different elastic solid model (ESM) of macromolecules that shares the same simplicity and efficiency as ENMs in producing the equilibrium dynamics and moreover, offers some significant new features that may greatly benefit the research community. ESM is different from ENM in that it treats macromolecules as elastic solids. Our particular version of ESM presented in this work, named αESM, captures the shape of a given biomolecule most economically using alpha shape, a well-established technique from the computational geometry community. Consequently, it can produce most economical coarse-grained models while faithfully preserving the shape and thus makes normal mode computations and visualization of extremely large complexes more manageable. Secondly, as a solid model, ESM’s close link to finite element analysis renders it ideally suited for studying mechanical responses of macromolecules under external force. Lastly, we show that ESM can be applied also to structures without atomic coordinates such as those from cryo-electron microscopy. The complete MATLAB code of αESM is provided.

Mass-spring models have been a standard approach in classical molecular modeling where atoms are modeled as spheres with a mass and their interactions modeled as springs. The models have been extremely successful. Thinking ahead, however, as molecular systems of our interest grow more quickly in size or dimension than what our computation resources can keep up with, some adjustments in methodology are timely. This work presents a vastly different elastic solid model (ESM) of macromolecules that shares the same simplicity and efficiency as mass-spring models in producing the equilibrium dynamics and moreover, offers some unique features that make it suitable for much larger systems. ESM is different from ENMs in that it treats macromolecules as elastic solids. Our particular version of ESM model presented in this work, named αESM, captures the shape of a given biomolecule most economically using alpha shape, a well-established technique from the computational geometry community. Consequently, it can produce most economical coarse-grained models while faithfully preserving the shape. ESM can be applied also to structures without atomic coordinates such as those from cryo-electron microscopy.

A time and memory efficient recipe for fast normal mode computations of complexes with icosahedral symmetry

With the recent breakthroughs in experimental technologies, structure determination of extremely large assemblies, many with icosahedral symmetry, has been rapidly accelerating. Computational studies of their dynamics are important to deciphering their functions as well as to structural refinement but are challenged by their extremely large size, which ranges from hundreds of thousands to even millions of atoms. Group theory can be used to significantly speed up the normal mode computations of these symmetric complexes, but the derivation is often obscured by the complexity of group theory and consequently is not widely accessible. To address this problem, this work presents an easy recipe for normal mode computations of complexes with icosahedral symmetry. The recipe details how the Hessian matrix in symmetry coordinates can be constructed in a few easy steps of matrix multiplications, without going through the complexity of group theory. All the "ingredient" matrices required in the recipe are fully provided in the Supplemental Information for easy reproduction. The work is timely considering the expected large in-flux of many more icosahedral assemblies in the near future. The recipe uses a minimum amount of memory and solves the normal modes in a significantly reduced amount of time, making it feasible to perform normal mode computations of these assemblies on most computer systems.

Slow normal modes of proteins are accurately reproduced across different platforms

The Protein data bank (PDB) (Berman et al 2000 Nucl. Acids Res. 28 235-42) contains the atomic structures of over 10⁵ biomolecules with better than 2.8 Å resolution. The listing of the identities and coordinates of the atoms comprising each macromolecule permits an analysis of the slow-time vibrational response of these large systems to minor perturbations. 3D video animations of individual modes of oscillation demonstrate how regions interdigitate to create cohesive collective motions, providing a comprehensive framework for and familiarity with the overall 3D architecture. Furthermore, the isolation and representation of the softest, slowest deformation coordinates provide opportunities for the development of mechanical models of enzyme function. The eigenvector decomposition, therefore, must be accurate, reliable as well as rapid to be generally reported upon. We obtain the eigenmodes of a 1.2 Å 34 kDa PDB entry using either exclusively heavy atoms or partly or fully reduced atomic sets; Cartesian or internal coordinates; interatomic force fields derived either from a full Cartesian potential, a reduced atomic potential or a Gaussian distance-dependent potential; and independently developed software. These varied technologies are similar in that each maintains proper stereochemistry either by use of dihedral degrees of freedom which freezes bond lengths and bond angles, or by use of a full atomic potential that includes realistic bond length and angle restraints. We find that the shapes of the slowest eigenvectors are nearly identical, not merely similar.

Kinematic Flexibility Analysis: Hydrogen Bonding Patterns Impart a Spatial Hierarchy of Protein Motion

Elastic network models (ENMs) and constraint-based, topological rigidity analysis are two distinct, coarse-grained approaches to study conformational flexibility of macromolecules. In the two decades since their introduction, both have contributed significantly to insights into protein molecular mechanisms and function. However, despite a shared purpose of these approaches, the topological nature of rigidity analysis, and thereby the absence of motion modes, has impeded a direct comparison. Here, we present an alternative, kinematic approach to rigidity analysis, which circumvents these drawbacks. We introduce a novel protein hydrogen bond network spectral decomposition, which provides an orthonormal basis for collective motions modulated by noncovalent interactions, analogous to the eigenspectrum of normal modes. The zero modes decompose proteins into rigid clusters identical to those from topological rigidity, while nonzero modes rank protein motions by their hydrogen bond collective energy penalty. Our kinematic flexibility analysis bridges topological rigidity theory and ENM, enabling a detailed analysis of motion modes obtained from both approaches. Analysis of a large, structurally diverse data set revealed that collectivity of protein motions, reported by the Shannon entropy, is significantly reduced for rigidity theory compared to normal mode approaches. Strikingly, kinematic flexibility analysis suggests that the hydrogen bonding network encodes a protein-fold specific, spatial hierarchy of motions, which goes nearly undetected in ENM. This hierarchy reveals distinct motion regimes that rationalize experimental and simulated protein stiffness variations. Kinematic motion modes highly correlate with reported crystallographic B factors and molecular dynamics simulations of adenylate kinase. A formal expression for changes in free energy derived from the spectral decomposition indicates that motions across nearly 40% of modes obey enthalpy-entropy compensation. Taken together, our results suggest that hydrogen bond networks have evolved to modulate protein structure and dynamics, which can be efficiently probed by kinematic flexibility analysis.

Fast Normal Mode Computations of Capsid Dynamics Inspired by Resonance

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

A critical assessment of finite element modeling approach for protein dynamics

Finite element (FE) modeling approach has emerged as an efficient way of calculating the dynamic properties of supramolecular protein structures and their complexes. Its efficiency mainly stems from the fact that the complexity of three-dimensional shape of a molecular surface dominates the computational cost rather than the molecular size or the number of atoms. However, no critical evaluation of the method has been made yet particularly for its sensitivity to the parameters used in model construction. Here, we make a close investigation on the effect of FE model parameters by analyzing 135 representative protein structures whose normal modes calculated using all-atom normal mode analysis are publicly accessible online. Results demonstrate that it is more beneficial to use a contour surface of electron densities as the molecular surface, in general, rather than to employ a solvent excluded surface, and that the solution accuracy is almost insensitive to the model parameters unless we avoid extreme values leading to an inaccurate depiction of the characteristic shapes.

A better prediction of conformational changes of proteins using minimally connected network models

Elastic network models have recently been used for studying low-frequency collective motions of proteins. These models simplify the complexity that arises from normal mode analysis by considering a simplified potential involving a few parameters. Two common parameters in most of the elastic network models are cutoff radius and force constant. Although the latter has been studied extensively and even elaborate new models were introduced, for the former usually an ad-hoc cutoff radius is considered. Moreover, the quality of the network models is usually assessed by evaluating their prediction against experimental B-factors. In this work, we consider various common elastic network models with different cutoff radii and assess them by their ability to predict conformational changes of proteins in complexes from unbound to bound state. This prediction is performed by perturbing the unbound structure using a number of low-frequency normal modes of its network model to optimally fit the bound structure. We evaluated a dataset of 30 proteins with distinct unbound and bound structures using this criterion. The results showed that, opposed to the common calibration process based on B-factors, a meaningful relationship exists between the quality of the prediction and model parameters. It was shown that the cutoff radius has a major role in this prediction and minimally connected network models, which use the shortest cutoff radius for which the network is stable, give the best results. It was also shown that by considering the first ten normal modes, the conformational changes can be predicted by about 25 percent. Hence, the evaluation process was extended to the case of considering the contribution of all normal modes in the prediction. The results indicated that minimally connected network models are superior, despite their simplicity, when any number of modes are considered in the prediction.

Predicting the functional motions of p97 using symmetric normal modes

p97 is a protein complex of the AAA+ family. Although functions of p97 are well understood, the mechanism by which p97 performs its unfolding activities remains unclear. In this work, we present a novel way of applying normal mode analysis to study this six-fold symmetric molecular machine. By selecting normal modes that are axial symmetric and give the largest movements at D1 or D2 pore residues, we are able to predict the functional motions of p97, which are then validated by experimentally observed conformational changes. Our results shed light and provide new understandings on several key steps of the p97 functional process that were previously unclear or controversial, and thus are able to reconcile multiple previous findings. Specifically, our results reveal that (i) a venous valve-like mechanism is used at D2 pore to ensure a one-way exit-only traffic of substrates; (ii) D1 pore remains shut during the functional process; (iii) the "swing-up" motion of the N domain is closely coupled with the vertical motion of the D1 pore along the pore axis; (iv) because of the shut D1 pore and the one-way traffic at D2 pore, it is highly likely that substrates enter the chamber through the gaps at the D1/D2 interface. The limited chamber volume inside p97 suggests that a substrate may be pulling out from D2 while at the same time being pulling in at the interface; (v) lastly, p97 uses a series of actions that alternate between twisting and pulling to remove the substrate. Proteins 2016; 84:1823-1835. © 2016 Wiley Periodicals, Inc.

On the sensitivity of protein data bank normal mode analysis: an application to GH10 xylanases

Protein data bank entries obtain distinct, reproducible flexibility characteristics determined by normal mode analyses of their three dimensional coordinate files. We study the effectiveness and sensitivity of this technique by analyzing the results on one class of glycosidases: family 10 xylanases. A conserved tryptophan that appears to affect access to the active site can be in one of two conformations according to x-ray crystallographic electron density data. The two alternate orientations of this active site tryptophan lead to distinct flexibility spectra, with one orientation thwarting the oscillations seen in the other. The particular orientation of this sidechain furthermore affects the appearance of the motility of a distant, C terminal region we term the mallet. The mallet region is known to separate members of this family of enzymes into two classes.

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.

Atomic torsional modal analysis for high-resolution proteins

We introduce a formulation for normal mode analyses of globular proteins that significantly improves on an earlier one-parameter formulation [M. M. Tirion, Phys. Rev. Lett. 77, 1905 (1996)] that characterized the slow modes associated with protein data bank structures. Here we develop that empirical potential function that is minimized at the outset to include two features essential to reproduce the eigenspectra and associated density of states in the 0 to 300cm-1 frequency range, not merely the slow modes. First, introduction of preferred dihedral-angle configurations via use of torsional stiffness constants eliminates anomalous dispersion characteristics due to insufficiently bound surface side chains and helps fix the spectrum thin tail frequencies (100-300cm-1). Second, we take into account the atomic identities and the distance of separation of all pairwise interactions, improving the spectrum distribution in the 20 to 300cm-1 range. With these modifications, not only does the spectrum reproduce that of full atomic potentials, but we obtain stable reliable eigenmodes for the slow modes and over a wide range of frequencies.