Hyuntae Na

Bridging between normal mode analysis and elastic network models

Normal mode analysis (NMA) has been a powerful tool for studying protein dynamics. Elastic network models (ENM), through their simplicity, have made normal mode computations accessible to a much broader research community and for many more biomolecular systems. The drawback of ENMs, however, is that they are less accurate than NMA. In this work, through steps of simplification that starts with NMA and ends with ENMs we build a tight connection between NMA and ENMs. In the process of bridging between the two, we have also discovered several high‐quality simplified models. Our best simplified model has a mean correlation with the original NMA that is as high as 0.88. In addition, the model is force‐field independent and does not require energy minimization, and thus can be applied directly to experimental structures. Another benefit of drawing the connection is a clearer understanding why ENMs work well and how it can be further improved. We discovered that ANM  can be greatly enhanced by including an additional torsional term and a geometry term. Proteins 2014; 82:2157–2168.

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.

Conventional NMA as a better standard for evaluating elastic network models

Normal mode analysis (NMA) is an important tool for studying protein dynamics. Because of the complexity of conventional NMA that uses an all‐atom model and a semi‐empirical force field, many simplified NMA models have been developed, some of which are known as elastic network models. The quality of these simplified NMA models was assessed mostly by evaluating their predictions against experimental B‐factors, and rarely by comparing them with the original NMA. In this work, we take the effort to create a publicly accessible dataset of proteins with their minimized structures, NMA modes, and mean‐square fluctuations. Then, for the first time, we evaluate the quality of individual normal modes of several widely used elastic network models by comparing them with the conventional NMA. Our results demonstrate that the conventional NMA presents a better and more complete evaluation measure of the quality of elastic network models. This realization should be very helpful in improving current or designing new, higher quality elastic network models. Moreover, using the conventional NMA as the standard of evaluation, a number of interesting and significant insights into the elastic network models are gained. Proteins 2015; 83:259–267.

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.

The performance of fine‐grained and coarse‐grained elastic network models and its dependence on various factors

In a recent work we developed a method for deriving accurate simplified models that capture the essentials of conventional all‐atom NMA and identified two best simplified models: ssNMA and eANM, both of which have a significantly higher correlation with NMA in mean square fluctuation calculations than existing elastic network models such as ANM and ANMr2, a variant of ANM that uses the inverse of the squared separation distances as spring constants. Here, we examine closely how the performance of these elastic network models depends on various factors, namely, the presence of hydrogen atoms in the model, the quality of input structures, and the effect of crystal packing. The study reveals the strengths and limitations of these models. Our results indicate that ssNMA and eANM are the best fine‐grained elastic network models but their performance is sensitive to the quality of input structures. When the quality of input structures is poor, ANMr2 is a good alternative for computing mean‐square fluctuations while ANM model is a good alternative for obtaining normal modes. Proteins 2015; 83:1273–1283.

The effective degeneracy of protein normal modes

Normal modes are frequently computed and used to portray protein dynamics and interpret protein conformational changes. In this work, we investigate the nature of normal modes and find that the normal modes of proteins, especially those at the low frequency range (0-600 cm(-1)), are highly susceptible to degeneracy. Two or more modes are degenerate if they have the same frequency and consequently any orthogonal transformation of them also is a valid representation of the mode subspace. Thus, degenerate modes can no longer characterize unique directions of motions as regular modes do. Though the normal modes of proteins are usually of different frequencies, the difference in frequency between neighboring modes is so small that, under even slight structural uncertainty that unavoidably exists in structure determination, it can easily vanish and as a result, a mode becomes effectively degenerate with its neighboring modes. This can be easily observed in that some modes seem to disappear and their matching modes cannot be found when the structure used to compute the modes is modified only slightly. We term this degeneracy the effective degeneracy of normal modes. This work is built upon our recent discovery that the vibrational spectrum of globular proteins is universal. The high density of modes observed in the vibrational frequency spectra of proteins renders their normal modes highly susceptible to degeneracy, under even the smallest structural uncertainty. Indeed, we find the degree of degeneracy of modes is proportional to the density of modes in the vibrational spectrum. This means that for modes at the same frequency, degeneracy is more severe for larger proteins. Degeneracy exists also in the modes of coarse-grained models, but to a much lesser extent than those of all-atom models. In closing, we discuss the implications of the effective degeneracy of normal modes: how it may significantly affect the ways in which normal modes are used in various normal modes-based applications.

A natural unification of GNM and ANM and the role of inter-residue forces

The Gaussian network model (GNM) and anisotropic network model (ANM) are two elastic network models that have been widely used to study protein fluctuation dynamics. Both models have strengths and weaknesses. Attempts have been made in the past to unify the two models but they had severe limitations. This work presents a novel theoretical result that shows how GNM and ANM can be unified through taking into account the effect of inter-residue forces. The unified model, called the force spring model, or FSM, is reduced to ANM when all the inter-residue forces are set to zero. Moreover, the unification reveals the role of inter-residue forces in protein fluctuation dynamics. Specifically, the effect of inter-residue forces is closely examined by studying the changes in mean-square fluctuations when inter-residue forces are present.

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.

Quantitative delineation of how breathing motions open ligand migration channels in myoglobin and its mutants

Ligand migration and binding are central to the biological functions of many proteins such as myoglobin (Mb) and it is widely thought that protein breathing motions open up ligand channels dynamically. However, how a protein exerts its control over the opening and closing of these channels through its intrinsic dynamics is not fully understood. Specifically, a quantitative delineation of the breathing motions that are needed to open ligand channels is lacking. In this work, we present and apply a novel normal mode‐based method to quantitatively delineate what and how breathing motions open ligand migration channels in Mb and its mutants. The motivation behind this work springs from the observation that normal mode motions are closely linked to the breathing motions that are thought to open ligand migration channels. In addition, the method provides a direct and detailed depiction of the motions of each and every residue that lines a channel and can identify key residues that play a dominating role in regulating the channel. The all‐atom model and the full force‐field employed in the method provide a realistic energetics on the work cost required to open a channel, and as a result, the method can be used to efficiently study the effects of mutations on ligand migration channels and on ligand entry rates. Our results on Mb and its mutants are in excellent agreement with MD simulation results and experimentally determined ligand entry rates. Proteins 2015; 83:757–770.

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.