Universality of vibrational spectra of globular proteins

Some Clues about Enzymes from Psychrophilic Microorganisms

Enzymes purified from psychrophilic microorganisms prove to be efficient catalysts at low temperatures and possess a great potential for biotechnological applications. The low-temperature catalytic activity has to come from specific structural fluctuations involving the active site region, however, the relationship between protein conformational stability and enzymatic activity is subtle. We provide a survey of the thermodynamic stability of globular proteins and their rationalization grounded in a theoretical approach devised by one of us. Furthermore, we provide a link between marginal conformational stability and protein flexibility grounded in the harmonic approximation of the vibrational degrees of freedom, emphasizing the occurrence of long-wavelength and excited vibrations in all globular proteins. Finally, we offer a close view of three enzymes: chloride-dependent α-amylase, citrate synthase, and β-galactosidase.

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.

Protein vibrations and their localization behaviour. A numerical scaling analysis

Using a classical force field, we investigate the localization properties of protein normal modes. For a set of eighteen proteins that cover five classes of increasing size, we compute the participation ratio as a measure of the spatial extent of protein vibrations. In this scaling analysis, we find extended low-frequency far-infrared and Terahertz modes, in contrast to localized high-frequency near-infrared vibrations. These regimes are separated by a broad crossover around a wave number of 260 cm^-1. Biophysical and biochemical implications are discussed, and the vibrational localization properties are compared to those of amorphous solids.

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.

Effects of spherical confinement and backbone stiffness on flexible polymer jamming

We use molecular simulations to study jamming of a crumpled bead-spring model polymer in a finite container and compare to jamming of repulsive spheres. After proper constraint counting, the onset of rigidity is seen to occur isostatically as in the case of repulsive spheres. Despite this commonality, the presence of the curved container wall and polymer backbone bonds introduce new mechanical properties. Notably, these include additional bands in the vibrational density of states that reflect the material structure as well as oscillations in local contact number and density near the wall but with lower amplitude for polymers. Polymers have fewer boundary contacts, and this low-density surface layer strongly reduces the global bulk modulus. We further show that bulk-modulus dependence on backbone stiffness can be described by a model of stiffnesses in series and discuss potential experimental and biological applications.

Protein-Ligand Complexes as Constrained Dynamical Systems

This study focuses on how the low-frequency end of the vibrational spectrum related to the functional motions changes as a protein binds to a small ligand(s). Our recently proposed residue-specific (RESPEC) elastic network model provides a natural laboratory for this aim due to its systematic mixed coarse-graining approach and parametrization. Current analysis on a large data set of protein-ligand complexes reveals a universal curve enclosing the frequency distributions, which bears the features of previous computational and experimental studies. We mostly observe positive frequency shifts in the collective modes of the protein upon ligand binding. This observation, conforming to the Rayleigh-Courant-Fisher theorem, points to a constraining effect imposed by ligands on protein dynamics, which may be accompanied by a negative vibrational entropy difference. Positive frequency shifts in the global modes can thus be linked to the harmonic well getting steeper, because of interactions with the ligand(s).

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.

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.

All-atom normal mode dynamics of HIV-1 capsid

Dynamics of biomolecular assemblies offer invaluable insights into their functional mechanisms. For extremely large biomolecular systems, such as HIV-1 capsid that has nearly 5 millions atoms, obtaining its normal mode dynamics using even coarse-grained models can be a challenging task. In this work, we have successfully carried out a normal mode analysis of an entire HIV-1 capsid in full all-atom details. This is made possible through our newly developed BOSE (Block of Selected Elasticity) model that is founded on the principle of resonance discovered in our recent work. The resonance principle makes it possible to most efficiently compute the vibrations of a whole capsid at any given frequency by projecting the motions of component capsomeres into a narrow subspace. We have conducted also assessments of the quality of the BOSE modes by comparing them with benchmark modes obtained directly from the original Hessian matrix. Our all-atom normal mode dynamics study of the HIV-1 capsid reveals the dynamic role of the pentamers in stabilizing the capsid structure and is in agreement with experimental findings that suggest capsid disassembly and uncoating start when the pentamers become destabilized. Our results on the dynamics of hexamer pores suggest that nucleotide transport should take place mostly at hexamers near pentamers, especially at the larger hemispherical end.

Supramolecular assemblies are large biomolecular complexes composed of hundreds or even thousands of protein chains. They function as molecular machines or as large containers that store or facilitate the chemical reactions of other molecules. Whatever they do, their functional mechanisms are tightly linked to their structures and intrinsic dynamics. Recently, due to breakthroughs in experimental techniques, many supramolecular assemblies have been determined, such as the capsid of human immunodeficiency virus (HIV) that is composed of nearly 5 millions of atoms. Computational studies of these systems are challenging due to their extremely large sizes. In this work, we have successfully carried out a dynamics study of an entire HIV capsid in full all-atom details. Our study reveals new insights into the dynamics of the N-terminal loops, the stabilizing role of the pentamers, and where the nucleotide transport may take place.

Universality and Specificity in Protein Fluctuation Dynamics

We investigate the universal scaling of protein fluctuation dynamics with a site-specific diffusive model of protein motion, which predicts an initial subdiffusive regime in the configurational relaxation. The long-time dynamics of proteins is controlled by an activated regime. We argue that the hierarchical free energy barriers set the time scales of biological processes and establish an upper limit to the size of single protein domains. We find it compelling that the scaling behavior for the protein dynamics is in close agreement with the Kardar-Parisi-Zhang scaling exponents.

Peptide-Protein Binding Investigated by Far-IR Spectroscopy and Molecular Dynamics Simulations

Molecular dynamics (MD) simulations and far-infrared (far-IR) spectroscopy were combined to study peptide binding by the second PDZ domain (PDZ1) of MAGI1, which has been identified as an important target for the Human Papilloma Virus. PDZ1 recognizes and binds to the C-terminal end of the E6 protein from high-risk Human Papilloma Virus. The far-IR spectra of two forms of the protein, an unbound APO form and a HOLO form (where the PDZ1 is bound to an 11-residue peptide derived from the C terminus of HPV16 E6), were obtained. MD simulations were used to determine the most representative structure of each form and these were used to compute their respective IR spectra by normal mode analysis. Far-UV circular dichroism spectroscopy was used to confirm the secondary structure content and the stability through temperature-dependent studies. Both the experimental and calculated far-IR spectra showed a red shift of the low-frequency peaks upon peptide binding. The calculations show that this is coincident with an increased number of hydrogen bonds formed as the peptide augments the protein β-sheet. We further identified the contribution of surface-bound water molecules to bands in the far-IR and, through the calculations, identified potential pathways for allosteric communication. Together, these results demonstrate the utility of combining far-IR experiments and MD studies to study peptide binding by proteins.

A comparison of the innate flexibilities of six chains in F1-ATPase with identical secondary and tertiary folds; 3 active enzymes and 3 structural proteins

The α and β subunits comprising the hexameric assembly of F1-ATPase share a high degree of structural identity, though low primary identity. Each subunit binds nucleotide in similar pockets, yet only β subunits are catalytically active. Why? We re-examine their internal symmetry axes and observe interesting differences. Dividing each chain into an N-terminal head region, a C-terminal foot region, and a central torso, we observe (1) that while the foot and head regions in all chains obtain high and similar mobility, the torsos obtain different mobility profiles, with the β subunits exhibiting a higher motility compared to the α subunits, a trend supported by the crystallographic B-factors. The β subunits have greater torso mobility by having fewer distributed, nonlocal packing interactions providing a spacious and soft connectivity and offsetting the resultant softness with local stiffness elements, including an additional β sheet. (2) A loop near the nucleotide binding-domain of the β subunits, absent in the α subunits, swings to create a large variation in the occlusion of the nucleotide binding region. (3) A combination of the softest three eigenmodes significantly reduces the root mean square difference between the open and closed conformations of the β subunits. (4) Comparisons of computed and observed crystallographic B-factors suggest a suppression of a particular symmetry axis in an α subunit. (5) Unexpectedly, the soft intra-monomer oscillations pertain to distortions that do not create inter-monomer steric clashes in the assembly, suggesting that structural optimization of the assembly evolved at all levels of complexity.

Far infrared spectroscopy of hydrogen bonding collective motions in complex molecular systems

Far infrared spectroscopy as a tool for the study of inter and intramolecular interactions in complex molecular structures.

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.