A theorem on amplitudes of thermal atomic fluctuations in large molecules assuming specific conformations calculated by normal mode analysis

An exact theorem is proved and its implication is discussed. The theorem states that, if a large molecule, typically biological macromolecules such as proteins, undergoes small-amplitude conformational fluctuations around its native conformation in such a way that within the range of conformational fluctuations at thermal equilibrium the conformational energy surface can be approximated by a multidimensional parabola, then the mass-weighted mean-square displacement of constituent atoms is given by the sum of the contributions from each normal mode of conformational vibration, which in turn is proportional to the inverse of the square of its frequency. This theorem provides a firm theoretical basis for the fact hitherto empirically recognized in the conformational dynamics of, for instance, native proteins that very-low-frequency normal modes make dominant contributions to the conformational fluctuations at thermal equilibrium. Discussion is given on the implication of this theorem, especially on the importance of the concept of the low-frequency normal modes, even in the case where the basic assumption of the harmonicity of the energy surface does not hold.     

Nullspace Sampling with Holonomic Constraints Reveals Molecular Mechanisms of Protein Gαs

Proteins perform their function or interact with partners by exchanging between conformational substates on a wide range of spatiotemporal scales. Structurally characterizing these exchanges is challenging, both experimentally and computationally. Large, diffusional motions are often on timescales that are difficult to access with molecular dynamics simulations, especially for large proteins and their complexes. The low frequency modes of normal mode analysis (NMA) report on molecular fluctuations associated with biological activity. However, NMA is limited to a second order expansion about a minimum of the potential energy function, which limits opportunities to observe diffusional motions. By contrast, kino-geometric conformational sampling (KGS) permits large perturbations while maintaining the exact geometry of explicit conformational constraints, such as hydrogen bonds. Here, we extend KGS and show that a conformational ensemble of the α subunit Gαs of heterotrimeric stimulatory protein Gs exhibits structural features implicated in its activation pathway. Activation of protein Gs by G protein-coupled receptors (GPCRs) is associated with GDP release and large conformational changes of its α-helical domain. Our method reveals a coupled α-helical domain opening motion while, simultaneously, Gαs helix α₅ samples an activated conformation. These motions are moderated in the activated state. The motion centers on a dynamic hub near the nucleotide-binding site of Gαs, and radiates to helix α₄. We find that comparative NMA-based ensembles underestimate the amplitudes of the motion. Additionally, the ensembles fall short in predicting the accepted direction of the full activation pathway. Taken together, our findings suggest that nullspace sampling with explicit, holonomic constraints yields ensembles that illuminate molecular mechanisms involved in GDP release and protein Gs activation, and further establish conformational coupling between key structural elements of Gαs.     

Normal mode calculations of icosahedral viruses with full dihedral flexibility by use of molecular symmetry

The study of the dynamics and thermodynamics of small icosahedral virus capsids is an active field of research. Normal mode analysis is one of the computational tools that can provide important insights into the conformational changes of the virus associated with cell entry or caused by changing of the physicochemical environment. Normal mode analysis of virus capsids has been limited due to the size of these systems, which often exceed 50,000 residues. Here we present the first normal mode calculation with full dihedral flexibility of several virus capsids, including poliovirus, rhinovirus, and cowpea chlorotic mottle virus. The calculations were made possible by applying group theoretical methods, which greatly simplified the calculations without any approximation beyond the all-atom force field representations in general use for smaller protein systems. Since a full Cartesian basis set was too large to be handled by the available computer memory, we used a basis set that includes all internal dihedral angles of the system with the exception of the peptide bonds, which were assumed rigid. The fluctuations of the normal modes are shown to correlate well with crystallographic temperature factors. The motions of the first several normal modes of each symmetry type are described. A hinge bending motion in poliovirus was found that may be involved in the mechanism by which bound small molecules inhibit conformational changes of the capsid. Fully flexible normal mode calculations of virus capsids are expected to increase our understanding of virus dynamics and thermodynamics, and can be useful in the refinement of cryo-electron microscopy structures of viruses.     

Allostery without conformational change

A general model is presented whereby lignand-induced changes in protein dynamics could produce allosteric communication between distinct binding sites, even in the absence of a macromolecular conformational change. Theoretical analysis, based on the statistical thermodynamics of ligand binding, shows that cooperative interaction free energies amounting to several kJ · mol^-1 may be generated by this means. The effect arises out of the possible changes in frequencies and amplitudes of macromolecular thermal fluctuations in response to ligand attachment, and can involve all forms of dynamic behaviour, ranging from highly correlated, low-frequency normal mode vibrations to random local anharmonic motions of individual atoms or groups. Dynamic allostery of this form is primarily an entropy effect, and we derive approximate expressions which might allow the magnitude of the interaction in real systems to be calculated directly from experimental observations such as changes in normal mode frequencies and mean-square atomic displacements. Long-range influence of kinetic processes at different sites might also be mediated by a similar mechanism. We suggest that proteins and other biological macromolecules may have evolved to take functional advantage not only of mean conformational states but also of the inevitable thermal fluctuations about the mean.     

A new spin on protein dynamics

Site-directed spin labeling is a general method for investigating structure and conformational switching in soluble and membrane proteins. It will also be an important tool for exploring protein backbone dynamics. A semi-empirical analysis of nitroxide sidechain dynamics in spin-labeled proteins reveals contributions from fluctuations in backbone dihedral angles and rigid-body (collective) motions of alpha helices. Quantitative analysis of sidechain dynamics is sometimes possible, and contributions from backbone modes can be expressed in terms of relative order parameters and rates. Dynamic sequences identified by site-directed spin labeling correlate with functional domains, and so nitroxide scanning could provide an efficient strategy for identifying such domains in high-molecular weight proteins, supramolecular complexes and membrane proteins.     

ProMode: a database of normal mode analyses on protein molecules with a full-atom model

MOTIVATION: Although information from protein dynamics simulation is important to understand principles of architecture of a protein structure and its function, simulations such as molecular dynamics and Monte Carlo are very CPU-intensive. Although the ability of normal mode analysis (NMA) is limited because of the need for a harmonic approximation on which NMA is based, NMA is adequate to carry out routine analyses on many proteins to compute aspects of the collective motions essential to protein dynamics and function. Furthermore, it is hoped that realistic animations of the protein dynamics can be observed easily without expensive software and hardware, and that the dynamic properties for various proteins can be compared with each other. RESULTS: ProMode, a database collecting NMA results on protein molecules, was constructed. The NMA calculations are performed with a full-atom model, by using dihedral angles as independent variables, faster and more efficiently than the calculations using Cartesian coordinates. In ProMode, an animation of the normal mode vibration is played with a free plug-in, Chime (MDL Information Systems, Inc.). With the full-atom model, the realistic three-dimensional motions at an atomic level are displayed with Chime. The dynamic domains and their mutual screw motions defined from the NMA results are also displayed. Properties for each normal mode vibration and their time averages, e.g. fluctuations of atom positions, fluctuations of dihedral angles and correlations between the atomic motions, are also presented graphically for characterizing the collective motions in more detail. AVAILABILITY: http://promode.socs.waseda.ac.jp     

Characterization of the Conformational Fluctuations in the Josephin Domain of Ataxin-3

As for a variety of other molecular recognition processes, conformational fluctuations play an important role in the cleavage of polyubiquitin chains by the Josephin domain of ataxin-3. The interaction between Josephin and ubiquitin appears to be mediated by the motions of α-helical hairpin that is unusual among deubiquitinating enzymes. Here, we characterized the conformational fluctuations of the helical hairpin by incorporating NMR measurements as replica-averaged restraints in molecular dynamics simulations, and by validating the results by small-angle x-ray scattering measurements. This approach allowed us to define the extent of the helical hairpin motions and suggest a role of such motions in the recognition of ubiquitin.     

Dynamics of transfer RNAs analyzed by normal mode calculation.

Normal mode calculation is applied to tRNAPhe and tRNAAsp, and their structural and vibrational aspects are analyzed. Dihedral angles along the phosphate-ribose backbone (alpha, beta, gamma, epsilon, zeta) and dihedral angles of glycosyl bonds (chi) are selected as movable parameters. The calculated displacement of each atom agrees with experimental data. In modes with frequencies higher than 130 cm-1, the motions are localized around each stem and the elbow region of the L-shape. On the other hand, collective motions such as bending or twisting of arms are seen in modes with lower frequencies. Hinge axes and bend angles are calculated without prior knowledge. Movements in modes with very low frequencies are combinations of hinge bending motions with various hinge axes and bend angles. The thermal fluctuations of dihedral angles well reflect the structural characters of transfer RNAs. There are some dihedral angles of nucleotides located around the elbow region of L-shape, which fluctuate about five to six times more than the average value. Nucleotides in the position seem to be influential in the dynamics of the entire structure. The normal mode calculation seems to provide much information for the study of conformational changes of transfer RNAs induced by aminoacyl-tRNA synthetase or codon during molecular recognition.     

A Physical Picture of Protein Dynamics and Conformational Changes

A physical model is reviewed which explains different aspects of protein dynamics consistently. At low temperatures, the molecules are frozen in conformational substates. Their average energy is 3/2RT. Solid-state vibrations occur on a time scale of femtoseconds to nanoseconds. Above a characteristic temperature, often called the dynamical transition temperature, slow modes of motions can be observed occurring on a time scale between about 140 and 1 ns. These motions are overdamped, quasidiffusive, and involve collective motions of segments of the size of an α-helix. Molecules performing these types of motion are in the “flexible state”. This state is reached by thermal activation. It is shown that these motions are essential for conformational relaxation. Based on this picture, a new approach is proposed to understand conformational changes. It connects structural fluctuations and conformational transitions.     

Normal mode refinement: crystallographic refinement of protein dynamic structure. II. Application to human lysozyme.

The dynamic structure of a protein, human lysozyme, is determined by the normal mode refinement of X-ray crystal structure. This method uses the normal modes of both internal and external motions to distinguish the real internal dynamics from the external terms such as lattice disorder, and gives an anisotropic and concerted picture of atomic fluctuations. The refinement is carried out with diffraction data of 5.0 to 1.8 A resolution, which are collected on an imaging plate. The results of the refinement show: (1) Debye-Waller factor consists of two parts, highly anisotropic internal fluctuations and almost isotropic external terms. The former is smaller than the latter by a factor of 0.72 in the scale of B-factor. Therefore, the internal dynamics cannot be recognized directly from the apparent electron density distribution. (2) The internal fluctuations show basically similar features as those predicted by the normal mode analysis, with almost the same amplitude and a similar level of anisotropy. (3) Correlations of fluctuations are detected between two lobes forming the active site cleft, which move simultaneously in opposite directions. This corresponds to the hinge-bending motion of lysozyme.     

Projection of Monte Carlo and molecular dynamics trajectories onto the normal mode axes: human lysozyme

A method is presented to describe the internal motions of proteins obtained from molecular dynamics or Monte Carlo simulations as motions of normal mode variables. This method calculates normal mode variables by projecting trajectories of these simulations onto the axes of normal modes and expresses the trajectories as a linear combination of normal mode variables. This method is applied to the result of the molecular dynamics and the Monte Carlo simulations of human lysozyme. The motion of the lowest frequency mode extracted from the simulations represents the hinge bending motion very faithfully. Analysis of the obtained motions of the normal mode variables provides an explanation of the anharmonic aspects of protein dynamics as due first to the anharmonicity of the actual potential energy surface near a minimum and second to trans-minimum conformational changes.     

Evolutionary Conserved Positions Define Protein Conformational Diversity

Conformational diversity of the native state plays a central role in modulating protein function. The selection paradigm sustains that different ligands shift the conformational equilibrium through their binding to highest-affinity conformers. Intramolecular vibrational dynamics associated to each conformation should guarantee conformational transitions, which due to its importance, could possibly be associated with evolutionary conserved traits. Normal mode analysis, based on a coarse-grained model of the protein, can provide the required information to explore these features. Herein, we present a novel procedure to identify key positions sustaining the conformational diversity associated to ligand binding. The method is applied to an adequate refined dataset of 188 paired protein structures in their bound and unbound forms. Firstly, normal modes most involved in the conformational change are selected according to their corresponding overlap with structural distortions introduced by ligand binding. The subspace defined by these modes is used to analyze the effect of simulated point mutations on preserving the conformational diversity of the protein. We find a negative correlation between the effects of mutations on these normal mode subspaces associated to ligand-binding and position-specific evolutionary conservations obtained from multiple sequence-structure alignments. Positions whose mutations are found to alter the most these subspaces are defined as key positions, that is, dynamically important residues that mediate the ligand-binding conformational change. These positions are shown to be evolutionary conserved, mostly buried aliphatic residues localized in regular structural regions of the protein like β-sheets and α-helix.     

Why are large conformational changes well described by harmonic normal modes?

Low-frequency normal modes generated by elastic network models tend to correlate strongly with large conformational changes of proteins, despite their reliance on the harmonic approximation, which is only valid in close proximity of the native structure. We consider 12 variants of the torsional network model (TNM), an elastic network model in torsion angle space, that adopt different sets of torsion angles as degrees of freedom and reproduce with similar quality the thermal fluctuations of proteins but present drastic differences in their agreement with conformational changes. We show that these differences are related to the extent of the deviations from the harmonic approximation, assessed through an anharmonic energy function whose harmonic approximation coincides with the TNM. Our results indicate that mode anharmonicity is more strongly related to its collectivity, i.e., the number of atoms displaced by the mode, than to its amplitude; low-frequency modes can remain harmonic even at large amplitudes, provided they are sufficiently collective. Finally, we assess the potential benefits of different strategies to minimize the impact of anharmonicity. The reduction of the number of degrees of freedom or their regularization by a torsional harmonic potential significantly improves the collectivity and harmonicity of normal modes and the agreement with conformational changes. In contrast, the correction of normal mode frequencies to partially account for anharmonicity does not yield substantial benefits. The TNM program is freely available at https://github.com/ugobas/tnm.     

Quasi-harmonic analysis of mode coupling in fluctuating native proteins

Mode coupling and anharmonicity in a native fluctuating protein are investigated in modal space by projecting the motion along the eigenvectors of the fluctuation correlation matrix. The probability distribution of mode fluctuations is expressed in terms of tensorial Hermite polynomials. Molecular dynamics trajectories of Crambin are generated and used to evaluate the terms of the polynomials and to obtain the modal energies. The energies of a few modes exhibit large deviations from the harmonic energy of kT/2 per mode, resulting from coupling to the surroundings, or to another specific mode or to several other modes. Slowest modes have energies that are below that of the harmonic, and a few fast modes have energies significantly larger than the harmonic. Detailed analysis of the coupling of these modes to others is presented in terms of the lowest order two-mode coupling terms. Finally, the effects of mode coupling on conformational properties of the protein are investigated.     

Chromatin Hydrodynamics

Following recent observations of large scale correlated motion of chromatin inside the nuclei of live differentiated cells, we present a hydrodynamic theory—the two-fluid model—in which the content of a nucleus is described as a chromatin solution with the nucleoplasm playing the role of the solvent and the chromatin fiber that of a solute. This system is subject to both passive thermal fluctuations and active scalar and vector events that are associated with free energy consumption, such as ATP hydrolysis. Scalar events drive the longitudinal viscoelastic modes (where the chromatin fiber moves relative to the solvent) while vector events generate the transverse modes (where the chromatin fiber moves together with the solvent). Using linear response methods, we derive explicit expressions for the response functions that connect the chromatin density and velocity correlation functions to the corresponding correlation functions of the active sources and the complex viscoelastic moduli of the chromatin solution. We then derive general expressions for the flow spectral density of the chromatin velocity field. We use the theory to analyze experimental results recently obtained by one of the present authors and her co-workers. We find that the time dependence of the experimental data for both native and ATP-depleted chromatin can be well-fitted using a simple model—the Maxwell fluid—for the complex modulus, although there is some discrepancy in terms of the wavevector dependence. Thermal fluctuations of ATP-depleted cells are predominantly longitudinal. ATP-active cells exhibit intense transverse long wavelength velocity fluctuations driven by force dipoles. Fluctuations with wavenumbers larger than a few inverse microns are dominated by concentration fluctuations with the same spectrum as thermal fluctuations but with increased intensity.     

Normal mode analysis of the horseradish peroxidase collective motions: correlation with spectroscopically observed heme distortions

Horseradish peroxidase C is a class III peroxidase whose structure is stabilized by the presence of two endogenous calcium atoms. Calcium removal has been shown to decrease the enzymatic activity of the enzyme and significantly affect the spectroscopically detectable properties of the heme, such as the spin state of the iron, heme normal modes, and distortions from planarity. In this work, we report on normal mode analysis (NMA) performed on models subjected to 2 ns of molecular dynamics simulations to describe the effect of calcium removal on protein collective motions and to investigate the correlation between active site (heme) and protein matrix fluctuations. We show that in the native peroxidase model, heme fluctuations are correlated to matrix fluctuations while they are not in the calcium-depleted model.     

Energy landscape of a native protein: Jumping-among-minima model

We have investigated energy landscape of human lysozyme in its native state by using principal component analysis and a model, jumping-among-minima (JAM) model. These analyses are applied to 1 nsec molecular dynamics trajectory of the protein in water. An assumption embodied in the JAM model allows us to divide protein motions into intra-substate and inter-substate motions. By examining intra-substate motions, it is shown that energy surfaces of individual conformational substates are nearly harmonic and mutually similar. As a result of principal component analysis and JAM model analysis, protein motions are shown to consist of three types of collective modes, multiply hierarchical modes, singly hierarchical modes, and harmonic modes. Multiply hierarchical modes, the number of which accounts only for 0.5% of all modes, dominate contributions to total mean-square atomic fluctuation. Inter-substate motions are observed only in a small-dimensional subspace spanned by the axes of multiplyhierarchical and singly hierarchical modes. Inter-substate motions have two notable time components: faster component seen within 200 psec and slower component. The former involves transitions among the conformational substates of the low-level hierarchy, whereas the latter involves transitions of the higher level substates observed along the first four multiply hierarchical modes. We also discuss dependence of the subspace, which contains conformational substates, on time duration of simulation. Proteins 33:496–517, 1998. © 1998 Wiley-Liss, Inc.     

The role of slow and fast protein motions in allosteric interactions

Allostery is fundamentally thermodynamic in nature. Long-range communication in proteins may be mediated not only by changes in the mean conformation with enthalpic contribution but also by changes in dynamic fluctuations with entropic contribution. The important role of protein motions in mediating allosteric interactions has been established by NMR spectroscopy. By using CAP as a model system, we have shown how changes in protein structure and internal dynamics can allosterically regulate protein function and activity. The results indicate that changes in conformational entropy can give rise to binding enhancement, binding inhibition, or have no effect in the expected affinity, depending on the magnitude and sign of enthalpy–entropy compensation. Moreover, allosteric interactions can be regulated by the modulation a low-populated conformation states that serve as on-pathway intermediates for ligand binding. Taken together, the interplay between fast internal motions, which are intimately related to conformational entropy, and slow internal motions, which are related to poorly populated conformational states, can regulate protein activity in a way that cannot be predicted on the basis of the protein’s ground-state structure.     

Normal mode refinement: crystallographic refinement of protein dynamic structure. I. Theory and test by simulated diffraction data.

A dynamic structure refinement method for X-ray crystallography, referred to as the normal mode refinement, is proposed. The Debye-Waller factor is expanded in terms of the low-frequency normal modes whose amplitudes and eigenvectors are experimentally optimized in the process of the crystallographic refinement. In this model, the atomic fluctuations are treated as anisotropic and concerted. The normal modes of the external motion (TLS model) are also introduced to cover the factors other than the internal fluctuations, such as the lattice disorder and diffusion. A program for the normal mode refinement (NM-REF) has been developed. The method has first been tested against simulated diffraction data for human lysozyme calculated by a Monte Carlo simulation. Applications of the method have demonstrated that the normal mode refinement has: (1) improved the fitting to the diffraction data, even with fewer adjustable parameters; (2) distinguished internal fluctuations from external ones; (3) determined anisotropic thermal factors; and (4) identified concerted fluctuations in the protein molecule.