Vibrational dynamics of icosahedrally symmetric biomolecular assemblies compared with predictions based on continuum elasticity

Scalable computation of anisotropic vibrations for large macromolecular assemblies

The Normal Mode Analysis (NMA) is a standard approach to elucidate the anisotropic vibrations of macromolecules at their folded states, where low-frequency collective motions can reveal rearrangements of domains and changes in the exposed surface of macromolecules. Recent advances in structural biology have enabled the resolution of megascale macromolecules with millions of atoms. However, the calculation of their vibrational modes remains elusive due to the prohibitive cost associated with constructing and diagonalizing the underlying eigenproblem and the current approaches to NMA are not readily adaptable for efficient parallel computing on graphic processing unit (GPU). Here, we present eigenproblem construction and diagonalization approach that implements level-structure bandwidth-reducing algorithms to transform the sparse computation in NMA to a globally-sparse-yet-locally-dense computation, allowing batched tensor products to be most efficiently executed on GPU. We map, optimize, and compare several low-complexity Krylov-subspace eigensolvers, supplemented by techniques such as Chebyshev filtering, sum decomposition, external explicit deflation and shift-and-inverse, to allow fast GPU-resident calculations. The method allows accurate calculation of the first 1000 vibrational modes of some largest structures in PDB ( > 2.4 million atoms) at least 250 times faster than existing methods.

Virtual indentation of the empty capsid of the minute virus of mice using a minimal coarse-grained model

A minimal coarse-grained model for T=1 viral capsids assembled from 20 protein rigid trimers has been designed by extending a previously proposed form of the interaction energy written as a sum of anisotropic pairwise interactions between the trimeric capsomers. The extension of the model has been performed to properly account for the coupling between two internal coordinates: the one that measures the intercapsomer distance and the other that gives the intercapsomer dihedral angle. The model has been able to fit with less than a 10% error the atomic force microscopy (AFM) indentation experimental data for the empty capsid of the minute virus of mice (MVM), providing in this way an admissible picture of the main mechanisms behind the capsid deformations. In this scenario, the bending of the intercapsomer dihedral angle is the angular internal coordinate that can support larger deformations away from its equilibrium values, determining important features of the AFM indentation experiments as the elastic constants along the three symmetry axes of the capsid and the critical indentations. From the value of one of the parameters of our model, we conclude that trimers in the MVM must be quite oblate tops, in excellent agreement with their known structure. The transition from the linear to the nonlinear regimes sampled in the indentation process appears to be an interesting topic for future research in physical virology.

Molecular Dynamics Simulations of Shockwave Affected STMV Virus to Measure the Frequencies of the Oscillatory Response

Acoustic shockwaves are of interest as a possible means of the selective inactivation of viruses. It has been proposed that such inactivation may be enhanced by driving the virus particles at frequencies matching the characteristic frequency corresponding to acoustic modes of the viral structures, setting up a resonant response. Characteristic frequencies of viruses have been previously studied through opto-mechanical techniques. In contrast to optical excitation, shockwaves may be able to probe acoustic modes without the limitation of optical selection rules. This work explores molecular dynamics simulations of shockwaves interacting with a single STMV virus structure, in full atomistic detail, in order to measure the frequency of the response of the overall structure. Shockwaves of varying energy were set up in a water box containing the STMV structure by assigning water molecules at the edge of the box with an elevated velocity inward—in the direction of the virus. It was found that the structure compressed and stretched in a periodic oscillation of frequency 65 ± 6.5 GHz. This measured frequency did not show strong dependency on the energy of the shockwave perturbing the structure, suggesting the frequency is a characteristic of the structure. The measured frequency is also consistent with values predicted from elastic theory. Additionally, it was found that subjecting the virus to repeated shockwaves led to further deformation of the structure and the magnitude of the overall deformation could be altered by varying the time delay between repeated shockwave pulses.

Minimal Design Principles for Icosahedral Virus Capsids

The geometrical structures of single- and multiple-shell icosahedral virus capsids are reproduced as the targets that minimize the cost corresponding to relatively simple design functions. Capsid subunits are first identified as building blocks at a given coarse-grained scale and then represented in these functions as point particles located on an appropriate number of concentric spherical surfaces. Minimal design cost is assigned to optimal spherical packings of the particles. The cost functions are inspired by the packings favored for the Thomson problem, which minimize the electrostatic potential energy between identical charged particles. In some cases, icosahedral symmetry constraints are incorporated as external fields acting on the particles. The simplest cost functions can be obtained by separating particles in disjoint nonequivalent sets with distinct interactions, or by introducing interacting holes (the absence of particles). These functions can be adapted to reproduce any capsid structure found in real viruses. Structures absent in Nature require significantly more complex designs. Measures of information content and complexity are assigned to both the cost functions and the capsid geometries. In terms of these measures, icosahedral structures and the corresponding cost functions are the simplest solutions.

A minimal coarse-grained model for the low-frequency normal mode analysis of icosahedral viral capsids

The main goal of this work is the design of a coarse-grained theoretical model of minimal resolution for the study of the physical properties of icosahedral virus capsids within the linear-response regime. In this model the capsid is represented as an interacting many-body system whose composing elements are capsid subunits (capsomers), which are treated as three-dimensional rigid bodies. The total interaction potential energy is written as a sum of pairwise capsomer-capsomer interactions. Based on previous work [Gomez Llorente et al., Soft Matter, 2014, 10, 3560], a minimal and complete anisotropic binary interaction that includes a full Hessian matrix of independent force constants is proposed. In this interaction model, capsomers have rotational symmetry around an axis of order n > 2. The full coarse-grained model is applied to analyse the low-frequency normal-mode spectrum of icosahedral T = 1 capsids. The model performance is evaluated by fitting its predicted spectrum to the full-atom results for the Satellite Tobacco Necrosis Virus (STNV) capsid [Dykeman and Sankey, Phys. Rev. Lett., 2008, 100, 028101]. Two capsomer choices that are compatible with the capsid icosahedral symmetry are checked, namely pentamers (n = 5) and trimers (n = 3). Both subunit types provide fair fits, from which the magnitude of the coarse-grained force constants for a real virus is obtained. The model is able to uncover latent instabilities whose analysis is fully consistent with the current knowledge about the STNV capsid, which does not self-assemble in the absence of RNA and is thermally unstable. The straightforward generalisability of the model beyond the linear regime and its completeness make it a promising tool to theoretically interpret many experimental data such as those provided by the atomic force microscopy or even to better understand processes far from equilibrium such as the capsid self-assembly.

MODE-TASK: large-scale protein motion tools

Summary

MODE-TASK, a novel and versatile software suite, comprises Principal Component Analysis, Multidimensional Scaling, and t-Distributed Stochastic Neighbor Embedding techniques using Molecular Dynamics trajectories. MODE-TASK also includes a Normal Mode Analysis tool based on Anisotropic Network Model so as to provide a variety of ways to analyse and compare large-scale motions of protein complexes for which long MD simulations are prohibitive. Beside the command line function, a GUI has been developed as a PyMOL plugin.

Availability and implementation

MODE-TASK is open source, and available for download from https://github.com/RUBi-ZA/MODE-TASK. It is implemented in Python and C++. It is compatible with Python 2.x and Python 3.x and can be installed by Conda.

Supplementary information

Supplementary data are available at Bioinformatics online.

Unraveling the Motions behind Enterovirus 71 Uncoating

Enterovirus 71 can be a severe pathogen in small children and immunocompromised adults. Virus uncoating is a critical step in the infection of the host cell; however, the mechanisms that control this process remain poorly understood. We applied normal mode analysis and perturbation response scanning to several complexes of the virus capsid and present a coarse-graining approach to analyze the full capsid. We show that our method offers an alternative to expressing the system as a set of rigid blocks and accounts for the interconnection between nodes within each subunit and protein interfaces across the capsid. In our coarse-grained approach, the modes associated with capsid expansion are captured in the first three nondegenerate modes and correspond to the changes observed in structural studies of the virus. We show that the resolution of the analysis may be modified without losing information on the global motions leading to uncoating. Perturbation response scanning revealed that a protomer cannot serve as a functional unit to explain deformations of the capsid. Instead, we define a pentamer as the minimum functional unit to investigate changes within the capsid. From the modal analysis and perturbation response scanning, we locate a hotspot region surrounding the fivefold axis. The range of the effect of these single, hotspot residues extend to 140 Å. The perturbation of internal capsid residues in this region displayed greatest propensity to capsid expansion, thus indicating the significant role that the RNA genome may play in triggering uncoating.

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.

TensorCalculator: exploring the evolution of mechanical stress in the CCMV capsid

A new computational methodology for the accurate numerical calculation of the Cauchy stress tensor, stress invariants, principal stress components, von Mises and Tresca tensors is developed. The methodology is based on the atomic stress approach which permits the calculation of stress tensors, widely used in continuum mechanics modeling of materials properties, using the output from the MD simulations of discrete atomic and [Formula: see text]-based coarse-grained structural models of biological particles. The methodology mapped into the software package TensorCalculator was successfully applied to the empty cowpea chlorotic mottle virus (CCMV) shell to explore the evolution of mechanical stress in this mechanically-tested specific example of a soft virus capsid. We found an inhomogeneous stress distribution in various portions of the CCMV structure and stress transfer from one portion of the virus structure to another, which also points to the importance of entropic effects, often ignored in finite element analysis and elastic network modeling. We formulate a criterion for elastic deformation using the first principal stress components. Furthermore, we show that von Mises and Tresca stress tensors can be used to predict the onset of a viral capsid's mechanical failure, which leads to total structural collapse. TensorCalculator can be used to study stress evolution and dynamics of defects in viral capsids and other large-size protein assemblies.

Normal mode analysis of Zika virus

In recent years, Zika virus (ZIKV) caused a new pandemic due to its rapid spread and close relationship with microcephaly. As a result, ZIKV has become an obvious global health concern. Information about the fundamental viral features or the biological process of infection remains limited, despite considerable efforts. Meanwhile, the icosahedral shell structure of the mature ZIKV was recently revealed by cryo-electron microscopy. This structural information enabled us to simulate ZIKV. In this study, we analyzed the dynamic properties of ZIKV through simulation from the mechanical viewpoint. We performed normal mode analysis (NMA) for a dimeric structure of ZIKV consisting of the envelope proteins and the membrane proteins as a unit structure. By analyzing low-frequency normal modes, we captured intrinsic vibrational motions and defined basic vibrational properties of the unit structure. Moreover, we also simulated the entire shell structure of ZIKV at the reduced computational cost, similar to the case of the unit structure, by utilizing its icosahedral symmetry. From the NMA results, we can not only comprehend the putative dynamic fluctuations of ZIKV but also verify previous inference such that highly mobile glycosylation sites would play an important role in ZIKV. Consequently, this theoretical study is expected to give us an insight on the underlying biological functions and infection mechanism of ZIKV.

Static and dynamic hidden symmetries of icosahedral viral capsids

Viral shells self-assemble from identical proteins, which tend to form equivalent environments in the resulting assembly. However, in icosahedral capsids containing more than 60 proteins, they are enforced to occupy not only the symmetrically equivalent locations but also the quasi-equivalent ones. Due to this important fact, static and dynamic symmetries of viral shells can include additional hidden components. Here, developing the Caspar and Klug ideas concerning the quasi-equivalence of protein environments, we derive the simplest hexagonal tilings, that in principle could correspond to the local protein order in viral shells, and apply the resulting theory to nucleocytoplasmic large dsDNA viruses. In addition, analyzing the dynamic symmetry of the P22 viral shell, we demonstrate that the collective critical modes responsible for the protein reorganization during the procapsid maturation are approximately equivalent to the normal modes of the isotropic spherical membrane with O(3) symmetry. Furthermore, we establish the relationship between the dynamic symmetry of the P22 procapsid and the protein arrangement regularities that appear only in the mature capsid.

Modeling global changes induced by local perturbations to the HIV-1 capsid

The HIV-1 capsid is a conical protein shell made up of hexamers and pentamers of the capsid protein. The capsid houses the viral genome and replication machinery, and its opening, or uncoating, within the host cell marks a critical step in the HIV-1 lifecycle. Binding of host factors such as TRIM5α and cyclophilin A (CypA) can alter the capsid's stability, accelerating or delaying the onset of uncoating and disrupting infectivity. We employ coarse-grained computational modeling to investigate the effects of point mutations and host factor binding on HIV-1 capsid stability. We find that the largest fluctuations occur in the low-curvature regions of the capsid, and that its structural dynamics are affected by perturbations at the inter-hexamer interfaces and near the CypA binding loop, suggesting roles for these features in capsid stability. Our models show that linking capsid proteins across hexamers attenuates vibration in the low-curvature regions of the capsid, but that linking within hexamers does not. These results indicate a possible mechanism through which CypA binding alters capsid stability and highlight the utility of coarse-grained network modeling for understanding capsid mechanics.

Fluctuating Nonlinear Spring Model of Mechanical Deformation of Biological Particles

The mechanical properties of virus capsids correlate with local conformational dynamics in the capsid structure. They also reflect the required stability needed to withstand high internal pressures generated upon genome loading and contribute to the success of important events in viral infectivity, such as capsid maturation, genome uncoating and receptor binding. The mechanical properties of biological nanoparticles are often determined from monitoring their dynamic deformations in Atomic Force Microscopy nanoindentation experiments; but a comprehensive theory describing the full range of observed deformation behaviors has not previously been described. We present a new theory for modeling dynamic deformations of biological nanoparticles, which considers the non-linear Hertzian deformation, resulting from an indenter-particle physical contact, and the bending of curved elements (beams) modeling the particle structure. The beams’ deformation beyond the critical point triggers a dynamic transition of the particle to the collapsed state. This extreme event is accompanied by a catastrophic force drop as observed in the experimental or simulated force (F)-deformation (X) spectra. The theory interprets fine features of the spectra, including the nonlinear components of the FX-curves, in terms of the Young’s moduli for Hertzian and bending deformations, and the structural damage dependent beams’ survival probability, in terms of the maximum strength and the cooperativity parameter. The theory is exemplified by successfully describing the deformation dynamics of natural nanoparticles through comparing theoretical curves with experimental force-deformation spectra for several virus particles. This approach provides a comprehensive description of the dynamic structural transitions in biological and artificial nanoparticles, which is essential for their optimal use in nanotechnology and nanomedicine applications.

Dynamic force experiments, which have become available to explore the physical properties of biological assemblies, oftentimes reveal results that are difficult to understand without theoretical framework. We employed a multiscale modeling approach—a combination of Molecular Dynamics simulations of atomic structures with Langevin simulations of coarse-grained models of virus shells—to characterize the degrees of freedom defining the deformation and structural collapse of biological particles tested mechanically. This enabled us to develop an analytical model that provides meaningful interpretation of force-deformation spectra available from single-particle nanoindentation experiments. The Fluctuating Nonlinear Spring (FNS) model of uniaxial particle’s deformation captures essential features of the force-deformation spectra as observed in nanomanipulations in vitro and in silico: initial non-linearity, then a subsequent force decrease transition due to structural collapse. Our theory uniquely combines the elements of continuum mechanics with the statistics of extremes, enabling one to gather mechanical and statistical characteristics of nanoparticles, which determine the Hertzian deformation of the particle’s protein layer, and bending deformation and structural damage to the particle structure. We have demonstrated how the FNS theory can accurately model the deformation of several viral shells, showing promising model applications for describing a variety of natural and synthetic nanoparticles.

Viruses as nanoparticles: structure versus collective dynamics

In order to test the application of the "nanoparticle" concept to viruses in terms of low-frequency dynamics, large viruses (140-190 nm) were compared to similar-sized polymer colloids using ultra-small-angle x-ray scattering and very-low-frequency Raman or Brillouin scattering. While both viruses and polymer colloids show comparable highly defined morphologies, with comparable abilities of forming self-assembled structures, their respective abilities to confine detectable acoustic vibrations, as expected for such monodisperse systems, differed. Possible reasons for these different behaviors are discussed.

The lumazine synthase/riboflavin synthase complex: shapes and functions of a highly variable enzyme system

The xylene ring of riboflavin (vitamin B2 ) is assembled from two molecules of 3,4-dihydroxy-2-butanone 4-phosphate by a mechanistically complex process that is jointly catalyzed by lumazine synthase and riboflavin synthase. In Bacillaceae, these enzymes form a structurally unique complex comprising an icosahedral shell of 60 lumazine synthase subunits and a core of three riboflavin synthase subunits, whereas many other bacteria have empty lumazine synthase capsids, fungi, Archaea and some eubacteria have pentameric lumazine synthases, and the riboflavin synthases of Archaea are paralogs of lumazine synthase. The structures of the molecular ensembles have been studied in considerable detail by X-ray crystallography, X-ray small-angle scattering and electron microscopy. However, certain mechanistic aspects remain unknown. Surprisingly, the quaternary structure of the icosahedral β subunit capsids undergoes drastic changes, resulting in formation of large, quasi-spherical capsids; this process is modulated by sequence mutations. The occurrence of large shells consisting of 180 or more lumazine synthase subunits has recently generated interest for protein engineering topics, particularly the construction of encapsulation systems.

PIM: phase integrated method for normal mode analysis of biomolecules in a crystalline environment

In this study, a normal mode analysis, named phase integrated method (PIM), is developed for computing modes of biomolecules in a crystalline environment. PIM can calculate low-frequency modes on one or a few asymmetric units (AUs) and generate exact modes of a whole unit cell according to space group symmetry, while the translational symmetry between unit cells is maintained via the periodic boundary condition. Therefore, the method can dramatically reduce computational cost in mode calculation in the presence of crystal symmetry. PIM also has an option to map modes onto a single AU to form an orthonormalized mode set, which can be directly applied to normal-mode-based thermal parameter refinement in X-ray crystallography. The performance of PIM was tested on all 65 space groups available in protein crystals (one protein for each space group) and on another set of 83 ultra-high-resolution X-ray structures. The results showed that considering space group symmetry in mode calculation is crucial for accurately describing vibrational motion in a crystalline environment. Moreover, the optimal inter-AU packing stiffness was found to be about 60% of that of intra-AU interactions (non-bonded interaction only).

Integrin and defensin modulate the mechanical properties of adenovirus

The propensity for capsid disassembly and uncoating of human adenovirus is modulated by interactions with host cell molecules like integrins and alpha defensins. Here, we use atomic force microscopy (AFM) nanoindentation to elucidate, at the single-particle level, the mechanism by which binding of these host molecules affects virus particle elasticity. Our results demonstrate the direct link between integrin or defensin binding and the mechanical properties of the virus. We show that the structure and geometry of adenovirus result in an anisotropic elastic response that relates to icosahedral symmetry. This elastic response changes upon binding host molecules. Whereas integrin binding softens the vertex regions, binding of a human alpha defensin has exactly the opposite effect. Our results reveal that the ability of these host molecules to influence adenovirus disassembly correlates with a direct effect on the elastic strength of the penton region. Host factors that influence adenovirus infectivity thus modulate the elastic properties of the capsid. Our findings reveal a direct link between virus-host interactions and capsid mechanics.

fSUB: normal mode analysis with flexible substructures

In this paper, we report a novel normal-mode analysis for supramolecular complexes, named fSUB. The method models a complex as a group of flexible substructures. The low-frequency substructure modes are first determined with substructures in isolation, and the motions of the whole complex are then calculated on the basis of substructure modes and substructure-substructure interactions. The calculation of modes in fSUB requires modal analysis without initial energy minimization, which is essential for maintaining energetic and structural consistency between substructures and whole complex. Compared with other coarse-grained methods, such as the RTB method, fSUB delivers much more accurate modes for the complex and allows for the choice of much larger substructures. The method can also accommodate any type of substructure arrangement including covalent bonds across the interface. In tests on molecular chaperonin GroEL (7350 residues) and HK97 capsid complex (118,092 residues), fSUB was shown to be much more efficient in terms of combined accuracy and demand of computing resources. Our results clearly demonstrated the vital importance of including substructure flexibility in complex modal analysis, as the deformational patterns of substructures were found to play an important role even in the lowest frequency modes of the whole complex.

On the morphology of viral capsids: elastic properties and buckling transitions

The morphology of icosahedral viruses ranges from highly spherical to highly faceted, and for some viruses a shape transition occurs during the viral life cycle. This phenomena is predicted from continuum elasticity, via the buckling transition theory by Nelson (Phys. Rev. E 2003, 68, 051910), in which the shape is dependent on the Foppl-von Kármán number (γ), which is a ratio of the two-dimensional Young's modulus (Y) and the bending modulus (κ). However, until now, no direct calculations have been performed on atomic-level capsid structures to test the predictions of the theory. In this study, we employ a previously described multiscale method by May and Brooks (Phys. Rev. Lett. 2011, 106, 188101) to calculate Y and κ for the bacteriophage HK97, which undergoes a spherical to faceted transition during its viral life cycle. We observe a change in γ consistent with the buckling transition theory and also a significant reduction in κ, which facilitates formation of the faceted state. We go on to examine many capsids from the T = 3 and 7 classes using only elastic network models, which allows us to calculate the ratio Y/κ, without the expense of all-atom molecular dynamics. We observe for the T = 7 capsids, there is strong correlation between the shape of the capsid and γ; however, there is no such correlation for the smaller T = 3 viruses.

Mechanics of bacteriophage maturation

Capsid maturation with large-scale subunit reorganization occurs in virtually all viruses that use a motor to package nucleic acid into preformed particles. A variety of ensemble studies indicate that the particles gain greater stability during this process, however, it is unknown which material properties of the fragile procapsids change. Using Atomic Force Microscopy-based nano-indentation, we study the development of the mechanical properties during maturation of bacteriophage HK97, a λ-like phage of which the maturation-induced morphological changes are well described. We show that mechanical stabilization and strengthening occurs in three independent ways: (i) an increase of the Young's modulus, (ii) a strong rise of the capsid's ultimate strength, and (iii) a growth of the resistance against material fatigue. The Young's modulus of immature and mature capsids, as determined from thin shell theory, fit with the values calculated using a new multiscale simulation approach. This multiscale calculation shows that the increase in Young's modulus isn't dependent on the crosslinking between capsomers. In contrast, the ultimate strength of the capsids does increase even when a limited number of cross-links are formed while full crosslinking appears to protect the shell against material fatigue. Compared to phage λ, the covalent crosslinking at the icosahedral and quasi threefold axes of HK97 yields a mechanically more robust particle than the addition of the gpD protein during maturation of phage λ. These results corroborate the expected increase in capsid stability and strength during maturation, however in an unexpected intricate way, underlining the complex structure of these self-assembling nanocontainers.

The effects of rigid motions on elastic network model force constants

Elastic network models provide an efficient way to quickly calculate protein global dynamics from experimentally determined structures. The model's single parameter, its force constant, determines the physical extent of equilibrium fluctuations. The values of force constants can be calculated by fitting to experimental data, but the results depend on the type of experimental data used. Here, we investigate the differences between calculated values of force constants and data from NMR and X-ray structures. We find that X-ray B factors carry the signature of rigid-body motions, to the extent that B factors can be almost entirely accounted for by rigid motions alone. When fitting to more refined anisotropic temperature factors, the contributions of rigid motions are significantly reduced, indicating that the large contribution of rigid motions to B factors is a result of over-fitting. No correlation is found between force constants fit to NMR data and those fit to X-ray data, possibly due to the inability of NMR data to accurately capture protein dynamics.

Virtual interface substructure synthesis method for normal mode analysis of super-large molecular complexes at atomic resolution

Normal mode analysis of large biomolecular complexes at atomic resolution remains challenging in computational structure biology due to the requirement of large amount of memory space and central processing unit time. In this paper, we present a method called virtual interface substructure synthesis method or VISSM to calculate approximate normal modes of large biomolecular complexes at atomic resolution. VISSM introduces the subunit interfaces as independent substructures that join contacting molecules so as to keep the integrity of the system. Compared with other approximate methods, VISSM delivers atomic modes with no need of a coarse-graining-then-projection procedure. The method was examined for 54 protein-complexes with the conventional all-atom normal mode analysis using CHARMM simulation program and the overlap of the first 100 low-frequency modes is greater than 0.7 for 49 complexes, indicating its accuracy and reliability. We then applied VISSM to the satellite panicum mosaic virus (SPMV, 78,300 atoms) and to F-actin filament structures of up to 39-mer, 228,813 atoms and found that VISSM calculations capture functionally important conformational changes accessible to these structures at atomic resolution. Our results support the idea that the dynamics of a large biomolecular complex might be understood based on the motions of its component subunits and the way in which subunits bind one another.

Viral capsid equilibrium dynamics reveals nonuniform elastic properties

The long wavelength, low-frequency modes of motion are the relevant motions for understanding the continuum mechanical properties of biomolecules. By examining these low-frequency modes, in the context of a spherical harmonic basis set, we identify four elastic moduli that are required to describe the two-dimensional elastic behavior of capsids. This is in contrast to previous modeling and theoretical studies on elastic shells, which use only the two-dimensional Young's modulus (Y) and the bending modulus (κ) to describe the system. Presumably, the heterogeneity of the structure and the anisotropy of the biomolecular interactions lead to a deviation from the homogeneous, isotropic, linear elastic shell theory. We assign functional relevance of the various moduli governing different deformation modes, including a mode primarily sensed in atomic force microscopy nanoindentation experiments. We have performed our analysis on the T = 3 cowpea chlorotic mottle virus and our estimate for the nanoindentation modulus is in accord with experimental measurements.

Determination of viral capsid elastic properties from equilibrium thermal fluctuations

We apply two-dimensional elasticity theory to viral capsids to develop a framework for calculating elastic properties of viruses from equilibrium thermal fluctuations of the capsid surface in molecular dynamics and elastic network model trajectories. We show that the magnitudes of the long wavelength modes of motion available in a simulation with all atomic degrees of freedom are recapitulated by an elastic network model. For the mode spectra to match, the elastic network model must be scaled appropriately by a factor which can be determined from an icosahedrally constrained all-atom simulation. With this method we calculate the two-dimensional Young's modulus Y, bending modulus κ, and Föppl-von Kármán number γ, for the T=1 mutant of the Sesbania mosaic virus. The values determined are in the range of previous theoretical estimates.

Normal mode analysis and applications in biological physics

Normal mode analysis has become a popular and often used theoretical tool in the study of functional motions in enzymes, viruses, and large protein assemblies. The use of normal modes in the study of these motions is often extremely fruitful since many of the functional motions of large proteins can be described using just a few normal modes which are intimately related to the overall structure of the protein. In this review, we present a broad overview of several popular methods used in the study of normal modes in biological physics including continuum elastic theory, the elastic network model, and a new all-atom method, recently developed, which is capable of computing a subset of the low frequency vibrational modes exactly. After a review of the various methods, we present several examples of applications of normal modes in the study of functional motions, with an emphasis on viral capsids.

Mechanism of cohesin loading onto chromosomes: a conformational dynamics study

The structure-function relationship of cohesin, an essential chromosome maintenance protein, is investigated by analyzing its collective dynamics and conformational flexibility, enhancing our understanding of the sister chromatid cohesion process. A three-dimensional model of cohesin has been constructed by homology modeling using both crystallographic and electron microscopy image data. The harmonic dynamics of the cohesin structure are calculated with a coarse-grained elastic network model. The model shows that the bending motion of the cohesin ring is able to adopt a head-to-tail conformation, in agreement with experimental data. Low-frequency conformational changes are observed to deform the highly conserved glycine residues at the interface of the cohesin heterodimer. Normal mode analysis further reveals that, near large globular structures such as nucleosome and accessory proteins docked to cohesin, the mobility of the coiled-coil regions is notably affected. Moreover, fully solvated molecular dynamics calculations, performed specifically on the hinge region, indicate that hinge opening starts from one side of the dimerization interface, and is coordinated by highly conserved glycine residues.

Atomistic modeling of the low-frequency mechanical modes and Raman spectra of icosahedral virus capsids

We describe a technique for calculating the low-frequency mechanical modes and frequencies of a large symmetric biological molecule where the eigenvectors of the Hessian matrix are determined with full atomic detail. The method, which follows order N methods used in electronic structure theory, determines the subset of lowest-frequency modes while using group theory to reduce the complexity of the problem. We apply the method to three icosahedral viruses of various T numbers and sizes; the human viruses polio and hepatitis B, and the cowpea chlorotic mottle virus, a plant virus. From the normal-mode eigenvectors, we use a bond polarizability model to predict a low-frequency Raman scattering profile for the viruses. The full atomic detail in the displacement patterns combined with an empirical potential-energy model allows a comparison of the fully atomic normal modes with elastic network models and normal-mode analysis with only dihedral degrees of freedom. We find that coarse-graining normal-mode analysis (particularly the elastic network model) can predict the displacement patterns for the first few (approximately 10) low-frequency modes that are global and cooperative.