A general method to quantify quasi-equivalence in icosahedral viruses

A quantitative, atom-based, method is described for comparing protein subunit interfaces in icosahedral virus capsids with quasi-equivalent surface lattices. An integrated, normalized value (between 0 and 1) based on equivalent residue contacts (Q-score) is computed for every pair of subunit interactions and scores that are significantly above zero readily identify interfaces that are quasi-equivalent to each other. The method was applied to all quasi-equivalent capsid structures (T=3, 4, 7 and 13) in the Protein Data Bank and the Q-scores were interpreted in terms of their structural underpinnings. The analysis allowed classification of T=3 structures into three groups with architectures that resemble different polyhedra with icosahedral symmetry. The preference of subunits to form dimers in the T=4 human Hepatitis B virus capsid (HBV) was clearly reflected in high Q-scores of quasi-equivalent dimers. Interesting differences between the classical T=7 capsid and polyoma-like capsids were also identified. Application of the method to the outer-shell of the T=13 Blue tongue virus core (BTVC) highlighted the modest distortion between the interfaces of the general trimers and the strict trimers of VP7 subunits. Furthermore, the method identified the quasi 2-fold symmetry in the inner capsids of the BTV and reovirus cores. The results show that the Q-scores of various quasi-symmetries represent a "fingerprint" for a particular virus capsid architecture allowing particle classification into groups based on their underlying structural and geometric features.     

Complementary approaches to structure determination of icosahedral viruses

Few biological macromolecular complexes exhibit the combination of massive size and hierarchical, symmetrical architecture embodied in icosahedral viruses. X-ray crystallography, electron cryomicroscopy and small-angle X-ray scattering provide complementary approaches to studying these remarkable structures. Through a combined approach, progress has been made towards providing detailed structures of highly complex and very large viruses, and towards imaging the dynamic structural changes performed by viruses at key stages in their life cycles.     

Homological series of icosahedral viruses and fullerenes

The icosahedral (spherical, i.e., of 3?5?m and 235 point group symmetry) fullerenes and viruses are shown to form a homological series. The mathematical system of icosahedral fullerenes is, at the same time, the morphological system of icosahedral viruses. It allows the prediction of their unknown forms and recognition of structural and, possibly, genetic relationships between them.     

Mechanical deformation of spherical viruses with icosahedral symmetry

Virus capsids and crystalline surfactant vesicles are two examples of self-assembled shells in the nano- to micrometer size range. Virus capsids are particularly interesting since they have to sustain large internal pressures while encapsulating and protecting the viral DNA. We therefore study the mechanical properties of crystalline shells of icosahedral symmetry on a substrate under a uniaxial applied force by computer simulations. We predict the elastic response for small deformations, and the buckling transitions at large deformations. Both are found to depend strongly on the number of elementary building blocks N (the capsomers in the case of viral shells), the F?ppl-von Kármán number gamma (which characterizes the relative importance of shear and bending elasticity), and the confining geometry. In particular, we show that whereas large shells are well described by continuum elasticity-theory, small shells of the size of typical viral capsids behave differently already for small deformations. Our results are essential to extract quantitative information about the elastic properties of viruses and vesicles from deformation experiments.     

The familiar and the unexpected in structures of icosahedral viruses

Viruses were the first large macromolecular assemblages to be visualized at high resolution. New virus structures continue to challenge our understanding of specificity in protein-protein "recognition". The evolution of virus structures has been even more opportunistic than previously imagined.     

Near-atomic resolution reconstructions of icosahedral viruses from electron cryo-microscopy

Nine different near-atomic resolution structures of icosahedral viruses, determined by electron cryo-microscopy and published between early 2008 and late 2010, fulfil predictions made 15 years ago that single-particle cryo-EM techniques could visualize molecular detail at 3-4? resolution. This review summarizes technical developments, both in instrumentation and in computation, that have led to the new structures, which advance our understanding of virus assembly and cell entry.     

Using UHF-dielectrometry to study protein structural transitions

UHF-dielectrometry method is based on the following facts: i) there is dispersion (i.e. dependence on frequency) of the dielectric permeability epsilon; ii) bound and free water have remarkable different epsilon, mobility and dispersion regions; iii) conformational changes in a macromolecule lead to redistribution of free and bound water and to change of the amount of free water molecules. Choosing the working frequency in the region of dispersion of free water molecules (9.2 GHz) we can detect conformational changes in proteins using free water as a marker. In this work the temperature dependencies of dielectric parameters of albumin and fibrinogen solutions were obtained in the temperature interval 5-40 degrees C. In contrast to dependencies for poor solvent, temperature dependencies of dielectric parameters for protein solutions are of non-monotonous character; they have a number of peculiarities in the temperature ranges of 8-10, 22-24 and 34-36 degrees C. At these temperatures redistribution of free and bound water in protein-water system occurs due to structural changes in protein molecules. In this work the mechanism of temperature changes of spatial organisation of protein molecules was proposed. Perhaps, this mechanism is responsible for maintenance of thermal stability of the functionally active conformation of native proteins.     

A probabilistic network model for structural transitions in biomolecules

Biological macromolecules often undergo large conformational rearrangements during a functional cycle. To simulate these structural transitions with full atomic detail typically demands extensive computational resources. Moreover, it is unclear how to incorporate, in a principled way, additional experimental information that could guide the structural transition. This article develops a probabilistic model for conformational transitions in biomolecules. The model can be viewed as a network of anharmonic springs that break, if the experimental data support the rupture of bonds. Hamiltonian Monte Carlo in internal coordinates is used to infer structural transitions from experimental data, thereby sampling large conformational transitions without distorting the structure. The model is benchmarked on a large set of conformational transitions. Moreover, we demonstrate the use of the probabilistic network model for integrative modeling of macromolecular complexes based on data from crosslinking followed by mass spectrometry.     

Group theory of icosahedral virus capsid vibrations: A top-down approach

We explore the use of a top-down approach to analyse the dynamics of icosahedral virus capsids and complement the information obtained from bottom-up studies of viral vibrations available in the literature. A normal mode analysis based on protein association energies is used to study the frequency spectrum, in which we reveal a universal plateau of low-frequency modes shared by a large class of Caspar-Klug capsids. These modes break icosahedral symmetry and are potentially relevant to the genome release mechanism. We comment on the role of viral tiling theory in such dynamical considerations.     

A multievent approach to estimating pair fidelity and heterogeneity in state transitions

Fidelity rates of pair-bonded individuals are of considerable interest to behavioral and population biologists as they can influence population structure, mating rates, population productivity, and gene flow. Estimates of fidelity rates calculated from direct observations of pairs in consecutive breeding seasons may be biased because (i) individuals that are not seen are assumed to be dead, (ii) variation in the detectability of individuals is ignored, and (iii) pair status must be known with certainty. This can lead to a high proportion of observations being ignored. This approach also restricts the way variation in fidelity rates for different types of individuals, or the covariation between fidelity and other vital rates (e.g., survival) can be analyzed. In this study, we develop a probabilistic multievent capture–mark–recapture (MECMR) modeling framework for estimating pair fidelity rates that accounts for imperfect detection rates and capture heterogeneity, explicitly incorporates uncertainty in the assessment of pair status, and allows estimates of state-dependent survival and fidelity rates to be obtained simultaneously. We demonstrate the utility of our approach for investigating patterns of fidelity in pair-bonded individuals, by applying it to 30 years of breeding data from a wild population of great tits Parus major Linnaeus. Results of model selection supported state-dependent recapture, survival, and fidelity rates. Recapture rates were higher for individuals breeding with their previous partner than for those breeding with a different partner. Faithful birds that were breeding with the same partner as in the previous breeding season (i.e., at t − 1) experienced substantially higher survival rates (between t and t + 1) and were also more likely to remain faithful to their current partner (i.e., to remain in the faithful state at t + 1). First year breeders were more likely to change partner than older birds. These findings imply that traditional estimates, which do not account for state-dependent parameters, may be both inaccurate and biased, and hence, inferences based on them may conceal important biological effects. This was demonstrated in the analysis of simulated capture histories, which showed that our MECMR model was able to estimate state-dependant survival and pair fidelity rates in the face of varying state-dependant recapture rates robustly, and more accurately, than the traditional method. In addition, this new modeling approach provides a statistically rigorous framework for testing hypothesis about the causes and consequences of fidelity to a partner for natural populations. The novel modeling approach described here can readily be applied, either in its current form or via extension, to other populations and other types of dyadic interactions (e.g., between nonpaired individuals, such as parent–offspring relationships, or between individuals and locations, such as nest-site fidelity).     

Classification of single-projection reconstructions for cryo-electron microscopy data of icosahedral viruses

We present a novel strategy for classification of heterogeneous electron microscopy data of icosahedral virus particles. The effectiveness of the procedure, which is based on classification of single-projection reconstructions (SPRs), is first investigated using simulated data. Of several reconstruction approaches examined, best results were obtained with algebraic reconstruction techniques (ART) when providing prior information about the reconstruction in the form of a starting volume. The results presented indicate that SPR-classification is sufficiently sensitive to classify assemblies with differences of only a few percent of the total mass. The usefulness of this procedure is illustrated by application to a heterogeneous cryo-electron microscopy dataset of adenovirus mutant dl313, lacking minor coat protein IX. These data were successfully divided into two distinct classes, in agreement with gel analysis and immuno-electron microscopy results. The classes yielded a wildtype-like reconstruction and a reconstruction representing the polypeptide IX-deficient dl313 virion. As the largest difference between these volumes is found at the location previously assigned to the external portion of minor coat protein polypeptide IIIa, questions arise concerning the current adenovirus model.     

A novel approach to structural alignment using realistic structural and environmental information

In the era of structural genomics, it is necessary to generate accurate structural alignments in order to build good templates for homology modeling. Although a great number of structural alignment algorithms have been developed, most of them ignore intermolecular interactions during the alignment procedure. Therefore, structures in different oligomeric states are barely distinguishable, and it is very challenging to find correct alignment in coil regions. Here we present a novel approach to structural alignment using a clique finding algorithm and environmental information (SAUCE). In this approach, we build the alignment based on not only structural coordinate information but also realistic environmental information extracted from biological unit files provided by the Protein Data Bank (PDB). At first, we eliminate all environmentally unfavorable pairings of residues. Then we identify alignments in core regions via a maximal clique finding algorithm. Two extreme value distribution (EVD) form statistics have been developed to evaluate core region alignments. With an optional extension step, global alignment can be derived based on environment-based dynamic programming linking. We show that our method is able to differentiate three-dimensional structures in different oligomeric states, and is able to find flexible alignments between multidomain structures without predetermined hinge regions. The overall performance is also evaluated on a large scale by comparisons to current structural classification databases as well as to other alignment methods.     

Affine extensions of the icosahedral group with applications to the three-dimensional organisation of simple viruses

Since the seminal work of Caspar and Klug on the structure of the protein containers that encapsulate and hence protect the viral genome, it has been recognised that icosahedral symmetry is crucial for the structural organisation of viruses. In particular, icosahedral symmetry has been invoked in order to predict the surface structures of viral capsids in terms of tessellations or tilings that schematically encode the locations of the protein subunits in the capsids. Whilst this approach is capable of predicting the relative locations of the proteins in the capsids, information on their tertiary structures and the organisation of the viral genome within the capsid are inaccessible. We develop here a mathematical framework based on affine extensions of the icosahedral group that allows us to describe those aspects of the three-dimensional structure of simple viruses. This approach complements Caspar-Klug theory and provides details on virus structure that have not been accessible with previous methods, implying that icosahedral symmetry is more important for virus architecture than previously appreciated.