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.     

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.     

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.     

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 soft-modeling approach to interpret thermodynamic and conformational transitions of polynucleotides

Multivariate outputs from the experimental monitoring of biochemical processes are usually difficult to interpret applying methods based on a priori chemical models. Curve resolution methods are model-free procedures, generally known as soft-modeling methods, which obtain the concentration profiles and instrumental responses of each individual species involved in a multivariate monitored process without making any kind of external assumption. Of the curve resolution methods available, the alternating least squares (ALS) is proposed here because of its ability to operate on one or on several matrices. Furthermore, ALS allows the introduction of information related to the internal data structure and to the general features of the concentration profiles and instrumental responses through the input of suitable constraints in the iterative resolution procedure. The ALS potential is tested on several data sets coming from the multivariate spectrometric monitoring of polyuridylic (polyU), polycytidylic (polyC), and polyadenylic (polyA) protonation equilibria in dioxane/water 30% (v/v). Information concerning the evolution of the concentration profiles and the spectra of each individual species involved in the acid-base equilibria, the presence and pattern of polyelectrolyte effects, and the presence of conformational transitions associated or not with the proton uptake process is presented.     

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.      

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.     

Ligand binding and structural perturbations in cytochrome c peroxidase. A crystallographic study

Crystal structures of the complexes formed between cytochrome c peroxidase and cyanide, nitric oxide, carbon monoxide, and fluoride have been determined and refined to 1.85 A. In all four complexes significant changes occur in the distal heme pocket due to movement of Arg-48, His-52, and a rearrangement of active site water molecules. In the cyanide, nitric oxide, and carbon monoxide complexes, Arg-48 moves away from the ligand while in the fluoride complex Arg-48 moves in toward the ligand to form a hydrogen bond or ion pair with the fluoride. More subtle changes occur on the proximal side of the heme. In an earlier study at lower resolution (Edwards, S. L., Kraut, J., and Poulos, T. L. (1988) Biochemistry 27, 8074-8081), we found that nitric oxide binding causes perturbations in the proximal domain involving Trp-191 which has been confirmed by the present study. Trp-191 is stacked parallel to and in contact with the proximal ligand, His-175. Nitric oxide binding results in a slight movement of Trp-191 away from His-175 and a large increase in crystallographic temperature factors indicating increased mobility of these residues on the proximal side of the heme. These proximal-side changes are unique to nitric oxide and are not related strictly to spin-state or oxidation state of the iron atom since similar changes were not observed in the cyanide (low-spin ferric), carbon monoxide (low-spin ferrous), or fluoride (high-spin ferric) complexes.     

Protein structural variation in computational models and crystallographic data

Normal mode analysis offers an efficient way of modeling the conformational flexibility of protein structures. We use anisotropic displacement parameters from crystallography to test the quality of prediction of both the magnitude and directionality of conformational flexibility. Normal modes from four simple elastic network model potentials and from the CHARMM force field are calculated for a data set of 83 diverse, ultrahigh-resolution crystal structures. While all five potentials provide good predictions of the magnitude of flexibility, all-atom potentials have a clear edge at prediction of directionality, and the CHARMM potential has the highest prediction quality. The low-frequency modes from different potentials are similar, but those computed from the CHARMM potential show the greatest difference from the elastic network models. The comprehensive evaluation demonstrates the costs and benefits of using normal mode potentials of varying complexity.     

Energetics of quasiequivalence: computational analysis of protein-protein interactions in icosahedral viruses.

Quaternary structure polymorphism found in quasiequivalent virus capsids provides a static framework for studying the dynamics of protein interactions. The same protein subunits are found in different structural environments within these particles, and in some cases, the molecular switching required for the polymorphic quaternary interactions is obvious from high-resolution crystallographic studies. Employing atomic resolution structures, molecular mechanics, and continuum electrostatic methods, we have computed association energies for unique subunit interfaces of three icosahedral viruses, black beetle virus, southern bean virus, and human rhinovirus 14. To quantify the chemical determinants of quasiequivalence, the energetic contributions of individual residues forming quasiequivalent interfaces were calculated and compared. The potential significance of the differences in stabilities at quasiequivalent interfaces was then explored with the combinatorial assembly approach. The analysis shows that the unique association energies computed for each virus serve as a sensitive basis set that may determine distinct intermediates and pathways of virus capsid assembly. The pathways for the quasiequivalent viruses displayed isoenergetic oligomers at specific points, suggesting that these may determine the quaternary structure polymorphism required for the assembly of a quasiequivalent particle.     

Dynamic approach to thermotropic and inotropic phase transitions

Thermotropic and inotropic phase transitions have been analysed with a dynamic theory on a self-organization. An equation of motion of a molecular assembly with strong interactions may be approximately described as: dQ/dt' congruent to -K1Q-K3Q3, where Q is a displacement from the equilibrium point Q0(identical to 0) in a vibrational state, K1 is a transition parameter. When the parameter K1 concerned with an internal driving force (partial system) changes from positive to negative through the potential bifurcation, the system transfers to a new stable state breaking down the symmetry. Such a sign change of K1 serves as a trigger to a phase transition. Using Weiss' approximation, we have evaluated the change of K1 by a function of temperature, kappa (T-TC), and have obtained the critical temperature TC of thermotropic phase transition. We have furthermore treated inotropic phase transition caused by the binding of divalent cations like Ca2+ using the function kappa (T-beta TC), where beta is a shift parameter of the critical temperature.