ABSTRACT
CONTENTS 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 11.2 Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
11.2.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 11.2.2 Calculation Details. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
11.3 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 11.3.1 (Dialanine)60 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 11.3.2 Poliovirus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 11.3.3 Rhinovirus and CCMV.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
11.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
In the past 20 years, many atomic structures of icosahedral viruses have become available. The VIPER database [1] currently lists 74 x-ray structures and 11 cryo-electron microscopy (cryo-EM) structures, corresponding to 65 different virus types and additional strain variations. In most structures only the virus protein capsid is present, whereas the RNA (or DNA) that is encapsulated by the capsid is not resolved. The icosahedral virus capsid consists of 60 or more identical subunits, called protomers, which in turn are made up of a number of identical or nonidentical proteins often called viral proteins (VPs). Computational analyses of virus capsid structures have included molecu-
lar dynamics (MD) calculations to study the binding of small molecules [2, 3],
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Poisson-Boltzmann calculations to study the structural basis of pH sensitivity [4], and normal mode (NM) calculations to study swelling pathways [5]. Because of the large size of these viruses all these calculations were simplified to reduce the complexity. The first MD studies were restricted to fully symmetric motions in which only one protomer is simulated and neighboring protomers are copies generated by icosahedral symmetry operations [2]. A different series of MD studies by Post and coworkers focused on a small molecule ligand binding site in rhinovirus in which only residues in a spherical region around the binding site were allowed to move [3,6-8]. The study of the pH-dependent stability of foot-and-mouth disease virus was reduced in complexity by calculating the electrostatic interactions between two protomers only [4]. TwoNMstudies on icosahedral viruses have been published, both of which included a full virus capsid. These, otherwise extremely large calculations, were greatly simplified by allowing the individual VPs to move as rigid bodies only [5], or by allowing symmetric motions only [9]. All these simplifications significantly limit the relevance of the results with
respect to the full virus capsid. To allow only fully symmetric motions [2, 9] the virus only explores a small fraction
) of its available phase space,
as the motion of a single protomer determines the motion of the complete system in this approach. Focusing on a small flexible region around a binding site within a large fixed structure [3] allows the simulation of local effects, but if the property to be studied involves larger scale motions, the results are not representative of the full virus. To use the interaction between two protomers as a model for all interprotomer interactions in the system, [4] ignores potentially significant multibody effects. The previously published NM studies on icosahedral viruses analyzed
conformational changes of cowpea chlorotic mottle virus (CCMV) [5] and bacteriophage HK97 [9], respectively. For CCMV, the x-ray structure of the native formwas used to calculate NMs of the virus. TheNMswere compared to the cryo-EM structure of the swollen form of CCMV to propose potential pathwaysof virus swelling. TheHK97 study calculated the symmetricNMsof a simplified atomic representation to predict a pathway of virus maturation. As mentioned earlier, both NM studies required significant simplifications to enable the calculations to be done with currently available computer power. Themotions allowed in the CCMV studywere rigid bodymotions of the individual VPs, resulting in dynamics analogous to tectonic plate motions. It is likely that internal flexibility of the VPs plays a significant role in large-scale motions of the system, and thus it would be desirable to be able to include more of the natural degrees of freedom in a NM calculation. The HK97 study only included fully symmetric motions, which constitute less than 2% of the motions of the system, and a full treatment of all motions would be a more complete representation of the dynamics of the system. The symmetry of icosahedral viruses enables a great simplification of the
NM calculation without any loss of accuracy. The approach uses the fact that the matrix of second derivatives of the potential energy (the Hessian) of a symmetric system is inherently symmetric [10]. By usingmethods developed
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in group theory the Hessian is expressed in “symmetry coordinates,” which results in the Hessian becoming block-diagonal. Because of this transformation, the size of the Hessian to be diagonalized
is reduced from 60N × 60N (where N is the number of degrees of freedom per protomer) to five blocks of sizes 5N × 5N, 4N × 4N, 3N × 3N, 2N × 2N, and N × N, respectively. Since the memory use of a NM calculation scales as N2 and CPU time scales approximately as N3 [11], the symmetry method enables a large increase in N without sacrificing any accuracy. The approach is completely general and any protomer basis set can be used as long as it can be expressed in Cartesian coordinate displacements. These include, for instance, the Cα-only basis set used in the elastic network model [12, 13], the rigid block basis sets of the rotation-translation block (RTB) model [14], and internal coordinate basis sets such as a dihedral basis set. The symmetry coordinate method has been applied to systems with cir-
cular symmetry such as the gramicidin channel dimer [15] and the tobacco mosaic virus protein disk (17-mer) [16]. For systems with icosahedral symmetry themethodhasbeenapplied tobuckminsterfullerene [17, 18], as aproof of concept to an artificial (Dialanine)60 system [19], and recently to icosahedral virus capsids [20, 21]. Here we describe the results of the NM calculations of the (Dialanine)60
system and several icosahedral virus capsids. The (Dialanine)60 system is a useful proof of concept calculation since it contains all types of intra-and intermolecular forces that are present in much larger icosahedral viruses, and because it allows a direct comparison between a full regular NM calculation and a calculation using symmetry coordinates. The calculations of icosahedral virus capsids focus on general NM properties such as frequency spectra, calculated fluctuations, and displacements of individual NMs. The symmetry method has great potential in the study of dynamic icosahedral virus properties such as large-scale conformational changes associated with cell entry, reversible swelling, or viral maturation. Another application is the study of the stabilizing effects of small molecule binding to the capsid conformation, which is thought to be entropic in nature [8, 22]. In addition, it has recently been shown that NMs can be used to improve the accuracy of cryo-EM structures of icosahedral viruses [23].
