Chandrajit Bajaj Geometric Modeling and Quantitative Visualization of Virus Ultra-structure Chandrajit Bajaj Department of Computer Science & Institute of Computational Engineering and Sciences, Center for Computational Visualization, University of Texas at Austin, 201 East 24th Street, ACES 2.324A, Austin, TX 78712-0027 Phone: +1 512-471-8870, Fax: +1 512-471-0982 Email : [email protected] Acknowledgements: This work was supported in part by NSF-ITR grant EIA-0325550, and grants from the NIH 0P20 RR020647, R01 GM074258 and R01 GM073087. Chap. 7 in “ Modeling Biology: Structures, Behaviors, Evolution”, MIT Press 2006, Ed. Manfred Laubichler , Gerd Mueller11. Introduction Viruses are one of the smallest parasitic nano-objects that are agents of human disease [White and Fenner 1994]. They have no systems for translating RNA, ATP generation, or protein, nucleic acid synthesis, and therefore need the subsystems of a host cell to sustain and replicate [White and Fenner 1994]. It would be natural to classify these parasites according to their eukaryotic or prokaryotic cellular hosts (e.g. plant, animal, bacteria, fungi, etc.), however there do exist viruses which have more than one sustaining host species [White and Fenner 1994]. Currently, viruses are classified simultaneously via the host species(Algae, Archae, Bacteria, Fungi, Invetebrates, Mycoplasma, Plants, Protozoa, Spiroplasma, Vetebrates), the host tissues that are infected, the method of virial transmission, the genetic organization of the virus (single or double stranded, linear or circular, RNA or DNA), the protein arrangement of the protective closed coats housing the genome (helical, icosahedral symmetric nucleo-capsids), and whether the virus capsids additionally have a further outer envelope covering (the complete virion)[White and Fenner 1994]. Table 1 summarizes a small yet diverse collection of viruses and virions [ICTV Database]. The focus of this article is on the computational geometric modeling and visualization of the nucleo-capsid ultrastructure of2plant and animal viruses exhibiting the diversity and geometric elegance of the multiple protein arrangements. Additionally, one computes a regression relationship between surface area v.s. enclosed volume for spherical viruses with icosahedral symmetric protein arrangements. The computer modeling and quantitative techniques for virus capsid shells ultra-structure that we review here are applicable for atomistic, high resolution (less than 4 A) model data, as well as medium (5 Ǻ to 15 Ǻ) resolution map data reconstructed from cryo-electron microscopy. 2. The Morphology of Virus Structures Minimally viruses consist of a single nucleocapsid made of proteins for protecting their genome, as well as in facilitating cell attachment and entry. The capsid proteins magically self-assemble, into often a helical or icosahedral symmetric shell (henceforth referred to as capsid shells). There do exist several examples of capsid shells which do not exhibit any global symmetry [ICTV Database], however we focus on only the symmetric capsid shells in the remainder of this article. Different virus morphologies that are known, (a small sampling included in Table 1) are distinguished by3optional additional outer capsid shells, the presence or lack of a surrounding envelope for these capsid shells (derived often from the host cell’s organelle membranes), as well as additional proteins within these optional capsids and envelopes, that are necessary for the virus lifecycle. The complete package of proteins, nucleic acids and envelopes is often termed a virion. Fig. 2.1. Organization of Helical Viruses The asymmetric structural subunit of a symmetric capsid shell may be further decomposable into simpler and smaller protein structure units termed protomers. Protomers could be a single protein in monomeric form (example TMV), or form homogeneous dimeric or trimeric structure units (example RDV). These structure units also often combine to form symmetric clusters, called capsomers, and are predominantly distinguishable in visualizations at even medium and low resolution virus structures. The capsomers and/or protomeric structure units pack to create the capsid shell in the form of either helical or icosahedral symmetric arrangements. Fig. 2.2. Organization of Icosahedral Viruses4The subsequent sub-sections dwell on the geometry of the individual protomers, and capsomers, as part of a hierarchical arrangement of symmetric capsid shells. 2.1 The Geometry of Helical Capsid Shells Helical symmetry can be captured by a 4 x 4 matrix transformation ),,( LaHφr parameterized by),,(zyxaaaa=r, a unit vector along the helical axis, byθ, an angle in the plane of rotation, and by the pitch L, the axial rise for a complete circular turn. ⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎣⎡+−+−−−−−+−+−+−−−+−=10002cos)cos1(sin)cos1(sin)cos1(2sin)cos1(cos)cos1(sin)cos1(2sin)cos1(sin)cos1(cos)cos1(222),,(πθθθθθθθπθθθθθθθπθθθθθθθφLaaaaaaaaLaaaaaaaaLaaaaaaaaHzzxzyyzxyxzyyzyxxyzxzyxxLar If P is the center of any atom of the protomer, then 'Pis the transformed center, andPHP *'=. Repeatedly applying this transformation to all atoms in a protomer yields a helical stack of protomeric units. The desired length of the helical nucleo-capsid shell5is typically determined by the length of the enclosed nucleic acids. The capsid shell of the tobacco mosaic virus (TMV) exhibits helical symmetry (Fig. 2.1, and 2.3), with the asymmetric protein structure unit or the protomer consisting of a single protein (pdb id 1EI7) Fig. 2.3 Helical Symmetry Axis 2.2 The Geometry of Icosahedral Capsid Shells In numerous cases the virus structure is icosahedrally symmetric. The advantage over the helical symmetry structure is the efficient construction of a capsid of a given size using the smallest protein subunits. An icosahedron has 12 vertices, 20 equilateral triangular faces, and 30 edges, and exhibits 5:3:2 symmetry. A 5-fold symmetry axis passes through each vertex, a 3-fold symmetry axis through the center of each face, and a 2-fold axis through the midpoint of each edge (see Fig. 2.4). Fig. 2.4 Icosahedral Symmetries and Axes6A rotation transformation around an axis ),,(zyxaaaa =r by an angle θ is described by the 4x4 matrix
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