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# FIELD BIOLOGY BIO 208L

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This 56 page Class Notes was uploaded by Ezequiel Orn on Sunday September 6, 2015. The Class Notes belongs to BIO 208L at University of Texas at Austin taught by Staff in Fall. Since its upload, it has received 31 views. For similar materials see /class/181737/bio-208l-university-of-texas-at-austin in Biology at University of Texas at Austin.

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Chandrajit Baj aj Geometric Modeling and Quantitative Visualization of Virus Ultra structure Chandrajit Bajaj Department of Computer Science amp Institute of Computational Engineering and Sciences Center for Computational Visualization University of Texas at Austin 20l East 24th Street ACES 2324A Austin TX 78712 0027 Phone 1 512 471 8870 Fax 1 512 471 0982 Email bajajcsutexasedu Total number of actual words Total number of figures Total word equivalents Acknowledgements This work was supported in part by NSE ITR grants ACT 022003 and ETA 0325550 and grants from the NIH OPZO RR020647 and ROl GMO74258 1 Introduction Viruses are one of the smallest parasitic nano organisms that are agents of human disease 89 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 89 It would be natural to classify these parasites according to their eukaryotic or prokaryotic cellular hosts eg plant animal bacteria fungi etc however there do exist viruses which have more than one sustaining host species 89 Currently viruses are classified simultaneously via the host speciesAlgae 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 virion89 Table l summarizes a small yet diverse collection of viruses and virions 98 The focus of this article is on the computational geometric modeling and visualization of the nucleo capsid ultrastructure of plant and animal viruses 2 exhibiting the diversity and geometric elegance of the multiple protein arrangements Additionally one computes a regression relationship between surface area vs 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 A to l5 A 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 98 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 l are distinguished by 3 optional additional outer capsid shells the presence or lack of a surrounding envelope for these capsid shells derived often from the host cell s organelles 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 2l 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 anomeric 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 andor protomeric structure units pack to create the capsid shell in the form of either helical or icosahedral symmetric arrangements with a greater propensity for icosahedral symmetry Fig 22 Organization of lcosahedral Viruses 4 The subsequent sub sections dwell on the geometry of the individual protomers and capsomers as part of a hierarchical arrangement of symmetric capsid shells 21 The Geometry of Helical Capsid Shells Helical symmetry can be captured by a 21 x 4 matrix transformation fJQL L parameterized byiia ay h a unit vector along the helical axis bye an angle in the plane of rotation and by the pitch Z the axial rise for a complete circular turn af1 co co axay l co az sinH axaz co ay sinH ayLG H axay1 co az s1n6 a 1 co co ayaz co ax S1116 11M 27 axaz co ay sinH ayaz1 co ax 1116 131 co co 6 7239 0 0 0 1 If I is the center of any atom of the protomer then P is the transformed center andl1 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 shell 5 is typically determined by the length of the enclosed nucleic acids The capsid shell of the tobacco mosaic virus TMV exhibits helical symmetry G g 21L W h the asymmetric protein structure unit or the protomer consisting of a single protein pdb id lEl7 Fig 23 Helical Symmetry Axis 22 The Geometry of Icosahedral Capsid Shells More often 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 l2 vertices 2O equilateral triangular faces and 30 edges and exhibits 5322 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 23 Fig 24 lcosahedral Symmetries and Axes 6 a axaaz A rotation transformation around an axis y by an angle 9 is described by the 4x4 matrix af1 cos cos axay1 0099 azsin6 axaz1 cos aysin6 0 R axay1 cos azsin6 a1 cos cos ayaz1 0099 axsin6 0 M axaz1 0099 aysin6 ayaz1 cos ax sin6 a221 cos 0099 0 0 0 0 1 The vertices of a canonical icosahedron are given by Orlr ili 0 i 0il where lt13 l52 is the golden ratio For a 5 fold symmetry transformation around the vertex Orlr the normalized axis of rotation is a0 03952573 03985064 and 921 the angle of rotation is 5 yielding a five fold symmetry transformation matrix 030902 080902 05000 0 080902 05000 030902 0 fo d 05000 030902 080902 0 0 0 0 1 R Similarly one is able to construct five fold symmetry transformation matrices for the other icosahedron vertices Using the generic rotational transformation R matrix 4 one is able to construct the three fold transformation matrices via the rotation axis passes through the centroid of the triangular faces of the7 72 icosahedron and an angle of rotation of ijy Consider the triangular face with corners at Ol O l and Ol The centroid is at 302 l3 and the normalized axis of rotation is 5 035682200934172 and w transformation matrix 030902 080902 05000 0 080902 05000 030902 0 W 05000 030902 080902 0 0 0 0 1 R A polyhedron with faces all equilateral triangles is called a deltahedron Deltahedra with icosahedral symmetry are classified as icosadeltahedra Any icosadeltahedron has 20T facets Where T is the triangulation number given by71f71 Wheref h2hkk2 for all pairs of integers h and k which do not have common factor and f is any integer l5 The possible values of P are L3ZI3J92L3L37 In Fig 24 we display triangles with different triangulation numbers for icosahedral Virus structures Fig 25 Architecture of lcosahedral Viruses Caspar Klug Triangulation Numbers Asymmetric structure units 8 The greater the T number the larger the size of the virus capsid Each triangular portion of the icosahedral virus capsid is easily subdivided into its three asymmetrical units with each unit containing some combination of protein structure units protomers In total an icosahedral virus capsid has 6OT asymmetrical units with numerous proteins structures inter twined to form a spherical mosaic In Fig 24 we see that when Tl each vertex is at the center of a pentagon and the capsid proteins are in an equivalent environment ie five neighbors cluster at a common vertex However for icosadeltahedra with larger triangulation numbers eg T3 there are pentagons and hexagons in the capsid mosaic Fig 25 Therefore even though the capsid proteins protomers may be chemically identical some cluster into a local 5 fold neighborhood and the others into a local 6 fold neighborhood Such locally symmetric clusterings of protomers are alternatively termed capsomers In these situations the proteins are no longer global symmetrically equivalent but only quasieequivalent l5 3 Surface and Volumetric Modeling and Visualization 31 Atomistic Resolution Mbdel Structures 9 Numerous schemes have been used to model and visualize bio molecules and their properties 47 All these different visual representation are often derived from an underlying geometric model constructed from the positions of atoms bonds chains and residues information deposited as part of an atomic resolution structure of the protein or nucleic acid in the Protein Data Bank PDB Hence structural models are designed to represent the primary sequence secondary eg d helices B sheets tertiary eg d 7 B barrels and quaternary fully folded geometric structures of the protein or nucleic acid An early approach to molecular modeling is to consider atoms as hard spheres and their union as an attempt to capture shape properties as well as spatial occupancy of the molecule This is similar to our perception of surfaces and volume occupancy of macroscopic objects The top two pictures in Figure 24 shows hard sphere model visualizations of the twin Rice Dwarf capsid shells with individual proteins colored differently Solvated versions of these 10 molecular surfaces have been proposed by Lee 7 Richards Connolly et al for use in computational biochemistry and biophysics Much of the preliminary work along with later extensions focussed on finding fast methods of triangulating this molecular surface or as sometimes referred to as the solvent contact surface Two prominent obstacles in Hmdeling are the correct handling of surface self intersections singularities and the high communication bandwidth needed when sending tessellated surfaces to the graphics hardware Figure 3l Analytic surface models of capsid shells of icosahedral viruses A more analytic and smooth description of molecular surfaces without singularities is provided by a suitable level set of the electron density representation of the molecule lsotropic Gaussian kernels have been traditionally used to describe atomic electron density due to their ability to 11 approximate electron orbitals The electron density of a molecule with M atoms centered at xjaj E can thus be written as M FBIecidens x Z Z 7Kx xj Where 7 and K are 11 typically chosen from a quadratic exponential description of atomic electron density d l a 2 2 d 39 gt 2 ltltx y r x y At0mx 6 V2 2 ede 2 AKqx y7elecidens x d r 2x x2 r2 gt gt e 7Kx xj The atomic electron density kernels are affected by the radius r of individual atoms and the decay parameter d Smooth and molecular surface models for individual proteins structure units as well as entire capsid shells can be easily constructed as a 12 gt M gt gt fixed level set of FelecidensocZzyj39KOC xj An 11 array of such structural molecular model visualizations are shown as Figures 2l 7 25 as well as figure 3l Some of them use transparency on the solvated leecular surface and show the protein back bone structure folded chains of d helices and B sheets 32 Structure Elucidation from 3D Maps Electron Microscopy EM and in particular single particle reconstruction using cryo EM has rapidly advanced over recent years such that several virus structures tertiary and secondary can be resolved routinely at low resolution 10 20 A and in some cases at sub nanometer intermediate resolution 7 l0 A 78 Figure 32 Structure Elucidation from 3D Maps of lcosahedral Viruses Symmetry of the virus capsid shells are exploited both in the 3D Map reconstruction from raw 2D EM images as 13 well in structure elucidation in the 3D Map In many cases the 3D maps are of spherical viruses with protein capsid shells exhibiting icosahedral symmetry In these cases the global symmetry detection can be simplified to computing the location of the 5 fold rotational symmetry axes passing through the twelve vertices of the icosahedron from which the 3 fold symmetry axis for the twenty icosahedron faces and the 2 fold symmetry axis for the thirty icosahedron edges can be easily derived However determining the local symmetries of the capsomers structure units are more complicated as they exhibit varied k fold symmetry and their detection requires a modified correlation based search algorithm 94 Volumetric segmentation methods are additionally utilized to partition color and thereby obtain a clearer view into the macromolecules architectural organization Furthermore electronically dissecting the local structure units from a 3D Map allows for further structural interpretation tertiary and secondary folds Visualizations from the afore mentioned local symmetry 14 detection and automatic segmentation applied to a 3D volumetric Map of the Turnip Yellow Mosaic virus pdbid lAUY are shown in Figure32 4 Quantitative Visualization The geometric modeling of virus capsids and the individual virus structure units can be further augmented by the computation of several global and local shape metrics While integral topological and combinatorial metrics capture global shape properties differential measures such as mean and Gaussian curvatures have also proved useful to an enhanced understanding and quantitative visualization of macromolecular structures 41 Integral Properties lntegral shape metrics include the area of the molecular capsid surface defining the capsid the volume enclosed by closed capsid shells and the gradient integral on the molecular capsid surface Given our smooth analytic level set definition of the 15 molecular surface from section 3 gt M gt gt Fezecidequot5x xJ39 consr for all the atoms 11 that make up either an individual structure unit or the entire virus capsid an efficient and accurate integration computation for these metrics is given by the contour spectrum 5 The surface integrations can be performed by adaptively sampling the capsid surface using a technique known as contouring 5 Contouring is often performed by first decomposing meshing the space surrounding the capsid surface into either a rectilinear Cartesian grid mesh a tetrahedral or a hexahedral mesh For a tetrahedral mesh the surface area for the portion of the level set inside a tetrahedron can be represented by a quadratic polynomial B spline 5 Summing these B splines over all of the tetrahedra containing the capsid surface yields the capsid surface area The volume enclosed by a closed capsid surface is determined by the definite integration of the surface area polynomial B splines 16 Fig 4l Area Volume Relationship for lcosahedral Viruses In Figure 4l we display the results of surface area and volume calculations and a regression relationship between the two for a selection of spherical icosahedral capsids for virus structures summarized in Table 2 The analytic molecular surfaces were first computed and then surface area and enclosed volume were estimated through B spline evaluation as stated above 42 Differential Properties The gradient function of our smooth analytic capsid gt M gt surface is simply VEIecidemxZijKx xj the j1 summation of the vector of first derivatives of the atomic electron density function This gradient function is non zero everywhere on the virus capsid surface ie no singularity The second derivatives of the Hblecular surface capture additional 17 differential shape properties and provide suitable metrics Popular metrics are the magnitudes of Mean Curvature H and the Gaussian curvature G These are given directly as H l imm kmax and G kmmkmax and are respectively the average and the product of the twin principal curvatures namely kmm and kmax also sometimes known as the minimum and maximum curvatures at a point on the surface Again for our level set based analytic molecular surface Felecidensxconszf the twin curvatures H and K can be evaluated as H Z fx2 fW fzz 2 Z fxfyfxy 2 Z fxz 1395 and G 2 2fx fy lt in fyze fxy in gtgt gt lt lt2 f gt gt2 gt where 2 represents a cyclic summation over x y and Z and Where additionally fX etc denotes partial differentiation with respect to those variables Displaying the magnitude of the gradient function and its variation as expressed by the mean and Gaussian curvature functions over a molecular surface helps guantitatively visualize the bumpiness or lack thereof 18 of an individual protomer a structure unit or the entire viral capsid ln Figures 2l the bottonl two pictures display the mean and Gaussian curvature functions of the Tobacco Mosaic virus asymmetric protomer surface exhibiting and enhancing the bumpiness of the surface 43 Topological and Combinatorial Properties Affine invariant topological structures of volumetric functions f such as our smooth analytic electron density function of section 3 include the Morse complex 28 56 and the contour tree CT 46 Both the Morse complex and contour tree are related to the critical points of the volumetric function f ie those points in the domain Al Where the function gradient vanishesz0 The functional range of f is the interval between the minimum and maximum values of the functionfU m m For a scalar value fmf the level set of the field f at the value ul is the subset of points LWCM such thatfxWVXELW 19 A level set may have several connected components called contours The topology of the level set 11w changes only at the critical points in l Whose corresponding functional values are called critical values A contour class is a maximal set of continuous contours which have the same topology and do not contain critical points Without loss of generality the critical points are assumed to be non degenerate ie only isolated critical points This assumption can be enforced by small perturbations of the function values If the critical points are non degenerate then the Hessian f a at a critical point a has non zero real eigenvalues The index of the critical point a is the number of negative eigenvalues off a For a 3D volumetric function there are four types of critical points index 0 minima indices l and 2 saddle points and index 3 maxima The contour tree CT was introduced by Kreveld et al 46 to find the connected components of level sets for contour generation The CT captures the topological changes of the level sets for the entire 20 functional range fmwfmm of f each node of the tree corresponds to a critical point and each arc corresponds to a contour class connecting two critical points As an example the contour tree for a virus capsid is shown in 42 Each leaf node of the CT represents the creation or deletion of a component at a local minimum or maximum and each interior node represents the joining andor splitting of two or more components or topology changes at the saddle points A cut on an arc of the tree 111112 T by an isovalue Vlgwsvz represents a contour of the level set Lw Therefore the number of connected components for the level set Lw is equal to the number of cuts to the CT at the value w The CT can be enhanced by tagging arcs with topological information such as the Betti numbers of the corresponding contour classes 46 Betti numbers k k0l intuitively measure the number of k dimensional holes of a virus capsid surface or of any individual structure unit Only the first three Betti numbers oa ly z of a smooth 21 surface are non zero corresponds to the number of connected components lzltxmresponds to the number of independent tunnels 2 represents the number of voids enclosed by the surface For example a sphere has the Betti numbers 03 13 2213031 While a torus has oa 1a 2121 Betti number computations for virus capsid surfaces provide useful topological and combinatorial structural information 5 Conclusion Ultra structure modeling and visualization of virus capsids are clearly just a couple of the steps in a computational modeling pipeline for determining structure to function relationships for such nano organisms Efforts are underway by several groups for virus energetics in solvated environs atomistic and coarse grained virus dynamics as well as interactions and binding of various ligands and proteins to the nucleo capsids 22 Acknowledgements Sincere thanks to my students S Goswami S Siddahanvalli J Wiggins and W Zhao for their help with this manuscript Thanks also to invaluable discussions with Dr Tim Baker at Univ of California San Diego 6 REFERENCES l Aloy P Russell RB 2002 The third dimension for protein 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the helix symmetry axis D with a transparent molecular surface and the protein backbone showing helix secondary structures E molecular surface of protomer with the mean curvature function F Gaussian curvature function Fig 22 Organization of Rice Dwarf Virus lUF2 with Icosahedral capsid shells A 2D texture based visualization of the outer capsid shell showing a single sphere per atom and colored by proteins B the outer capsid shell shown as a smooth analytic molecular surface while the inner capsid surface is displayed using 2D texture maps of a union of spheres and colored C shows the outer capsid D displays the inner capsid E shows the icosahedral asymmetric structure unit of the outer unit F displays the icosahedral asymmetric structure unit of the inner unit G shows the protein backbone of the structure unit shown in E and H shows the protein backbone of the structure unit show in F Fig 23 Helical Symmetry Axis Sifold axis a 01 U WSDJMHM 3 fold axis Fig 24 Icosahedral Transformations showing 5fold and 3fold Symmetry Axis 49 Fig 25 Architecture of lcosahedral Viruses A Caspar Klug Triangulation Number T via a hexagonal lattice Green triangle has T 1 while yellow represents T 21 B shows the asymmetric unit of an icosahedron C asymmetric structure units of the capsid shall D a single asymmetric structure unit E asymmetric unit colored by protein as well as showing protein backbone F a capsomere consisting of three proteins VIRUS PDB lGW8 lauy lCWp ldnv lej 6 1mlc lohf lqgt lsva 25tv 2cas thv lihm Figure 31 Portions of Capsid Shells of Icosahedral Viruses visualized using molecular surface visualization Fig 3 2 PDBID lAUY Size 256 Resolution N4A A Blurred map outside View B Blurred map inside View C Symmetry detected including global and local 3fold symmetry axes D Segmented trimers outside View with randomly assigned colors E Segmented trimers inside View F One of the segmented trimers leftbottom outside View righttop inside View 7 1n VLo ma Am 359 gt Wus r I t m7 m7 7 y g 2 g 1 x 7 V 7 m7 7 5 BS 9 95 10 ms 1 115 12 125 13 1 5 1 Area Hg4l Area Volume Relationship for Icosahedral Viruses given in Table l Hgne42 The contour tree upper left and the contour spectrum bottom for the Human Rhihovirus serotype 2 pdbid l FPNThe red color in the spectrum curve is the graph of molecular surface area while the blue and green curves are the excluded and enclosed volume 54 Appendix LIST OF TABLES Table l Helical and lcosahedral Viruses 1 Name and structure reference of virus given in square brackets H Family nomencleature from the TCTV database 3 Host types are P for Plant V for vertebrate l for Invertebrate F for Fungi 4 Virus Nucleic Acid NA type is single stranded RNA sR or DNA sD double stranded RNA dR or DNA dB and linear L or circular C 5 Capsid symmetry is Helix He or lcosahedral Tc with the triangulation number of each capsid shell in parenthesis 6 The number of capsid shells and whether enveloped E or not n 7 The acquisition modality Xray feature resolution and PDB id in parenthesis Shell E Tombusviridae X25 25W Totiviridae X36 1m1c Parvoviridae X33 Zoas disease VLP MAb ES Bromoviridae Calioiviridae Parvoviridae Pa povaviridae Reoviridae Phycod naviridae Hepadnaviridae X25 1khv X37 1dnv X32 1ksx X238 1lj2

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