Journal of Mathematical Biology

, Volume 59, Issue 3, pp 287–313

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

Article

Abstract

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.

Keywords

Virus structure Symmetry group 

Mathematics Subject Classification (2000)

92B05 20F55 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Crick FHC, Watson JD (1956) The structure of small viruses. Nature 177: 473–475CrossRefGoogle Scholar
  2. 2.
    Caspar DLD, Klug A (1962) Physical principles in the construction of regular viruses. Cold Spring Harb Symp Quant Biol 27: 1–24Google Scholar
  3. 3.
    Rayment I et al (1982) Polyoma virus capsid structure at 22.5 Ȧ resolution. Nature 295: 110CrossRefGoogle Scholar
  4. 4.
    Liddington RC et al (1991) Structure of Simian Virus 40 at 3.8 Ȧ resolution. Nature 354: 278CrossRefGoogle Scholar
  5. 5.
    Twarock R (2004) A tiling approach to virus capsid assembly explaining a structural puzzle in virology. J Theor Biol 226: 477CrossRefMathSciNetGoogle Scholar
  6. 6.
    Twarock R (2005) The architecture of viral capsids based on tiling theory. J Theor Med 6: 87–90MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Keef T, Twarock R (2007) Blueprints for viral capsids in the family of Papovaviridae. J Theor Biol (submitted)Google Scholar
  8. 8.
    Senechal M (1996) Quasicrystals and geometry. Cambridge University Press, LondonGoogle Scholar
  9. 9.
    Shechtman D, Blech I, Gratias D, Cahn JW (1984) Metallic phase with long-range order and no translational symmetry. Phys Rev Lett 53: 1951–1953CrossRefGoogle Scholar
  10. 10.
    Bamford DM, Burnett RM, Stuart DI (2002) Evolution of viral structure. Theor Popul Biol 61: 461CrossRefGoogle Scholar
  11. 11.
    Bamford DH, Grimes JM, Stuart DI (2005) What does structure tell us about virus evolution? Curr Opin Struct Biol 15: 655CrossRefGoogle Scholar
  12. 12.
    Twarock R (2002) New group structures for carbon onions and carbon nanotubes via affine extensions of noncrystallographic Coxeter groups. Phys Lett A 300: 437–444CrossRefMathSciNetGoogle Scholar
  13. 13.
    Patera J, Twarock R (2002) Affine extensions of noncrystallographic Coxeter groups and quasicrystals. J Phys A 35: 1551–1574MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Janner A (2006) Towards a classification of icosahedral viruses in terms of indexed polyhedra. Acta Crystallogr A 62: 319CrossRefMathSciNetGoogle Scholar
  15. 15.
    Janner A (2006) Crystallographic structural organization of human rhinovirus serotype 16, 14, 3, 2 and 1A. Acta Crystallogr A 62: 270CrossRefMathSciNetGoogle Scholar
  16. 16.
    Tang L, Johnson K, Ball L, Lin T, Yeager M, Johnson J (2001) The structure of Pariacoto virus reveals a dodecahedral cage of duplex RNA. Nat Struct Biol 8: 77–83CrossRefGoogle Scholar
  17. 17.
    Reddy VS, Natarajan P, Okerberg B, Li K, Damodaran KV, Morton RT, Brooks CL III, Johnson JE (2001) Virus particle explorer (VIPER), a website for virus capsid structures and their computational analyses. J Virol 75: 11943–11947CrossRefGoogle Scholar
  18. 18.
    Keef T, Toropova K, Ranson NA, Stockley PG, Twarock R (2007) A new paradigm for symmetry reveals hidden features in the architecture of simple viruses (in preparation)Google Scholar
  19. 19.
    Valegard K, Liljas L, Fridborg K, Unge T (1990) The three-dimensional structure of the bacterial virus MS2. Nature 345: 36CrossRefGoogle Scholar
  20. 20.
    Valegard K, Murray JB, Stockley PG, Stonehouse NJ, Liljas L (2002) Crystal structure of a bacteriophage RNA coat protein operator system. Nature 371: 623CrossRefGoogle Scholar
  21. 21.
    Golmohammadi R, Valegard K, Fridborg K, Liljas L (1993) The refined structure of bacteriophage MS2 at 2.8 A resolution. J Mol Biol 234: 620CrossRefGoogle Scholar
  22. 22.
    Toropova K, Basnak G, Twarock R, Stockley PG, Ranson NA (2007) The three-dimensional structure of genomic RNA in bacteriophage MS2: implications for assembly. J Mol Biol 375(3): 824–836CrossRefGoogle Scholar
  23. 23.
    Grayson N, Keef T, Severini S, Twarock R (2007) Assembly pathways for bacteriophage MS2 based on a Hamilton path approach (in preparation)Google Scholar
  24. 24.
    Keef T, Taormina A, Twarock R (2005) Assembly models for Papovaviridae based on tiling theory. Phys Biol 2: 175–188CrossRefGoogle Scholar
  25. 25.
    Keef T, Micheletti C, Twarock R (2006) Master equation approach to the assembly of viral capsids. J Theor Biol 242: 713–721CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of YorkYorkUK
  2. 2.Department of BiologyUniversity of YorkYorkUK

Personalised recommendations