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Virus shapes and buckling transitions in spherical shells*
DAVID R. NELSON
Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA
We show that the icosahedral packings of protein
capsomeres proposed by Caspar and Klug for spherical
viruses become unstable to faceting for sufficiently large
virus size, in analogy with the buckling instability of
disclinations in two-dimensional crystals. The faceted shape
depends only on the dimensionless Foppl-von Karman
number g ¼ YR2=k; where Y is the two-dimensional
Young’s modulus of the protein shell, g is its bending
rigidity and R is the mean virus radius. Our model, based on
the nonlinear physics of thin elastic shells, produces
excellent one parameter fits in real space to the full three-
dimensional shape of large spherical viruses such as HK97
ðg ¼ 1480Þ and Yeast L-A ðg ¼ 547Þ: The shape can be
parameterized more quantitatively in terms of a spherical
harmonic expansion. We also investigate elastic shell theory
for extremely large g, 103 , g , 108; and find results
applicable to icosahedral shapes of large vesicles studied
with freeze fracture and electron microscopy. Finally, we
discuss an instability towards grain boundaries in shells with
an exceptionally large bending rigidity, which has been
observed experimentally in related problems in spherical
crystallography such as the packing of colloidal particles on
the surface of water droplets in oil [1].
References
[1] Bausch, A.R., Bowick, M.J., Cacciuto, A., Dinsmore, A.D., Hsu,M.F., Nelson, D.R., Nikolaides, M.G., Travesset, A. and Weitz, A.,2003, Grain boundary scars and spherical crystallography. Science,299, 1716.
Journal of Theoretical Medicine
ISSN 1027-3662 print/ISSN 1607-8578 online q 2005 Taylor & Francis Group Ltd
http://www.tandf.co.uk/journals
DOI: 10.1080/10273660500148838
*Work on viruses carried out in collaboration with Jack Lidmar and Leonid Mirny: More information can be found in Phys. Rev. E 68, 051910 (2003)or http://www.arxiv.org/PS_cache/cond-mat/pdf/0306/0306741.pdf
Journal of Theoretical Medicine, Vol. 6, No. 2, June 2005, 137
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