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Statistical analysis of sizes and shapes of virus capsids and their resulting elastic properties

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Abstract

From the analysis of sizes of approximately 130 small icosahedral viruses we find that there is a typical structural capsid protein, having a mean diameter of 5 nm and a mean thickness of 3 nm, with more than two thirds of the analyzed capsid proteins having thicknesses between 2 nm and 4 nm. To investigate whether, in addition to the fairly conserved geometry, capsid proteins show similarities in the way they interact with one another, we examined the shapes of the capsids in detail. We classified them numerically according to their similarity to sphere and icosahedron and an interpolating set of shapes in between, all of them obtained from the theory of elasticity of shells. In order to make a unique and straightforward connection between an idealized, numerically calculated shape of an elastic shell and a capsid, we devised a special shape fitting procedure, the outcome of which is the idealized elastic shape fitting the capsid best. Using such a procedure we performed statistical analysis of a series of virus shapes and we found similarities between the capsid elastic properties of even very different viruses. As we explain in the paper, there are both structural and functional reasons for the convergence of protein sizes and capsid elastic properties. Our work presents a specific quantitative scheme to estimate relatedness between different proteins based on the details of the (quaternary) shape they form (capsid). As such, it may provide an information complementary to the one obtained from the studies of other types of protein similarity, such as the overall composition of structural elements, topology of the folded protein backbone, and sequence similarity.

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Notes

  1. This is the basis of the Caspar-Klug classification scheme [14] that classifies different capsid icosahedra in terms of the triangulation number T, which is, roughly, a way to divide the icosahedron in similar, nearly equivalent parts that represent individual proteins.

  2. The triangulation numbers denoted with p (pseudo) do not completely conform to the Caspar-Klug principle of quasi-equivalence since the basic unit is composed of different (but morphologically similar) proteins.

  3. The positions of the amino acids are represented either by the positions of their centers-of-mass (see also [18]), or, where this data was unavailable, by the positions of their C α atoms.

  4. There are 60T proteins in a virus of a given T-number.

  5. The analysis of available space and restricted length of the genome is somewhat different in ssRNA viruses, where the ssRNA molecule is held within the capsid mostly by electrostatic interactions with the capsid proteins. In this case, the quantity of information that can be stored in the capsid scales as \(\overline{R}^2\) [4], so that the production of large proteins always requires the same percentage of the ssRNA information, irrespectively of the radius of the virus.

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Acknowledgements

We thank Alberto Vianelli for informing us about [20]. We also thank the two anonymous reviewers for their suggestions on improving the manuscript. A.L.B. acknowledges the support from the Slovene Agency for Research and Development under the young researcher grant. A.Š. acknowledges support from the Ministry of Science, Education, and Sports of Republic of Croatia (Grant No. 035-0352828-2837). R.P. acknowledges the support from the Slovene Agency for Research and Development (Grant No. P1-0055 and J1-4297).

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Authors and Affiliations

  1. Department of Theoretical Physics, Jožef Stefan Institute, 1000, Ljubljana, Slovenia

    Anže Lošdorfer Božič & Rudolf Podgornik

  2. Institute of Physics, 10001, Zagreb, Croatia

    Antonio Šiber

  3. Department of Physics, Faculty of Mathematics and Physics, 1000, Ljubljana, Slovenia

    Rudolf Podgornik

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  1. Anže Lošdorfer Božič
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  2. Antonio Šiber
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  3. Rudolf Podgornik
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Correspondence to Anže Lošdorfer Božič.

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Lošdorfer Božič, A., Šiber, A. & Podgornik, R. Statistical analysis of sizes and shapes of virus capsids and their resulting elastic properties. J Biol Phys 39, 215–228 (2013). https://doi.org/10.1007/s10867-013-9302-3

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  • DOI: https://doi.org/10.1007/s10867-013-9302-3

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