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From the analysis of sizes of approximately 130 small icosahedral viruses wefind that there is a typical structural capsid protein, having a mean diameterof 5 nm and a mean thickness of 3 nm, with more than two thirds of the analyzedcapsid proteins having thicknesses between 2 nm and 4 nm. To investigatewhether, in addition to the fairly conserved geometry, capsid proteins showsimilarities in the way they interact with one another, we examined the shapesof the capsids in detail. We classified them numerically according to theirsimilarity to sphere and icosahedron and an interpolating set of shapes inbetween, all of them obtained from the theory of elasticity of shells. In orderto make a unique and straightforward connection between an idealized,numerically calculated shape of an elastic shell and a capsid, we devised aspecial shape fitting procedure, the outcome of which is the idealized elasticshape fitting the capsid best. Using such a procedure we performed statisticalanalysis of a series of virus shapes and we found similarities between thecapsid elastic properties of even very different viruses. As we explain in thepaper, there are both structural and functional reasons for the convergence ofprotein sizes and capsid elastic properties. Our work presents a specificquantitative scheme to estimate relatedness between different proteins based onthe details of the (quaternary) shape they form (capsid). As such, it mayprovide an information complementary to the one obtained from the studies ofother types of protein similarity, such as the overall composition ofstructural elements, topology of the folded protein backbone, and sequencesimilarity.

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