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“We have given you, O Adam, no visage proper to yourself, nor endowment properly your own, in order that whatever place, whatever form, whatever gifts you may, with premeditation, select, these same you may have and possess through your own judgement and decision […] We have made you a creature neither of heaven nor of earth, neither mortal nor immortal as the free and proud shaper of your own being, fashion yourself in the form you may prefer.”
Oration on the Dignity of Man – Giovanni Pico della Mirandola

The geometry of viruses

published on 7.10.2020

Russian dolls, also called matryoshka dolls, are characterized by their different sizes and fit inside each other to the innermost and most protected that is usually a baby. They were created in 1890 as a set of 8 units, but can now be found with a variable number, usually between 5 and 12. According to the Guinness Book, the world record is a set of 51 units.
Viruses are also arranged in concentric layers protecting the genetic material in the innermost part. The number of layers varies depending on the virus. The human immunodeficiency virus (HIV) is made up of five layers. From the outside we find: a spherical lipid bilayer, an icosahedral protein matrix, a conical protein capsid, a helical protein capsid and also helical genetic material (RNA). The coronavirus, on the other hand, consists of three layers, and the tobacco mosaic virus, in two. The outermost layer type determines the geometric shape of the virus. The viral capsid is a protein structure made up of a series of monomers called “capsomers”. In general there are few different types of proteins in capsomeres; The reason is that the more protein, the more genetic information, and the genome of a virus is greatly reduced. The capsids that are attached to the genetic material, take their shape, so the virus has a helical shape. Protein capsids that are not directly linked to the genetic material can adopt very limited geometries, the most common being the icosahedral, consisting of twenty triangular faces, the regular polyhedron with the largest number of faces, and therefore the closest to the sphere. When the outermost layer is a lipid bilayer, the virus is said to be enveloped, and the most common shape is usually the sphere or spheroid. The reason is that the bonds between lipids are much more flexible than between proteins and allow a greater degree of curvature. Somehow, as in Russian dolls, the layers of viruses tend to close inside each other in rounded shapes. But why?
Because the sphere is the geometry that emerges spontaneously in nature under conditions of isotropy, that is, when the same physical properties occur in all directions of space. The planets and stars are spherical due to the fact that gravity is exerted from a central point in all directions. River pebbles are spherical because they receive impacts that erode them in all directions. But in addition, the sphere is the smallest surface that encloses a volume. This can be easily understood if we go from 3 to 2 dimensions. In two dimensions the circumference is the shortest perimeter enclosing a flat surface. A square of 10×10 cm has an area of ​​100 cm2 and a perimeter of 40 cm, while to draw a circle of 100 cm2, we need a radius of 5.64 cm, and we will obtain a perimeter of 35 cm. The circumference saves 5 cm of perimeter with respect to the square.
With a sphere, a volume is closed, encapsulated, to protect the genetic material with the smallest number of molecules. That’s why viruses and cells are usually spherical or spheroidal. Even the simplest multicellular organisms, and the early stages of embryonic development are a sphere of spheres, like a blackberry.
The ancient Greeks already discovered some properties of the sphere and considered it the perfect shape. Even today, when we want to indicate something well done, we say that it has come out “round”.

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