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Durham University

Department of Mathematical Sciences


Discovery of Paradoxical Geometries published in Nature

(9 May 2019)

Image attribution: Heddle Lab, Jagiellonian University

An international group of researchers from the Jagiellonian University (Krakow, Poland), led by Prof J. Heddle, have produced a super-stable artificial protein ball that apparently defies the rules of geometry.

The nano-cage is made out of 24 regular hendecagons (polygons with 11 edges) but this is known to be mathematically impossible.

Bernard Piette (Department of Mathematical Sciences, Durham University) and visiting PhD Student Agnieszka Kowalczyk (Department of Mathematics and Malopolska Centre of Biotechnology, Jagiellonian University) have shown that the shape realised in the lab is possible if one distorts the edges and the angles of the polygons by 0.5% and 0.27% respectively.

The image above is an electron density representation of the "mathematically impossible" protein cage, around 22 nanometres in diameter. The cage is made from 24 copies of an 11-sided, ring-shaped protein each shown in a different colour and held together by single gold atoms (yellow balls). The background shows models of other "impossible" cages predicted by the work.

Piette and Kowalczyk have also identified several other so called paradoxical geometries by defining cages which are made out of nearly regular polygons as well as some, less regular, holes.

Beside their mathematical interest, these geometries are candidates to create artificial protein cages which can be used for drug delivery. The drug is trapped inside protein cages which are themselves dressed with ligands that can bind to the receptors of target cells, such as cancer cells. The advantage is to reduce side effects as the drug is only delivered to cells that need treatment. This can also lead to reduced cost, especially for very expensive drugs, as smaller amounts of the chemical are needed.

Protein cages exist in nature, as viruses for example, but the advantage of the protein cage made by the Heddle lab is that it is made of a single protein. In nature the many proteins that form a protein cage are held together by a complex network of chemical bonds and these are very difficult to predict and simulate. This makes engineering new cages of that type very difficult. The cage created by the Heddle Lab, on the other hand, is made out of a single protein that forms so called rings, the polygonal faces of the cage. These rings are then linked together by gold atoms to form a cage with a 22nm diameter.

The identification of other paradoxical geometries, as found by Bernard Piette and Agnieszka Kowalczyk, will help biochemists select proteins which can form "rings" from which other types of cages can be made.

Caption:The paradoxical cage made out of 24 nearly regular hendecagonal faces. The faces edge lengths are identical within 0.5% and the angles within 0.27%.

The research has been published in Nature, which can also be accessed from here.

Interestingly, Jonathan Heddle, who led the discovery, is originally from Bishop Auckland and is a regular visitor of the Department of Mathematical Sciences and the Biophysical Science Institute in Durham.