The Gynoecium
Nigel Fleming
July 11, 2023
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“The more I study nature, the more I become impressed […] that the contrivances and beautiful adaptations [acquired through natural selection] transcend in an incomparable degree [those] which the most fertile imagination of the most imaginative man could suggest with unlimited time at his disposal.” (Charles Darwin).
http://www.sciencedirect.com/science/article/pii/S0960982214003881
Viruses have always been named haphazardly, relating variously to the diseases they cause, the tissues they inhabit, or where they were first discovered. HK97 is no exception. HK97 is a bacteriophage virus that preys on E. Coli, and was found in Hong Kong (hence the ‘HK’), in 1997 (hence the ‘97’), and living in pig feces (fortunately, not specified). Not an auspicious beginning for a new virus, one might think. But this curious virus is becoming an exemplar for a new geometric understanding of viral proteins and for its armored motifs that confer enormous strength.
Viruses can be crystallized like common table salt and do not possess the machinery to reproduce outside their host cell; for this reason, long-eared and quarrelsome taxonomists have long made the case for their lack of inclusion in any definition of life, and there it largely sits today. Viruses remain in the same box of odds-and-sods as transposons, those raw bits of autonomous DNA, and with prions, those blobs of infectious proteins, both of which reproduce and evolve.
Viruses are one of the most numerous creatures on earth, have a mutation rate a million times faster than man, and have been responsible for — or at the very least participated in — most of the giant steps of evolution across many species. Their fossilized remains form a confounding proportion of all genes, including our own. Viruses are also the largest single source of genetic variation on the planet, a veritable tool box for innovation for living things.
Viruses are notoriously hard to classify due to their enormous mutation rates, existing in shimmering quasi-species that morph continuously. Rather than exclude it from a definition of life, a virus with outsourced metabolism and streamlined genome is rather more of a very advanced creature than a bag of chemicals.
Viruses are Euclidian beings that live in a hyperbolic and fractal world. Normally, Nature abhors straight lines, perfect circles, triangles and polygons — indeed everything that makes up classical Euclidian geometry. Flowers, fleas, and noses are curved, non-geometric in form, fractal and hyperbolic in nature. Viruses are radical departures from the forms we usually associate with life. Viruses are fiercely geometric in design, simply Euclidean.
The deconstruction of the geometry behind the protein configurations of the HK97 capsid, the head, is confounding. Arranging the capsid head icosahedron in a two dimensional planar origami format permits a linear representation of the capsid head protein geometry.
The capsid head proteins are arranged in an offset skew spiral design that winds down the Platonic solid, contained in a series of hidden circular motif arrangements that did not fit neatly into the equilateral triangles, but rather, slop over the edge of one triangle into the adjacent one.
Hexagons and pentagons contained within the circular motifs form the underlying geometric tiling that frames the position of the capsid proteins. These hexagons and pentagons are contained within identical sized circles. Within each circle, are six proteins for the hexagons, and five for the pentagons, whose shape traces the underlying hexagon and pentagon.
One of the curious properties of the structure is the transition between the six-sided hexagons in the centers and the five sided pentagons at each corner of the capsid icosahedron.
The underlying geometry of the hexagons and pentagons that form the underpinning for the whole capsid head protein location is a tessellated pattern of tilings, that are based on a small number of repetitive units that do not overlap nor leave gaps. Tessellations mean that only three repetitive motifs are necessary to construct the whole HK97 capsid, when arranged in an alternating fashion. These are the hexagon, the pentagon, and the chainmail motifs.
Regular pentagons in two dimensional origami format do not tessellate because their internal angles at junctures do not sum to 360’. However, when mixed with hexagons, whose internal angles at junctures do sum to 360, and when configured on the 3D icosahedron, they do tessellate perfectly, and form tilings.
The same pattern of tessellated tilings found in HK97 is also present in Buckminster Fullerenes, with pentagon corners and hexagonal faces on a regular icosahedron, conferring these molecules with extraordinary properties. Another example in nature is Giants’ Causeway , where pentagons and hexagons exist in a tessellated form, and exist as a mixture of polygons in three dimensions.
The peculiar skew geometry of the proteins on the HK97 capsid head is geometrically anchored on a grid of equilateral triangles, and offset on this grid at a slant of a defined angle. This is what orients the hexagons and pentagons in the skew format.
The capsid head family of proteins, and their hexagons, are contained within an identical circular geometry that rotate in orientation in a complex alternating pattern.
Each six-sided face that contains identical circular rotating motifs that come together to make anchor corners of five sides. The six protein molecules that make the hexagons are composed of two that are the same type of protein, and four that differ, but are all neatly tucked around in a sleeping cat design, with their tails embracing the next protein.
All the five proteins of the corner hexagons are the same protein. The corner pentagons of each equilateral triangle apex are also in a sleeping cat configuration.
The HK97 virus turns out to have a feature that is widely adopted in nature, called the HK97 fold, or chainmail. Around each underlying hexagonal and pentagonal geometry is a very unusual protein structure that resembling mediaeval armor chainmail, or Borromean rings. The proteins that compose the chainmail are entwined to embrace the clusters of proteins in the hexagonal and pentagonal motifs. The chainmail proteins form a super-strong structure on the capsid head.
Buckminster Fuller famously wrote that “any polygon with more than three sides is unstable. Only the triangle is inherently stable”. All of the unusual properties of strength, combined with lightness and modular design, geodesic domes result from these triangles that divide and distribute loads without structural deformation. So also, the HK97 is a geodesic dome in structure.
The many unusual geometric properties of the regular icosahedron have been documented for centuries. They contain Golden proportion rectangles that derive the related Golden spirals based driven by the number Pi.
All Gothic tracery is based on the equilateral triangle. In order to design Gothic tracery, stonemasons relied on a compass and ruler, and start with a hidden underlying geometric grid of hexagons, pentagons, and triangles. This tessellated underlying geometric logic allows a few simple motifs to be repeated to make a complex final design. This meant that the stonemasons only needed to carefully fabricate an iterative number of basic units to make their constructions. And so it is with the HK97 capsid head: few motifs with few proteins repeat in tessellation tilings to make the final complex structure. And this is genetically efficient.
Is the HK97 an ironic message in the bottle to a rather stupid humanity? Not only had this virus chosen a very potent geometric shape, the icosahedron, that have obsessed mathematicians for 2,500 years with its embodiment of Phi, but it thumbed its nose at us with its skew offset geometry, a true brain-mangler for an advanced species to figure out and make, let alone in the wet and warm chaos of a host cell using borrowed machinery.
It may be that the HK97 virus has had the last laugh. Maybe, just maybe, us foolish humans will grant this mere piece of chemistry a place on the tree of life. Our geocentric nonsense has given us blinkers to not see a very advanced life form, outsourced and streamlined, living right under our very noses.
Perhaps, one day, when we get to Mars, we may have the wisdom to recognize life not as it is, but as it could be.