Pentagrammarspin: why twelve pentagons?

This post has been in my drafts folder for a while. With the World Cup here, it’s time to post it!

It’s a rule that a 3D assembly of hexagons must have at least twelve pentagons in order to be a closed polyhedral shape. This post takes a look at why this is true.

First, some examples from nature. The stinkhorn fungus Clathrus ruber, has a largely hexagonal layout, with pentagons inserted. The core of HIV has to contain twelve pentagons (shown in red, in this image from the Briggs group) amongst many hexagonal units. My personal favourite, the clathrin cage, can assemble into many buckminsterfullerene-like shapes, but all must contain at least twelve pentagons with a variable number of hexagons.

The case of clathrin is particularly interesting because clathrin triskelia can assemble into a flat hexagonal lattice on membranes. If clathrin is going to coat a vesicle, that means 12 pentagons need to be introduced. So there needs to be quite a bit of rearrangement in order to do this.

You can see the same rule in everyday objects. The best example is a football, or soccer ball, if you are reading in the USA.

The classic design of football has precisely twelve pentagons and twenty hexagonal panels. The roadsign for football stadia here in the UK shows a weirdly distorted hexagonal array that has no pentagons. 22,543 people signed a petition to pressurise the authorities to change it, but the Government responded that it was too costly to correct this geometrical error.

So why do all of these assemblies have 12 pentagons?

In the classic text “On Growth and Form” by D’Arcy Wentworth Thompson, polyhedral forms in nature are explored in some detail. In the wonderfully titled On Concretions, Specules etc. section, the author notes polyhedral forms in natural objects.

One example is Dorataspis, shown left. The layout is identical to the D6 hexagonal barrel assembly of a clathrin cage shown above. There is a belt of six hexagons, one at the top, one at the bottom (eight total) and twelve pentagons between the hexagons. In the book, there is an explanation of the maths behind why there must be twelve pentagons in such assemblies, but it’s obfuscated in bizarre footnotes in latin. I’ll attempt to explain it below.

To shed some light on this we need the help of Euler’s formulae. The surface of a polyhedron in 3D is composed of faces, edges and vertices. If we think back to the football the faces are the pentagons and hexgonal panels, the edges are the stitching where two panels meet and the vertices are where three edges come together. We can denote faces, edges and vertices as f, e and v, respectively. These are 2D, 1D and zero-dimensional objects, respectively. Euler’s formula which is true for all polyhedra is:

\(f – e + v = 2\)

If you think about a cube, it has six faces. It has 12 edges and 8 vertices. So, 6 – 12 + 8 = 2. We can also check out a the football above. This has 32 faces (twelve pentagons, twenty hexagons), 90 edges and sixty vertices. 32 – 90 + 60 = 2. Feel free to check it with other polyhedra!

Euler found a second formula which is true for polyhedra where three edges come together at a vertex.

\(\sum (6-n)f_{n} = 12\)

in this formula, \(f_{n}\) means number of n-gons.

So let’s say we have dodecahedron, which is a polyhedron made of 12 pentagons. So \(n\) = 5 and \(f_{n}\) = 12, and you can see that \((6-5)12 = 12\).

Let’s take a more complicated object, like the football. Now we have:

\(((6-6)20) + ((6-5)12) = 12\)

You can now see why the twelve pentagons are needed. Because 6-6 = 0, we can add as many hexagons as we like, this will add nothing to the left hand side. As long as the twelve pentagons are there, we will have a polyhedron. Without them we don’t. This is the answer to why there must be twelve pentagons in a closed polyhedral assembly.

So how did Euler get to the second equation? You might have spotted this yourself for the f, e, v values for the football. Did you notice that the ratio of edges to vertices is 3:2? This is because each edge has two vertices at either end (it is a 1D object) and remember we are dealing with polyhedra with three edges at each vertex. so \(v = \frac{2}{3}e\). Also, each edge is at the boundary of two polygons. So \( e = \frac{1}{2}\sum n f_{n}\). You can check that with the values for the cube or football above. We know that \(f = \sum f_{n}\), this just means that the number of faces is the sum of all the faces of all n-gons. This means that:

\(f – e + v = 2\)

Can be turned into

\(f – (1/3)e = \sum n f_{n} – \frac{1}{6}\sum n f_{n} = 2\)

Let’s multiply by 6 to get, oh yes

\(\sum (6-n)f_{n} = 12\)

There are some topics for further exploration here:

  • You can add 0, 2 or 10000 hexagons to 12 pentagons to make a polyhedron, but can you add just one?
  • What happens when you add a few heptagons into the array?

Image credits (free-to-use/wiki or):

Clathrus ruber – tineye search didn’t find source.

HIV cores – Briggs Group

Exploded football – Quora

The post title comes from “Pentagrammarspin” by Steve Hillage from the 2006 remaster of his LP Fish Rising

Outer Limits

This post is about a paper that was recently published. It was the result of a nice collaboration between me and Francisco López-Murcia and Artur Llobet in Barcelona.

The paper in a nutshell
The availability of clathrin sets a limit for presynaptic function

Background
Clathrin is a three legged protein that forms a cage around membranes during endoctosis. One site of intense clathrin-mediated endocytosis (CME) is the presynaptic terminal. Here, synaptic vesicles need to be recaptured after fusion and CME is the main route of retrieval. Clathrin is highly abundant in all cells and it is generally thought of as limitless for the formation of multiple clathrin-coated structures. Is this really true? In a neuron where there is a lot of endocytic activity, maybe the limits are tested?
It is known that strong stimulation of neurons causes synaptic depression – a form of reversible synaptic plasticity where the neuron can only evoke a weak postsynaptic response afterwards. Is depression a vesicle supply problem?

What did we find?
We showed that clathrin availability drops during stimulation that evokes depression. The drop in availability is due to clathrin forming vesicles and moving away from the synapse. We mimicked this by RNAi, dropping the clathrin levels and looking at synaptic responses. We found that when the clathrin levels drop, synaptic responses become very small. We noticed that fewer vesicles are able to be formed and those that do form are smaller. Interestingly, the amount of neurotransmitter (acetylcholine) in the vesicles was much less than the volume of the vesicles as measured by electron microscopy. This suggests there is an additional sorting problem in cells with lower clathrin levels.

Killer experiment
A third reviewer was called in (due to a split decision between Reviewers 1 and 2). He/she asked a killer question: all of our data could be due to an off-target effect of RNAi, could we do a rescue experiment? We spent many weeks to get the rescue experiment to work, but a second viral infection was too much for the cells and engineering a virus to express clathrin was very difficult. The referee also said: if clathrin levels set a limit for synaptic function, why don’t you just express more clathrin? Well, we would if we could! But this gave us an idea… why don’t we just put clathrin in the pipette and let it diffuse out to the synapses and rescue the RNAi phenotype over time? We did it – and to our surprise – it worked! The neurons went from an inhibited state to wild-type function in about 20 min. We then realised we could use the same method on normal neurons to boost clathrin levels at the synapse and protect against synaptic depression. This also worked! These killer experiments were a great addition to the paper and are a good example of peer review improving the paper.

People
Fran and Artur did almost all the experimental work. I did a bit of molecular biology and clathrin purification. Artur and I wrote the paper and put the figures together – lots of skype and dropbox activity.
Artur is a physiologist and his lab like to tackle problems that are experimentally very challenging – work that my lab wouldn’t dare to do – he’s the perfect collaborator. I have known Artur for years. We were postdocs in the same lab at the LMB in the early 2000s. We tried a collaborative project to inhibit dynamin function in adrenal chromaffin cells at that time, but it didn’t work out. We have stayed in touch and this is our first paper together. The situation in Spain for scientific research is currently very bad and it deteriorated while the project was ongoing. This has been very sad to hear about, but fortunately we were able to finish this project and we hope to work together more in the future.

We were on the cover!
25.coverNow the scientific literature is online, this doesn’t mean so much anymore, but they picked our picture for the cover. It is a single cell microculture expressing GFP that was stained for synaptic markers and clathrin. I changed the channels around for artistic effect.

What else?
J Neurosci is slightly different to other journals that I’ve published in recently (my only other J Neurosci paper was published in 2002). For the following reasons:

  1. No supplementary information. The journal did away with this years ago to re-introduce some sanity in the peer review process. This didn’t affect our paper very much. We had a movie of clathrin movement that would have gone into the SI at another journal, but we simply removed it here.
  2. ORCIDs for authors are published with the paper. This gives the reader access to all your professional information and distinguishes authors with similar names. I think this is a good idea.
  3. Submission fee. All manuscripts are subject to a submission fee. I believe this is to defray the costs of editorial work. I think this makes sense, although I’m not sure how I would feel if our paper had been rejected.

Reference:

López-Murcia, F.J., Royle, S.J. & Llobet, A. (2014) Presynaptic clathrin levels are a limiting factor for synaptic transmission J. Neurosci., 34: 8618-8629. doi: 10.1523/JNEUROSCI.5081-13.2014

Pubmed | Paper

The post title is taken from “Outer Limits” a 7″ Single by Sleep ∞ Over released in 2010.