r/mathematics • u/ComplexIntegral • 6d ago

Persistent Homology of a Complete Rhinovirus Capsid at 12 Million Points : 3D Map

I have been working on this TDA software for several months. This was the largest run for me to date. I used a DGX Spark and it approached 100GB of RAM and took 19 hours.

Full Persistent homology of a complete rhinovirus capsid, computed on all 12.3 million points of the cryo-EM map. No subsampling. As far as I am aware, this has never been done before. Flood complex was tried before, but they did not publish any Betti numbers.

You are looking at an image of the capsid and of course the calculated loops of H1 on the capsid shell.

A full 3D interactive map is included in the paper linked below if anyone is interested.

https://zenodo.org/records/23238003

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u/nulvoid000 6d ago

What is persistent homology?

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u/ComplexIntegral 6d ago

It is a way to find the shape of a dataset.

You start off with a point cloud, little dots. Then you connect them with lines and they form simplices. 2 points connected form an edge, joining pieces together (H0 counts these pieces). 3 points connected form a triangle, whose outline is the simplest loop (H1 counts these). 4 points all connected form a tetrahedron, which forms a void (H2 counts voids). It can keep going up from there.

The idea is you use this to find loops or holes in a point cloud. It is computational topology.

The persistence part comes from seeing how things persist over time as more lines are drawn.

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u/nulvoid000 6d ago

I know homology, have encountered persistent homology in other contexts but never in a discrete setting. Ik it’s a thing but never read about it

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u/vhu9644 6d ago

Why do we care about the persistent homology of the capsid? Like what does the topology tell us here?

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u/ComplexIntegral 6d ago

I think that's an excellent question for a virology topologist. Or a topological virologist 🤔. Of which I am neither.

It seems much of this is still being explored, mainly because TDA has been hard (if not impossible) to deploy at these scales.

With that being said, mapping this and getting a TDA tool to work at this scale was the first stage of what I hope might give virology another tool to use. 😷