Persistent Homology of a Complete Rhinovirus Capsid at Twelve Million Points

Abstract

We compute the persistent homology in degrees zero and one of a complete virus capsid, taken directly from its cryo-electron microscopy density map with no subsampling. The input is the deposited map EMD-50844 of the empty capsid of rhinovirus A89, a 1000^3 grid at 0.413 Å per voxel, from which every voxel above a fixed density threshold contributes one point: n = 12,313,265 points in R^3. The Ordinal Locality Filtration was computed through 515 rounds on a single workstation (NVIDIA DGX Spark, CPU only) in 19.1 h with a peak memory of 99.9 GiB. The complex is connected from round 44. The first Betti number rises to 1.24 x 10^7, drains through a second maximum of 82,673, and settles at beta_1 = 173, which holds unchanged over the eighteen rounds k = 405-422, again over the eighteen rounds k = 432-449, and a third time over the eight rounds k = 478-485; the two eighteen-round runs are the widest stable plateaus of the computation. Besides the Betti numbers, the computation records the birth and death round of every loop that lives five rounds or more and writes out one representative cycle for every loop present at thirteen chosen rounds. Within each of the three runs of 173 the loops are the same 173 loops at every round. Between consecutive runs, 24 loops die and 24 are born, and 125 loops are present in all three. Sixty-seven of the loops present at round 405 are still present at round 515. An interactive three-dimensional rendering of the loops on the capsid accompanies the paper. At round 515 the complex has 2.68 x 10^9 edges and 2.18 x 10^11 triangles, against the 3.11 x 10^20 candidate triangles of a Vietoris-Rips complex on the same points. We are not aware of any previous persistent homology computation on a complete virus density map at this point count. Keywords: persistent homology, Betti numbers, barcodes, representative cycles, topological data analysis, cryo-electron microscopy, virus capsid