Some Stuff

EDIT : Updated with new image. Well, unless I can make a more horrifying diagram than this: ..but I think that's unlikely. An explanation - if such a thing is possible - is that the red/purple arrows correspond to a kind of inside-out operation. So the triangular map (M3) at the bottom can be converted to the square (M4) by choosing the purple darts (2, 14, 16, 5) as the boundary, and putting everything else on the inside. The cycle (1, 4, 9, 20, 22) is red because it is the boundary of the pentagonal map (M5). So, it seems like ϕ(M4) is α(ϕ(M5)) - so the cycle (0,8,23,21,5) in ϕ(M4) is (1,9,22,20,4) in ϕ(M5). That is, because (α[0] = 1, α[8] = 9, α[23] = 22 ...). However ϕ(M3) = ϕ(M5), which is confusing. Oh well So I have updated the image so that all three 'maps' have the same cycles. Which means they have the same ϕ and therefore the same σ. I guess this means I misunderstood what a CM actually is : you can get 'different' embeddings with the same σ. Here is a s...

So, to continue about combinatorial maps, here are some more intricate diagrams. Firstly, an embedding of cuneane with a 4-cycle as the outer face: The permutations below are just for reference. Anyway, with this embedding, the cycles of ϕ are (0,8,23,21,5)(1,2,11,7)(3,4,17,15)(6,12,9)(10,14,18,22,13)(16,20,19) which does indeed have one for each face, including the boundary. If you use a different embedding, naturally you get a different map: Which is quite different, and has ϕ of (0,6,10,3)(1,4,20,22,9)(2,14,16,5)(7,8,13)(11,12,23,19,15)(17,18,21) - again, 6 cycles for the 6 rings. And one to rule them all, and in the darkness bind them, of course. The cycles of darts are shown in this composite image: Some of the triangular faces are missing the dart labels, as they were getting too crowded. Well, the whole thing is too crowded, but still. Highlighted in red in each are the darts corresponding to the outer face of the other embedding. Not sure what it means, though. Finally, some ...

So the traditional way to say it is probably 'planar embedding', but I've gone for the alternate terminology of 'squashing flat'. For many molecules, this is a fairly easy process : some decisions have to be made about rotatable bonds, but quite a few drugs are just things sticking off a benzene ring or two. For fully 3-dimensional molecular graphs ( for example ) it is trickier. I don't yet know a simple algorithm for choosing different possible embeddings ... er ... squashings. Perhaps they are all complicated; they seem to involve tree data structures with names like "SPQR Tree" (see "Optimizing over all combinatorial embeddings of a planar graph"). Far easier, though, is the intermediate data structure between a graph and some 2D coordinates - known as a combinatorial map . These mathematical objects store the detail of the embedding by recording the order of 'half bonds' called flags (or darts?) around a vertex. Maybe a flag is t...

Previously... After using setFormalCharge instead of setCharge, some of the null atom types disappeared. Specifically, the quaternary (SP3?) nitrogens, that have to have a +1 formal charge. What is left? Mostly FeS clusters, and other metals . Turns out that the height-1 signature is clearer for 'clustering' the atoms into types. This is because a signature of this height records only the immediate neighbours of the atom. One fairly frequent example is "[Co]([N][N][N][N])", but fortunately there is a patch for this based on a bug report. The atom is found in cobalamine and other cobalt-haemes. Still, there are quite a few 'odd' ligands - either due to unusual elements, like Iridium ; or unusual coordinations, like this iron with 6 oxygen neighbours . Hmmm, PDBeChem's image (at the bottom of the page) doesn't show all six bonds. It's clearer in Jmol from PDBSum , where the C3 symmetry of the iron ligands makes it seem (to me) that it is 6-coordi...