Some Stuff

One diagram I've often wanted to make was filled-in rings in molecules : Mainly for the purposes of highlighting rings without highlighting the atoms involved. This image was made using the CDK renderbasic module, and a small toy AWTRenderingVisitor that fills in paths. I'm not sure if the current one does this... The gist for the main drawing method is here but is really just stuff seen before. The custom generator for rings is probably more interesting - if very simple - and is here . Note that it doesn't look too nice with inner-ring double bonds: and would look nicer if the double bonds were symmetric. EDIT : Coloring by ring equivalence classes didn't do what I expected... Shouldn't all the outer rings be in the same class? Steran is how I expect, though:

So, there was a question on BioStar about calculating the 'external symmetry number' of a molecule - something I hadn't heard of, but turns out to be something like the subgroup of rotations and reflections of the automorphism group of a graph. Since I have some code to calculate the automorphism group, I naïvely thought it would be simple... The questioner - Nick Vandewiele - kindly provided some test cases, which ended up as this code . Although many of these tests now pass, they only do so because I commented out the hydrogen adding! :) On the one hand, there are some recent improvements that try to handle vertex and edge 'colors' - in other words, element symbols and bond orders. For example, consider the improbable molecule C1OCO1 : These are the three permutations that leave the carbons and the oxygens in the same positions; when you include the identity, that makes 4. Cyclobutane (without hydrogens!) has a symmetry group of order 8. Similarly, cyclobutadien...

A couple of different sources pointed me to a paper by Dehof et al (Bioinformatics, 2011; doi: 10.1093/bioinformatics/btq718) about bond order assignment by various methods. One of them is the A*-algorithm , which finds paths. Not in the molecule graph, but in a kind of tree of possible bond combinations. Precisely, the tree has a layer for each decision, and a leaf for each combination of decisions. For example, here is the usual example of a square four-cycle (G): with the tree shown below. If we pick a particular path through the tree - say to leaf 5 - we get the sequence of colors in the box in the upper center of the image. This corresponds to the assignment on the upper right. Clearly, it has too many double bonds to be carbon skeleton, but still. So, it's a nice, simple way to represent the set of solutions. Well, it could be even simpler but representing it as a tree lets the paper authors assign weights to the edges based on the chemical rules outlined in work they refere...

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...